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What do we mean when we say that something grows exponentially?

As an example of exponential growth, imagine a little biological organism, such as a bacterium. Imagine that a single organism divides into daughter organisms within the space of an hour. Over the subsequent hour, the two daughter organisms then also divide into organisms each, leaving us with organisms. During the third hour, these four organisms also divide into organisms each, leaving us with organisms, and so on.

This tree diagram shows how the number of organisms grows over the first three hours.

In general, after the hour, there will organisms. The growth in the number of organisms is called exponential because we can describe the quantity that changes over time as raised to the exponent . Exponential growth can be very rapid. In our example, within less than 33 hours there will be more organisms than there are people on Earth!

The picture below indicates what happens if we continue our tree diagram above for many generations. We have to make the lines much thinner and rearrange them on the page to make it all fit in. But still, there eventually are so many of them, they form a dense canopy.

This graphic beautifully illustrates exponential growth: the stem splits into two branches, which each split into two branches and so on. Soon there are so many branches, they form a dense canopy.

Exponential growth involves repeated multiplication. We don't come across this kind of growth too often in everyday life. We're much more familiar with repeated addition, that is, calculations of the form rather than . The growth you get from addition is a lot slower, which is why it's easy to underestimate the speed of exponential growth. (See this video for an illustration of the difference between additive and multiplicative growth.)

An important feature of exponential growth is that it speeds up as time goes on. In our example above, the number of organisms only increases by during the first hour and by during the second hour, but you only have to wait for hours to see an hourly increase of over ! This acceleration doesn't happen for additive growth.

What we have looked at so far was only an example, and there are several generalisations we can make. Firstly, the base of the exponential expression doesn't have to be , it could be any real number that is greater than Secondly, rather than thinking of time as proceeding in discrete hourly steps, we can think of it as being continuous. This can be captured by the expression where and where represents time which can run from all the way up to infinity.

We can also multiply the time variable by some constant and the entire expression by some positive constant This now gives the expression

Any such expression for , and real numbers with and describes exponential growth. The Geogebra applet below lets you explore this for and . You can vary the value of using the slider. You will see that when becomes negative, the quantity given by our formula decreases over time and we get what is called exponential decay.

This is the graph of the function f(t)=. Use the slider to vary the value of c.

When it comes to describing quantities that grow exponentially, people often use expressions involving the Euler's constant , such as To find out why, and to find out more about , see e for exponential.
This article is part of our collaboration with JUNIPER, the Joint UNIversities Pandemic and Epidemiological Research network. JUNIPER is a collaborative network of researchers from across the UK who work at the interface between mathematical modelling, infectious disease control and public health policy. You can see more content produced with JUNIPERhere.

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What do we mean when we say that something grows exponentially?

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X marks the unknown in algebra – but X’s origins are a math mystery

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X marks the unknown in algebra – but X’s origins are a math mystery

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Even though x is one of the least-used letters in the English alphabet, it appears throughout American culture - from Stan Lee's X-Men superheroes to " The X-Files" TV series. The letter x often symbolizes something unknown, with an air of mystery that can be appealing - just look at Elon Musk with SpaceX, Tesla's Model X, and now X as a new name for Twitter.

You might be most familiar with x from math class. Many algebra problems use x as a variable, to stand in for an unknown quantity. But why is x the letter chosen for this role? When and where did this convention begin?

There are a few different explanations that math enthusiasts have put forward - some citing translation, others pointing to a more typographic origin. Each theory has some merit, but historians of mathematics, like me, know that it's difficult to say for sure how x got its role in modern algebra.

Ancient unknowns

Algebra today is a branch of math in which abstract symbols are manipulated, using arithmetic, to solve different kinds of equations. But many ancient societies had well-developed mathematical systems and knowledge with no symbolic notation.

All ancient algebra was rhetorical. Mathematical problems and solutions were completely written out in words as part of a little story, much like the word problems you might see in elementary school.

Ancient Egyptian mathematicians, who are perhaps best known for their geometric advances, were skilled in solving simple algebraic problems. In the Rhind papyrus, the scribe Ahmes uses the hieroglyphics referred to as "aha" to denote the unknown quantity in his algebraic problems. For example, problem 24 asks for the value of aha if aha plus one-seventh of aha equals 19. "Aha" means something like "mass" or "heap."

The ancient Babylonians of Mesopotamia used many different words for unknowns in their algebraic system - typically words meaning length, width, area or volume, even if the problem itself was not geometric in nature. One ancient problem involved two unknowns termed the "first silver thing" and the "second silver thing."

Mathematical know-how developed somewhat independently in many lands and in many languages. Limitations in communication prevented any immediate standardization of notation. However, over time some abbreviations crept in.

In a transitional syncopated phase, authors used some symbolic notation, but algebraic ideas were still presented mainly rhetorically. Diophantus of Alexandria used a syncopated algebra in his great work Arithmetica. He called the unknown "arithmos" and used an archaic Greek letter similar to s for the unknown.

Indian mathematicians made additional algebraic discoveries and developed what are essentially the modern symbols for each of the decimal digits. One especially influential Indian mathematician was Brahmagupta, whose algebraic techniques could handle any quadratic equation. Brahmagupta's name for the unknown variable was yãvattâvat. When additional variables were required, he instead used the initial syllable of color names, like kâ from kâlaka (black), ya from yavat tava (yellow), ni from nilaka (blue), and so on.

Islamic scholars translated and preserved a great deal of both Greek and Indian scholarship that has contributed immensely to the world's mathematical, scientific and technical knowledge. The most famous Islamic mathematician was al-Khowarizmi, whose foundational book Al-jabr wa'l muqabalah is at the root of the modern word "algebra."

So what about x?

One theory of the genesis of x as the unknown in modern algebra points to these Islamic roots. The theory contends that the Arabic word used for the quantity being sought was al-shayun, meaning "something," which was shortened to the symbol for its first "sh" sound. When Spanish scholars translated the Arabic mathematical treatises, they lacked a letter for the "sh" sound and instead chose the "k" sound. They represented this sound by the Greek letter χ, which later became the Latin x.

It's not unusual for a mathematical expression to come about through convoluted translations - the trigonometric word "sine" started as a Hindu word for a half-chord but, through a series of translations, ended up coming from the Latin word "sinus," meaning bay. However, there is some evidence that casts doubt upon the theory that using x as an unknown is an artifact of Spanish translation.

The Spanish alphabet includes the letter x, and early Catalonian involved several pronunciations of it depending on context, including a pronunciation akin to the modern sh. Although the sound changed pronunciation over time, there are still vestiges of the sh sound for x in Portuguese, as well as in Mexican Spanish and its use in native place names. By this reasoning, Spanish translators conceivably could have used x without needing to resort first to the Greek χ and then to the Latin x.

Moreover, although the letter x may have been used in mathematics during the Middle Ages sporadically, there is no consistent use of it dating back that far. Western mathematical texts over the next several centuries still used a variety of words, abbreviations and letters to represent the unknown.

For instance, a typical problem in the algebra book " Sumario Compendioso of Juan Diez," published in Mexico in 1556, uses the word "cosa" - meaning "stuff" or "thing" - to stand in for the unknown.

I think that the most plausible explanation is to credit the influential French scholar René Descartes for the modern use of x. In an appendix to his major work "Discourse" in the 17th century, Descartes introduced a version of analytic geometry - in which algebra is used to solve geometric problems. For unspecified constants he chose the first few letters of the alphabet, and for variables he chose the last letters in reverse order.

Although scholars may never know for sure, some theorize that Descartes may have chosen the letter x to appear often since the printer had a large cache of x's because of its scarcity in the French language. Whatever his reasons for choosing x, Descartes greatly influenced the development of mathematics, and his mathematical writings were widely circulated.

Xtending beyond algebra

Even if the origins of x in algebra are uncertain, there are some instances in which historians do know why x is used. The X in Xmas as an abbreviation for Christmas definitely does come from the Greek letter χ. The Greek word for Christ is Christos, written χριστοσ and meaning "anointed." The χ monogram was used as a shorthand for Christ in both Roman Catholic and Eastern Orthodox writings dating back as far as the 16th century.

There are also some contexts in which x was chosen specifically to indicate something unknown or extra, such as when the German physicist Wilhelm Roentgen accidentally discovered X-rays in 1895 while experimenting with cathode rays and glass.

But there are other cases in which scholars can only guess about the origins of x's role, such as the phrase "X marks the spot." And there are other contexts - such as Elon Musk's affinity for the letter - that may just be a matter of personal taste.

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X marks the unknown in algebra - but X's origins are a math mystery

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How did the letter x get its enduring role as a symbol of the unknown? A mathematician explains why it's hard to say for sure.

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Cryptography - How Mathematicians Helped Win WWII

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Cryptography - How Mathematicians Helped Win WWII

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Learn a skill a day for 20 days in order to get the most out of this graphic display calculator

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A Rubik’s cube, you’ll be pleased to hear, can always be solved in at most 20 moves, no matter how badly it was scrambled up to start with. Mathematicians have proved that that’s true. But what if you’re wrestling with a larger cube that has more than three little cubes in a row?

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By Jim Frost

Who would've thought that an old TV game show could inspire a statistical problem that has tripped up mathematicians and statisticians with Ph.Ds? The Monty Hall problem has confused people for decades. In the game show, Let's Make a Deal, Monty Hall asks you to guess which closed door a prize is behind. The answer is so puzzling that people often refuse to accept it! The problem occurs because our statistical assumptions are incorrect.

The Monty Hall problem's baffling solution reminds me of optical illusions where you find it hard to disbelieve your eyes. For the Monty Hall problem, it's hard to disbelieve your common sense solution even though it is incorrect!

The comparison to optical illusions is apt. Even though I accept that square A and square B are the same color, it just doesn't seem to be true. Optical illusions remain deceiving even after you understand the truth because your brain's assessment of the visual data is operating under a false assumption about the image.

I consider the Monty Hall problem to be a statistical illusion. This statistical illusion occurs because your brain's process for evaluating probabilities in the Monty Hall problem is based on a false assumption. Similar to optical illusions, the illusion can seem more real than the actual answer.

To see through this statistical illusion, we need to carefully break down the Monty Hall problem and identify where we're making incorrect assumptions. This process emphasizes how crucial it is to check that you're satisfying the assumptions of a statistical analysis before trusting the results.

What is the Monty Hall Problem?

Monty Hall asks you to choose one of three doors. One of the doors hides a prize and the other two doors have no prize. You state out loud which door you pick, but you don't open it right away.

Monty opens one of the other two doors, and there is no prize behind it.

At this moment, there are two closed doors, one of which you picked.

The prize is behind one of the closed doors, but you don't know which one.

Monty asks you, "Do you want to switch doors?"

The majority of people assume that both doors are equally like to have the prize. It appears like the door you chose has a 50/50 chance. Because there is no perceived reason to change, most stick with their initial choice.

Time to shatter this illusion with the truth! If you switch doors, you double your probability of winning!

What!?

How to Solve the Monty Hall problem

When Marilyn vos Savant was asked this question in her Parade magazine column, she gave the correct answer that you should switch doors to have a 66% chance of winning. Her answer was so unbelievable that she received thousands of incredulous letters from readers, many with Ph.D.s! Paul Erdős, a noted mathematician, was swayed only after observing a computer simulation.

It'll probably be hard for me to illustrate the truth of this solution, right? That turns out to be the easy part. I can show you in the short table below. You just need to be able to count to 6!

It turns out that there are only nine different combinations of choices and outcomes. Therefore, I can just show them all to you and we calculate the percentage for each outcome.

You Pick Prize Door Don't Switch Switch

3 Wins (33%)

6 Wins (66%)

Here's how you read the table of outcomes for the Monty Hall problem. Each row shows a different combination of initial door choice, where the prize is located, and the outcomes for when you "Don't Switch" and "Switch." Keep in mind that if your initial choice is incorrect, Monty will open the remaining door that does not have the prize.

The first row shows the scenario where you pick door 1 initially and the prize is behind door 1. Because neither closed door has the prize, Monty is free to open either and the result is the same. For this scenario, if you switch you lose; or, if you stick with your original choice, you win.

For the second row, you pick door 1 and the prize is behind door 2. Monty can only open door 3 because otherwise he reveals the prize behind door 2. If you switch from door 1 to door 2, you win. If you stay with door 1, you lose.

The table shows all of the potential situations. We just need to count up the number of wins for each door strategy. The final row shows the total wins and it confirms that you win twice as often when you take up Monty on his offer to switch doors.

Why the Monty Hall Solution Hurts Your Brain

I hope this empirical illustration convinces you that the probability of winning doubles when you switch doors. The tough part is to understand why this happens!

To understand the solution, you first need to understand why your brain is screaming the incorrect solution that it is 50/50. Our brains are using incorrect statistical assumptions for this problem and that's why we can't trust our answer.

Typically, we think of probabilities for independent, random events. Flipping a coin is a good example. The probability of a heads is 0.5 and we obtain that simply by dividing the specific outcome by the total number of outcomes. That's why it feels so right that the final two doors each have a probability of 0.5.

However, for this method to produce the correct answer, the process you are studying must be random and have probabilities that do not change. Unfortunately, the Monty Hall problem does not satisfy either requirement.

Related post: How Probability Theory Can Help You Find More Four-Leaf Clovers

How the Monty Hall Problem Violates the Randomness Assumption

The only random portion of the process is your first choice. When you pick one of the three doors, you truly have a 0.33 probability of picking the correct door. The "Don't Switch" column in the table verifies this by showing you'll win 33% of the time if you stick with your initial random choice.

The process stops being random when Monty Hall uses his insider knowledge about the prize's location. It's easiest to understand if you think about it from Monty's point-of-view. When it's time for him to open a door, there are two doors he can open. If he chose the door using a random process, he'd do something like flip a coin.

However, Monty is constrained because he doesn't want to reveal the prize. Monty very carefully opens only a door that does not contain the prize. The end result is that the door he doesn't show you, and lets you switch to, has a higher probability of containing the prize. That's how the process is neither random nor has constant probabilities.

Here's how it works.

The probability that your initial door choice is wrong is 0.66. The following sequence is totally deterministic when you choose the wrong door. Therefore, it happens 66% of the time:

Because this process occurs 66% of the time and because it always ends with the prize behind the door that Monty allows you to switch to, the "Switch To" door must have the prize 66% of the time. That matches the table!

Related post: Luck and Statistics: Do You Feel Lucky, Punk?

1. You pick the incorrect door by random chance. The prize is behind one of the other two doors.

2. Monty knows the prize location. He opens the only door available to him that does not have the prize.

3. By the process of elimination, the prize must be behind the door that he does not open.

If Your Assumptions Aren't Correct, You Can't Trust the Results

The solution to Monty Hall problem seems weird because our mental assumptions for solving the problem do not match the actual process. Our mental assumptions were based on independent, random events. However, Monty knows the prize location and uses this knowledge to affect the outcomes in a non-random fashion. Once you understand how Monty uses his knowledge to pick a door, the results make sense.

Ensuring that your assumptions are correct is a common task in statistical analyses. If you don't meet the required assumptions, you can't trust the results. This includes things like checking the residual plots in regression analysis, assessing the distribution of your data, and even how you collected your data.

For more on this problem, read my follow up post: Revisiting the Monty Hall Problem with Hypothesis Testing.

As for the Monty Hall problem, don't fret, even expert mathematicians fell victim to this statistical illusion!

Related

Filed Under: Tagged With: Fun assumptions

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The Monty Hall Problem: A Statistical Illusion - Statistics By Jim

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By Jim Frost Who would've thought that an old TV game show could inspire a statistical problem that has tripped up mathematicians and statisticians with Ph.Ds? The Monty Hall problem has confused people for decades. In the game show, Let's Make a Deal, Monty Hall asks you to guess which closed door a prize is behind.

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Monopoly: The Mathematical Secret Behind Winning - Samvart

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Monopoly: The Mathematical Secret Behind Winning - Samvart

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Monopoly: The Mathematical Secret Behind Winning

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To CLOSE MENUS, hit ESC.
To UNDO an entry, hit CTRL - ESC (and repeat to remove numerous entries)
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using the graphing calculator in secondary mathematics

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Using the TI-Nspire Graphing Calculator - Table of Contents

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To CLOSE MENUS, hit ESC. To UNDO an entry, hit CTRL - ESC (and repeat to remove numerous entries) To DELETE/CLEAR variables within a document, type DelVar (followed by the variable name) on a Calculator page.

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The word “algorithm” has probably got more usage over the last few years than it has in its entire history. But what exactly is an algorithm?

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Quantum mechanics is the physics of the extremely small. With something so far outside our everyday experience it’s not surprising to find mathematics at the heart of it all. But at the quantum scale nothing in life is certain... Peter Landshoff explains.

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Ice, food, and invisibility cloaks — meet the maths that links them together!

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One thing that's great about maths is that it can cope with the tricky concept of infinity. An example of this comes from infinite sums, also known as infinite series.

An example is

As the dots suggest, the series goes on for ever, with each new term four times smaller than the previous one.

Yet the "sum" is finite, in fact, we can say that it's equal to . I would now like to prove this in a slightly unconventional way . . .using chocolate.

Imagine, if you will, that you are a shopkeeper, and selling chocolate bars on the special offer shown below.

Now, each bar consists of 1 unit of actual chocolate and 1 coupon. And a customer with 4 coupons can claim 1 bar free. The key to proving our result turns out to be the following question:

What is one coupon worth, in terms of actual chocolate?

In search of an infinite series

Well, 4 coupons are worth 1 bar, and therefore 1 coupon is, in a sense, worth bar.

So 1 coupon is worth

where "chocolate" here is just my shorthand for the amount of actual chocolate in one bar.

Now, the coupon in brackets is itself worth bar, so 1 coupon can also be said to be worth

And if we continue like this for ever, we find that 1 coupon is worth

units of actual chocolate.

To prove our result, then, we just need to prove that 1 coupon is also worth unit of actual chocolate.

In search of 1/3

To put it another way, we need to prove that 3 coupons are worth 1 unit of actual chocolate. And to do this, imagine that I have saved up 3 coupons, come to your shop, and say:

How much chocolate do you get for a coupon?

"Good morning. I should like to buy a bar of chocolate. I intend to eat the chocolate straight away, in your shop, and I will pay you when I have eaten it."

Your first thought will doubtless be "another crazy mathematician!" but, being of generous spirit, you give me a bar of chocolate.

I open it, and eat the chocolate.

Next, I take out the coupon that's inside, and add it to the 3 that I have already, so that I now have 4 coupons!

I then present these to you, saying: "Here are 4 coupons. I claim a bar of chocolate!"

At which point, you say: "You've got a short memory, haven't you? I've just given you a bar of chocolate!"

Somewhat embarrassed, I reply: "Of course you have. How silly of me. So you don't owe me anything, and I don't owe you anything, either. Our business is complete. Thank you very much. Good morning."

Now, I arrived with 3 coupons and no chocolate, and went away with no coupons and 1 unit of chocolate. So 1 coupon is worth unit of chocolate, and therefore

And whatever we may think of such an unusual proof, the result itself contains a very important idea: it is possible for infinitely many positive numbers to have a finite sum.

Further reading

To find out more about the wonderful concept of infinite series, how to cope with them, and how they can turn our tricky, see these Plus articles.

About this article

This article is an excerpt of David Acheson's new book The spirit of mathematics, published by Oxford University Press in March 2023, available in hardback and eBook formats, £14.99. © David Acheson.

David Acheson is Emeritus Fellow of Jesus College, Oxford, and was the University's first winner of a National Teaching Fellowship in 2004. He was President of the Mathematical Association from 2010 to 2011, and now lectures widely on mathematics to young people and the general public. In 2013, Acheson was awarded an Honorary D.Sc. by the University of East Anglia for his outstanding work in the popularisation of mathematics. His books include 1089 and All That (OUP, 2002), The Calculus Story, (OUP, 2017), and The Wonder Book of Geometry, (OUP, 2020).

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One thing that's great about maths is that it can cope with the tricky concept of infinity. An example of this comes from infinite sums, also known as infinite series. An example is

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The growth rate of a disease - Augustina 

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The growth rate of a disease - Augustina 

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The growth rate of a disease is a natural way to capture how quickly the number of infections are changing day by day. The growth of cases of a disease is modelled using an exponential curve:

Here is the number of cases, which depends on time measured in days, and (pronounced "lambda") is what is called the growth rate of the disease per day. (The number is a mathematical constant approximately equal to 2.719 and intimately connected to exponential growth.)

For example, the growth rate for HIV is =0.002 per day and for measles is =0.06 per day. You can use the interactivity below to explore how the progression of a diseases changes for different growth rates (use the slider to change the value of ).

If the growth rate is positive, the number of new cases each day is increasing. And if the growth rate is 0, the number of new cases stays constant. What is needed to keep an epidemic under control is for the growth rate to be negative and hence the number of new cases to be decreasing.

But what does the value of the number actually mean? If the number of new cases has decreased by 3% since yesterday, then the growth rate is, approximately, = -0.03 per day. (This isn't exactly equivalent but a good approximation for typical values of . The growth rate actually works like compound interest which you can read about here.) Conversely, for small values of (eg ), the value of lambda approximates the percentage growth in cases per day: means that cases will increase by around 1% a day.

However, for larger values of lambda (eg ) this approximation is no longer accurate. In this case the percentage growth from one day to the next is : for example for that's ). (For a more detailed explanation of the relationship between the growth rate and the percentage growth per day read The doubling time of a disease.)

Why isn't R enough?

If you have already studied a bit of epidemiology, you have probably come across the concept of R, the reproduction ratio (or sometimes called the reproduction number) of a disease. That's the number of people infected, on average, by a single infected person. As explained here, R helps us understand what is happening with the disease: R>1 means that the epidemic will grow, R=1 means we are plateauing, R<1 means that the epidemic will decline.

One thing that R does not tell us, though, is how quickly things are changing. This is because R is not a rate, there is no timescale involved. For example, if R=2 for some disease then we know the epidemic will grow (because R>1), but we cannot tell how quickly. For diseases like HIV or TB, where there can be months or years between one person infecting the next person, even R=2 means slow growth over time. However for influenza or measles, where the infection is much faster, on the scale of days, R=2 means very rapid growth.

Here are two example curves for the growth of infection, both with a reproduction ratio R=2. The difference is the time between new infections - several months for HIV but just days for measles.

Which is better: R or the growth rate?

Both the reproduction ratio , and the growth rate, , are valid measures for understanding the growth of a disease. They each have their uses, as outlined below:

Both the reproduction ratio and the growth rate are particularly hard to estimate when the number of cases is small, for example if the incidence of the disease is very low, or if the community you are studying has a very small population. In that case, day to day fluctuations can easily swamp the underlying patterns of the disease, so you will have greater uncertainty about the growth rate (and so expect wider confidence intervals).

How do you get from R to the growth rate and vice versa?

The precise relationship between R and the growth rate is not straightforward: it needs to take into account the timings of each infection to the next. A crude approximation is

where

is the mean generation time: the time from one infection to the next.

I can cope with some advanced maths, tell me more!

OK! All of this supposes that the control measures and number of people susceptible to the disease are not changing too quickly.

Following one infected person, denote their time since infection by (in days). They go on to infect others on average. For each of these, their timing of infection is distributed with probability density function . Then (exercise for maths undergraduates!) and are related as follows:

and yes, this is very closely related to a Laplace transform, or a moment generating function for the generation time distribution.

For specific distributions for the generation time, for example a gamma distribution, this can sometimes be simplified. Taking the generation time to be exactly constant, say , we recover the , but this is a rather crude approximation for many infectious diseases in practice.

Note, depends on a variety of things, including biological things like the incubation period, and on social factors like whether you still mix with others when you have symptoms or if you self-isolate.

See this paper by Wallinga and Lipsitch for more details.

About this article

Rachel Thomas and Marianne Freiberger are the Editors of Plus. This article was produced with Julia Gog, Professor of Mathematical Biology at the University of Cambridge and co-lead of the JUNIPER network. Gog is also a member of the Scientific Pandemic Infections group on Modelling (SPI-M), an advisory group of the Department of Health and Social Care which provides expert advice to the UK government based on infectious disease modelling and epidemiology and fed results into the Scientific Advisory Group for Emergencies (SAGE) during the COVID-19 pandemic.
This article is part of our collaboration with JUNIPER, the Joint UNIversities Pandemic and Epidemiological Research network. JUNIPER is a collaborative network of researchers from across the UK who work at the interface between mathematical modelling, infectious disease control and public health policy. You can see more content produced with JUNIPERhere.

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The growth rate of a disease is a natural way to capture how quickly the number of infections are changing day by day. The growth of cases of a disease is modelled using an exponential curve:

# data[].attributes.attachment_link.readable_text
One thing that's great about maths is that it can cope with the tricky concept of infinity. An example of this comes from infinite sums, also known as infinite series.

An example is

As the dots suggest, the series goes on for ever, with each new term four times smaller than the previous one.

Yet the "sum" is finite, in fact, we can say that it's equal to . I would now like to prove this in a slightly unconventional way . . .using chocolate.

Imagine, if you will, that you are a shopkeeper, and selling chocolate bars on the special offer shown below.

Now, each bar consists of 1 unit of actual chocolate and 1 coupon. And a customer with 4 coupons can claim 1 bar free. The key to proving our result turns out to be the following question:

What is one coupon worth, in terms of actual chocolate?

In search of an infinite series

Well, 4 coupons are worth 1 bar, and therefore 1 coupon is, in a sense, worth bar.

So 1 coupon is worth

where "chocolate" here is just my shorthand for the amount of actual chocolate in one bar.

Now, the coupon in brackets is itself worth bar, so 1 coupon can also be said to be worth

And if we continue like this for ever, we find that 1 coupon is worth

units of actual chocolate.

To prove our result, then, we just need to prove that 1 coupon is also worth unit of actual chocolate.

In search of 1/3

To put it another way, we need to prove that 3 coupons are worth 1 unit of actual chocolate. And to do this, imagine that I have saved up 3 coupons, come to your shop, and say:

How much chocolate do you get for a coupon?

"Good morning. I should like to buy a bar of chocolate. I intend to eat the chocolate straight away, in your shop, and I will pay you when I have eaten it."

Your first thought will doubtless be "another crazy mathematician!" but, being of generous spirit, you give me a bar of chocolate.

I open it, and eat the chocolate.

Next, I take out the coupon that's inside, and add it to the 3 that I have already, so that I now have 4 coupons!

I then present these to you, saying: "Here are 4 coupons. I claim a bar of chocolate!"

At which point, you say: "You've got a short memory, haven't you? I've just given you a bar of chocolate!"

Somewhat embarrassed, I reply: "Of course you have. How silly of me. So you don't owe me anything, and I don't owe you anything, either. Our business is complete. Thank you very much. Good morning."

Now, I arrived with 3 coupons and no chocolate, and went away with no coupons and 1 unit of chocolate. So 1 coupon is worth unit of chocolate, and therefore

And whatever we may think of such an unusual proof, the result itself contains a very important idea: it is possible for infinitely many positive numbers to have a finite sum.

Further reading

To find out more about the wonderful concept of infinite series, how to cope with them, and how they can turn our tricky, see these Plus articles.

About this article

This article is an excerpt of David Acheson's new book The spirit of mathematics, published by Oxford University Press in March 2023, available in hardback and eBook formats, £14.99. © David Acheson.

David Acheson is Emeritus Fellow of Jesus College, Oxford, and was the University's first winner of a National Teaching Fellowship in 2004. He was President of the Mathematical Association from 2010 to 2011, and now lectures widely on mathematics to young people and the general public. In 2013, Acheson was awarded an Honorary D.Sc. by the University of East Anglia for his outstanding work in the popularisation of mathematics. His books include 1089 and All That (OUP, 2002), The Calculus Story, (OUP, 2017), and The Wonder Book of Geometry, (OUP, 2020).

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One thing that's great about maths is that it can cope with the tricky concept of infinity. An example of this comes from infinite sums, also known as infinite series. An example is

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Deduction of the average kinetic energy of an atom. --ToNy

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Deduction of the average kinetic energy of an atom. --ToNy

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This follows from the equipartition theorem. The equipartition theorem states that in thermal equilibrium, the average energy of each degree of freedom (each independent way the system can move) is $k_B T/2$, where $T$ is the temperature and $k_B$ (or just $k$) is called the Boltzmann constant. There are three independent directions in which a gas particle can move (three independent components of velocity), so the total kinetic energy is $3\times k_B T/2$. It is important to realise that this is a statistical formula that is only true on average for a large number of particles in equilibrium: each individual particle may in fact have a kinetic energy different from this. The Boltzmann constant provides a link between the microscopic and macroscopic worlds, by relating the typical average energies of microscopic particles to the energies required to change the temperature of a macroscopic mass by a measurable amount.

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It says in my course notes for undergraduate environmental physics that a particle has so-called "kinetic energy" $$E=\frac{3}{2}kT=\frac{1}{2}mv^{2}$$ Where does this formula come from? What is ...

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Suppose you're a hotel manager and your hotel is full. That's great, of course, but there's always the temptation to squeeze in more guests. In real life, this might mean clearing out the broom cupboards and getting bad reviews on Trip Advisor, but in the world of maths it's no problem. As long as your hotel has infinitely many rooms, that is.

Welcome to Hilbert's hotel!

The idea goes back to the German mathematician David Hilbert, who used the example of a hotel to demonstrate the counter-intuitive games you can play with infinity. Suppose that your hotel has infinitely many rooms, numbered 1, 2, 3, etc. All rooms are occupied, when a new guest arrives and asks to be put up. What do you do? It's easy. Ask the guest in room 1 to move to room 2, the guest in room 2 to move into room 3, the guest in room 3 to move into room 4, and so on. If there were only finitely many rooms, the guest in the last room would have nowhere to go, but since there are infinitely many, everybody will find a new abode. You'll have to ask the guests to move simultaneously though, because if you ask them to move one after the other, the move might take an infinite amount of time, since infinitely many guests have to move.

Using this trick you can actually accommodate any finite number of new guests. If new guests arrive, simply ask each existing guest to move to the room whose number is plus the number of their existing room. As an example, if there are new guests, then the guest currently in room needs to move into room .

But things get better still. Suppose an infinite number of new guests arrive, forming an orderly queue outside the hotel. In this case, ask each existing guest to move into the room whose number is twice the number of their current room. So, a guest staying in room moves to room . After this manoeuvre only the even numbered rooms are occupied: rooms , , , and so on. The odd numbered rooms are all free, so you can put your first new guest into room 1, the second new guest into room 3, the third new guest into room 5, and so on. Everybody is happy.

Infinite layering

This isn’t all. Suppose an infinite number of coaches arrive, each carrying an infinite number of new guests. Assume, for simplicity, that the coaches are numbered 1, 2, 3, etc, and that the seats in each coach are also numbered 1, 2, 3, etc. You start by asking each existing guest to move into the room whose number is twice the number of their current room, as before. This leaves the odd numbered rooms free again. Now tell the passenger of coach 1 with seat 1 to move into room 3, the passenger of coach 1 with seat 2 into room the passenger of coach 1 with seat 3 into room , and so on. In other words, the passenger of coach 1 with seat moves into room . Any power of is odd, so all these rooms are guaranteed to be free.

There's room for everyone.

What about people arriving in the second coach? Here’s how to accommodate them: the guest with seat 1 in coach 2 moves into room 5, the guest with seat 2 in coach 2 moves into room , the guest with seat 3 in coach 3 moves into room , and so on. In other words, the guest with seat in coach 2 moves into room

How do you continue this for the third coach? Well, 3 and 5 are consecutive prime numbers. The next prime number along is 7, so we put the passenger of coach 3 with seat number into room . The next prime number after 7 is 11, so we put the passenger of coach 4 with seat number into the room . Generally, you put the passenger of coach with seat number into the room , where is the prime number.

Are all these rooms guaranteed to be free? The answer is yes. All the room numbers of the new guests are powers of prime numbers. There's a beautiful result known as the fundamental theorem of arithmetic, which says that every whole number can be written as a product of primes in a unique way.

This means that if room number is a power of some prime, then it can’t be a power of another prime. For example, we know that guest number 5 from coach number 1 is allocated room Now if there were another guest allocated to the same room, say guest 3 from coach 2, then would also have to be a power of another prime, eg , which, according to the fundamental theorem it can’t be (and isn’t since ).

We’ll carry on for one more level. Suppose an infinite number of ships arrive, each carrying an infinite number of coaches, each carrying an infinite number of guests. Each guest now comes with three numbers attached to them: the number of their ship, the number of their coach and the number of their seat in the coach. You start by moving all the existing guests to the even-numbered rooms as before. Now suppose a guest arrived on ship , coach and seat . We start as before: find the prime number, call it , and raise it to the power of the seat number. That gives . Now for the ship number, find the prime number, call it , and raise that to the number you already have, to get

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As an example, if the passenger arrived on ship 1, coach 1 and had seat number 2, then they should move into room

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Numbers get very large very quickly with this approach. The guest from ship and coach with seat number should go into room

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but if you try to work out that number, you’ll find that your calculator will give up. In theory, though, it’s possible to accommodate all the guests (another appeal to the fundamental theorem of arithmetic will prove this, see the box) and since this is maths, theory is all that counts.

So far we have managed to accommodate guests that arrive in three layers of infinity: infinitely many ships, each carrying infinitely many coaches, each carrying infinitely many guests. Can we go further? The answer is yes. You could accommodate guests arriving in any finite number of layers of infinity. What you can't always do, however, is accommodate guests arriving in infinitely many layers of infinity. But then, that's never going to happen anyway.

You can find out more about infinity on our What is infinity? page.

About the author

Marianne Freiberger is Editor of Plus.

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How to squeeze infinitely many new guests into a full hotel.

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Imagine you're a gambler and someone proposes to play the following game. You start with a fortune of £1. You then flip a fair coin.

If it lands heads-up you gain £1

If it lands tail-up you lose £1

You continue to flip the coin, gaining or losing a pound with each toss. If you go broke the game ends. As long as you don't go broke, you keep on playing indefinitely, potentially amassing a huge fortune.

Should you accept to play the game?

An infinity of ways

To find out, let's try to figure out the probability that you eventually end up broke. First note that there are an infinite number of ways in which this could happen.

On your very first toss of the coin, you could lose your £1, immediately rendering yourself broke. Alternatively you could gain £1 on your first toss before losing £2 on the two subsequent tosses. Or you could gain £2 on your first two tosses before losing £3 on the three subsequent tosses. More generally, if you gain on your first tosses and then lose on the subsequent tosses you will go broke.

This already gives you an infinite number of heads and tail sequences that mean you end up broke. But there are even more ways. That's because you may also end up first increasing your fortune, then decreasing it but not all the way down to zero, then increasing it again, etc, before your fortune finally drops down to zero and you're broke.

Your first turn will either leave you broke or with £2. After that anything can happen...

On the up-side, you may also end up never going broke at all. Your fortune could even end up approaching infinity (you wish).

Considering each unique way of ending up with £0 is impossible, simply because there are infinitely many sequences of heads and tails that can make this happen. So let's encompass all the possible "routes" from £1 to £0, and their respective probabilities, in . Let be the probability that, starting from £1 you eventually end up with £0. The probability is the number we are interested in finding.

Universal p

Now, the probability of going broke after your first turn is

However, after your first turn it is equally probable that you could have risen from £1 to £2. From £2 what then is the probability of going broke? Going broke from £2 would require you to first get from £2 down to £1. (Whatever route we take from £2 to £0, we will inevitably end up at £1 along the way.) Now if a sequence of heads and tails gets you from £2 to £1, then the same sequence of heads and tails would also get you from £1 to £0, and vice versa. The starting and end points of the journey determined by the coin tosses would simply be shifted along by a distance of 1. This means that the probability of eventually going from £2 to £1 is the same as the probability of eventually going from £1 to £0. That probability is .

The same reasoning works more generally. The probability of going from to is equal to for all positive integers The probability is universal in this way.

This means that the probability of dropping from £2 to £1 and then from £1 to £0 is . Therefore, the probability of eventually going broke, starting with £1 and first going up to £2 is That's because:

The probability of going up to £2 from £1 is

The probability of going from £2 to £0 is

Now the probability of going broke at all is the probability of going broke at the first toss of the coin (which is equal to ) plus the probability of going broke after first having gone up to £2. In other words,

But wait a second... Haven't we just found two expressions for the same thing? One the one hand, the probability of going broke at all, starting from £1, is equal to . But on the other, as we have just shown, it is also equal to

Since these two expressions are equal, we can form the equation

Multiplying through by gives

Bringing all terms over to the right hand side gives

Solving this equation for gives .

This means that the probability of eventually going broke is 1.

What does this mean?

You'll almost surely lose all of these...

Usually, if something has a probability of 1 this means that it's guaranteed to happen: no other outcome is possible. So does our result mean that you going broke is the only possible outcome of this game? Clearly not. For example, it is theoretically possible that you only ever toss heads, thereby steadily increasing your fortune. It is also possible for your fortune to go up and down forever, without ever decreasing enough to go broke.

To see what's going on, notice that you can imagine playing the game against an opponent: if heads comes up your opponent gives you £1 and if tails comes up you give the opponent £1. To ensure that the game only stops when you go broke, your opponent needs to have infinite wealth.

You can also imagine playing the game against an opponent who only has finite wealth to start with. In this case the game ends when you go broke, or when your opponent goes broke, whichever happens first. It is possible to show that in this situation someone definitely will eventually go broke. In other words, the game can't go on forever without anyone ever hitting £0. Intuitively, it seems clear that, the larger your opponent's initial fortune, the higher the chance that you go broke first, and the lower the chance that your opponent goes broke first.

This is indeed the case. If you start with £1 and your opponent with , for some natural number , then the probability of you going broke is

and the probability of your opponent going broke is

If is a very large number, then is very close to 1 and is very close to 0. In fact, if you increase your opponent's wealth towards infinity, the probability of you going broke converges to 1, and the probability of your opponent going broke converges to 0. That's the situation we considered in this article.

This phenomenon, that an outcome can have probability 1 but still not be the only outcome that can occur, can happen in situations where infinity is involved. Mathematicians say the outcome occurs "almost surely", or "almost certainly" in this case. Such situations don't occur in real life: we are always dealing with finite amounts of time or space, and a finite level of precision. It's amazing, though, that mathematics allows us to think through these hypothetical situations.

The game we have played is an example of something called gambler's ruin in action. The gambler's ruin is a thought experiment which, as the name suggests, looks at games where a gambler is (almost) sure to go broke, even though it's not at first obvious that's the case.

About this article

Maryam Naheem

Maryam is a final year student at the Tiffin Girls' School in Kingston Upon Thames. She enjoys mathematical problem solving, created a school Maths Journal to share fun mathematical ideas with other students, and is looking forward to studying Maths at university.

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Imagine you're a gambler and someone proposes to play the following game. You start with a fortune of £1. You then flip a fair coin. If it lands heads-up you gain £1 If it lands tail-up you lose £1 You continue to flip the coin, gaining or losing a pound with each toss.

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Mathematics plays a central role in understanding how infectious diseases spread: it can help us assess whether an outbreak is likely to turn into an epidemic, which parts of society and the country are likely to be most affected, and what we can do to stop or slow down the spread.

This collection of articles looks at some basic concepts in epidemiology to help you understand this fascinating and important field, and set you up for further study.
To see all our content relating to epidemiology, including content produced during the COVID-19 pandemic, click here.
The doubling time of a disease - The doubling time of a disease estimates how long it'll take for the number of infected people to double. Find out how to calculate it from the growth rate.

e for exponential - When modelling the beginning of an epidemic the number e plays an important role. But why does e appear in descriptions of exponential growth?

How can maths fight an epidemic? - This article provides a first conceptual look at the use of mathematical models in epidemiology, exploring the famous SIR model.
This content is part of our collaboration with JUNIPER, the Joint UNIversities Pandemic and Epidemiological Research network. JUNIPER is a collaborative network of researchers from across the UK who work at the interface between mathematical modelling, infectious disease control and public health policy. You can see more content produced with JUNIPERhere.

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Mathematics plays a central role in understanding how infectious diseases spread: it can help us assess whether an outbreak is likely to turn into an epidemic, which parts of society and the country are likely to be most affected, and what we can do to stop or slow down the spread.

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The Science of Encryption: Prime numbers and Mod n Arithmetic - Samvart

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The Science of Encryption: Prime numbers and Mod n Arithmetic - Samvart

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Bayes' theorem

Suppose that a particular type of cancer affects 1% of the population. There is a test for this cancer but it's not perfect: although the test gives a positive result for 90% of people who have the cancer, it also gives a positive result for 5% of the people who are cancer-free. You have just received a positive test result – what is the probability you have cancer?

Many of us would say there is now a 90% chance that we have cancer. But this isn't correct – your chances are closer to 15%.

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A few months ago, my brother-in-law, who is a biologist, asked me a nice mathematical question. He had developed a blood test for detecting a certain minor abnormality in infants. The presence of a particular substance in the blood would indicate the presence of the condition and vice versa. His test was sufficiently sensitive that it would be possible to detect the presence of one or more affected children within a group of 100 or so, by testing a mixture containing a small amount of blood from each child: by testing a pooled sample of blood.
Obviously if you have taken blood samples from 100 children, you could find out precisely which children have the abnormality by running 100 separate tests. The question was this: can you cut down the number of tests significantly, by pooling samples of blood? If so, what is the best way to do it?

This question is a perfect example of a problem in the branch of mathematics known as information theory. We are given a random system (100 samples from randomly chosen babies) and we want to use experiments to identify which of the system's possible states has actually occurred.

Yes/no questions

Partitioning of the states

The basic insight with which information theory begins is this. Whatever random system we are dealing with, and whatever experiments we are doing, each experiment is just a way of dividing up the collection of states into parts. An experiment asks which states produce outcome A, which produce outcome B and so on. In the diagram to the right the lines represent a partitioning of the collection of states by experiments.

In our case the possible outcomes of each test are:

At least one of the contributors to the pool does show the abnormality;

All contributors are free of the abnormality.

So each test divides the possible states into two parts: those states that yield P and those that yield N.

As we perform more experiments, we refine our partition further. After 2 tests we will have divided the state-space into 4 pieces, after 3 into 8 and so on.

Yes/No questions

In general, after m tests we will have partitioned the space of states into pieces. Our task is to identify which state has actually occurred. So we want to invent a testing protocol that will partition the states completely: into individual items. If we are asking questions with two possible answers (Yes or No) then the maximum number of possibilities that we can guarantee to distinguish with m questions is 2.

Usually it is more convenient to express the number of experiments that you need in terms of the number of states that you are asked to distinguish, rather than the other way around. If the number of states is n=2m then the number of questions is the base 2 logarithm

The rough principle involved here can be written:

(Number of Y/N questions) is approximately equal to log ₂(Number of states).

This looks promising since it tells us that the number of questions we need (the uncertainty of our system) is much smaller than the number of states.

However, if there are n children we might have to run n tests in order to find out precisely which ones are affected by the abnormality, since one possibility is that all n children are affected. In that case, we are certainly going to have to test each child's blood separately to confirm that he or she is affected. However, if the abnormality is rare, then it is extremely unlikely that all the children will have it. Our hope is that for a rare abnormality there will be a good chance that we will be able to eliminate large numbers of children with just one test. What this means is that instead of asking:

How many tests might we need to determine who is affected?

we should be asking:

Can we design a testing protocol that will determine who is affected, but will usually use far fewer than n tests?

The entropy

A measure of uncertainty

The number log ₂ n that we found earlier is the uncertainty of a collection of n equally likely states. In the case of the blood samples, the states are not all equally likely: the possibility that all the children are affected is very remote. So we need to refine our measure of uncertainty by taking account of the likelihoods of the different states.

To do that, let's rewrite our earlier estimate in terms of probabilities. The number log ₂ n is the uncertainty of a collection of n equally likely states: a collection of states, each of which has probability 1/ n of occurring. If we call this probability p, so that n=1/ p, then the uncertainty is

If we are given a system with equally likely states of probability p it will take log ₂1/ p questions to identify the state.

Now suppose that we have a collection of states with different probabilities

The idea is to think of each of them as if it belonged to a collection of equally likely states. The i th state has probability , so we will need log questions to identify it (if it occurs). To find the average number of questions that we will need, we have to add up these numbers log , but weighted according to their likelihoods of occurring: to calculate the weighted average of these log values.

So we get a measure of uncertainty,

which is called the entropy of the system. It is possible to prove that the minimum average number of Y/N questions required to identify the state cannot be smaller than the entropy of the set of possible states: so the entropy is indeed a good measure of uncertainty. The entropy measures a "hard core" of uncertainty that we cannot expect to circumvent however cleverly we design our testing protocol.

In the case of n blood samples from a population in which the disease occurs with probability p, the entropy can be calculated quite easily: it is

This quantity is proportional to n, the number of samples, which is what we would expect: twice as many people should need twice as many tests. The number of tests per sample is the proportion

Entropy per sample

We would like this proportion to be less than 1 - to need fewer tests than there are samples. If we plot a graph of this quantity against p we get the diagram on the left.

Sure enough, the quantity is always below 1 and if p is very small, if the disease is rare, the entropy is close to zero: there is less uncertainty. This is reassuring.

The entropy measures the minimum average number of tests that we might need in order to determine which infants are affected. But it doesn't tell us how to run the tests: it doesn't tell us whether there is any protocol that will work with so few tests, let alone what that protocol is. This situation is more or less unavoidable. The question of whether there is an efficient way to test a random system depends rather delicately on what that system is and on what questions we are allowed to ask. To get an upper estimate, we have to design an explicit protocol which does a good job.

A binary protocol for blood testing

One possible way to test the samples is to use a "binary" division protocol: repeatedly dividing the possibilities into two. We collect n samples and to simplify matters let's assume that n is a power of 2, . It may be that none of the samples exhibits the abnormality, so it might make sense to begin by testing a pool of all n samples. If this test comes up negative then we have eliminated all n samples in one test. Otherwise, if the first test comes up positive then we proceed to divide the samples into 2 batches of n/2 = 2m-1. Since there might be more than one affected sample, we have to test both of these batches. If either batch comes up negative we can eliminate all the samples in that batch. We continue in this way, eliminating any batch that tests negative and subdividing any batch that tests positive.

The binary protocol

As you can see in the diagram to the right, the process of repeatedly dividing samples into two can be represented by what looks like a family tree shows such a tree for n=8. Each batch is divided into two "daughter" batches on the level below. The diagram also shows what happens if one of the 8 samples exhibits the abnormality (marked with a cross). The shaded batches are the ones that we actually have to test. Each time we test a batch without the cross in it, we get a negative result and we do not have to test any sample in this batch any further. Each time we test a batch which does have the cross in it, we have to go on and test its two daughter batches.

We can calculate the expected number of tests used by this protocol. If we start with n=2 samples and the probability of each sample being affected is p, then the expected number of tests used by the binary protocol turns out to be

Tests per sample for p=0.01

The graph to the left shows this expected number divided by n (the number of tests per sample) plotted against n for the value p = 0.01. The dots show the number of tests per sample that we expect to use with the binary protocol while the horizontal line shows the entropy bound: the minimum number of tests that we could possibly hope to get away with. The binary protocol only takes about twice as many tests as the optimum, so it does a fair job.

The pattern of the dots looks a bit strange: it initially decreases, then increases, the minimum occurring at the dot corresponding to 64 samples. The reason is that for a small number of samples no protocol is very efficient: with just one sample you are bound to need 1 test: not a fraction of a test. For a large number of samples the protocol also has a small inefficiency: with a large number of samples it is highly unlikely that all the samples will be clear. This means that if our collection of blood samples is very large, there is no point in testing them all at once: the test would almost certainly be positive and we would have to go ahead and divide up the samples. There is no point carrying out a test if you are almost certain what result it will give, because the test is highly unlikely to give you any useful information.

If we use a refined testing protocol in which we divide up the samples into batches of size 64 before running any tests and then use the binary division protocol on each of these batches, then for p = 0.01 the protocol will not use more than about 13 tests per hundred children, on average. (In general, for a condition with prevalence p the optimal initial batch size is about 1/ p.) This is a considerable improvement upon the 100 tests that would be needed without pooling.

About the author

Keith Ball's new book, reviewed in this issue of Plus.

Keith Ball studied mathematics at Cambridge University (both BA and PhD). After some years working at Universities in the US he returned to the UK and is currently a Professor of Maths at University College London. His mathematical interests include high-dimensional geometry, probability and information theory and Diophantine approximation.

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A biologist has developed a blood test for detecting a certain minor abnormality in infants. Obviously if you have blood samples from 100 children, you could find out which children are affected by running 100 separate tests. But mathematicians are never satisfied by the obvious answer.

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https://docs.google.com/presentation/d/1AbZNYErMW1zw1SWhDbRycR8qTtSsXLKDmPgqhWxds70/edit#slide=id.g262efc87276_0_18

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Veganism, vegetarianism, and flexitarianism have been booming. There are three main reasons for people turning away from meat and dairy: concerns about their own health, about the environmental impact of the meat and dairy industries, and about animal cruelty.

However, while studies in 2020 suggested that the number of people who consume plant-based alternatives had roughly doubled over the previous ten years, recent reports suggest that sales in the UK have since stagnated. The cost of living crisis may have something to do with this - plant-based alternatives tend to be expensive - but perhaps it's also to do with flavour. If you like meat, then vegan burgers and sausages may not cut it.

If you like your meat then plant based alternatives may not cut it.

This is where mathematician Ed Steele and his biologist friend Max Jamilly have spotted an opportunity. Their company Hoxton Farms is developing processes for growing stem cells from pigs into fat cells. The resulting fat can then be used as an ingredient in food, alongside plant proteins.

"We recognised that the traditional meat industry is broken," says Steele. "It's killing us and it's killing the planet. [At the same time] we saw the demand for meat alternatives soar. But we think that plant-based meat still isn't good enough yet. And that's because it's missing this one key ingredient: fat. Fat is what gives meat its flavour. If you cook it in the right way it gives you all the magic that you need for a really meaty dish."

The result won't be a vegan or vegetarian product, of course, it'll be animal-derived. "But it's made without harming animals and avoids the environmental damage of traditional animal agriculture," says Steele. "It ticks the boxes that [many] people are looking for in meat alternatives. We call the ingredient we make 'cultivated fat'."

Hoxton Farms is still in the research and development phase, with a team of 35 working in a lab in London on optimising the process and scaling it up. But a larger manufacturing facility is planned for the next couple of years, and Steele thinks that their product may be commercially available within a few years.

What gives Hoxton Farms the edge over competitors working on similar processes, Steele thinks, is that mathematics plays a central role in all that is involved: from feeding the stem cells to optimising the bioreactor they grow in. "We have this awesome way of working internally, where mathematicians and statisticians work very closely with biologists. It's one of the ways we think we have a superpower as a company: the interdisciplinary nature of all the work that we do."

Feeding cells

An important aspect of growing stem cells is to find the cells' favourite food. The cells are fed growth media, which can be made up of a large number of components. "The media [are] the most expensive part of the process of growing cells at large scale," says Steele.

Ed Steele is co-founder of Hoxton Farms.

Feeding the cells the optimal growth media is vital, but the number of possible recipes is so large that finding the best ones through experimentation is impossible. Biological theory doesn't help either. "It's relatively difficult to build a biological understanding of how the different media components will interact with each other and interact with the cells."

Here Hoxton Farms uses the help of machine learning, whose powerful ability to spot patterns in data has been all over the news lately. You can think of the collection of all possible growth media recipes as a landscape, with the optimal ones dotted throughout. The people at Hoxton Farms explore a small patch of this landscape experimentally, feeding the recipes to the stem cells and watching how they grow into fat cells. The resulting data about their growth is then fed to a machine learning algorithm which, based on this information, suggests which part of the landscape might be good to explore next. The corresponding recipes are tried out on the stem cells, the data is again fed to the algorithm, which again makes optimal use of the information to suggest a new patch to explore. Combined with the expertise of human biologists, this feedback approach (technically known as Bayesian optimisation) guides the way through the recipe landscape to find those that perform best.

"[This approach] completely changes the way we think about media optimisation," says Steele. "Being able to use a modelling approach, rather than a guess work or a biological approach, means we can move much faster and end up with solutions that are much better."

Seeing cells

Carefully monitoring the development of the cells is obviously crucial. "One of the main ways we get information about our cells is by taking lots of images using microscopes and then analysing those images," says Steele. There are many pieces of information to record from a single image: how many cells there are, what their shape is, and how large fat droplets are inside them, for example. If you have ever looked through a microscope at a bunch of cells, you can imagine that this is a painstaking, time-consuming task for a human.

Hoxton Farms is currently hiring software engineers and modelling engineers! Find out more here.

Again it's machine learning that helps. By training machine learning algorithms to recognise important features in an image of cells, mathematicians at Hoxton Farms have built a bespoke computer vision tool that can do much of the work for the team. "The [tool works] really well, much better than the scientists would themselves," says Steele.

Stirring cells

The whole process of stem cells growing into fat cells happens in a bioreactor, a big tank, which has the cells suspended in the growth media, which is a liquid. "The tank has an impeller which spins around to mix everything together. Oxygen bubbles through the bottom of the tank. Everything needs to be well mixed so that all of the cells get oxygen and different [components] of the media. You don't want pockets where certain waste products or media components build up."

Pigs will be spared if cultivated fat becomes the norm.

There are many different ways of setting up the tank with its impeller to mix things around, and again it's important to find the optimum. Rather than using trial and error experimentation, Hoxton Farms use computational fluid dynamics. Based on the equations that describe the flow of liquids, this approach uses computers to simulate its behaviour in different settings.

"Then we can change things, for example we can see what happens if we change the rate of the impeller, or the head of impeller, or push oxygen through the [tank] in a slightly different way. Computational fluid dynamics is an important tool for us to model [what happens in the bioreactor]."

These are just some examples of how a mathematical approach can help with the work that is being done at Hoxton Farms. The fact that the team includes mathematicians and statisticians alongside biologists, rather than only biologists with maths and stats training, is vital, Steele thinks. "We are constantly thinking about how we can use people on the team with different skill sets, in particular on the mathematical modelling side and the statistics side, to make a difference to the work we do."

Is it safe?

Once the manufacturing process is optimised, the next step will then be to have the product approved by regulators. "We need to demonstrate to the regulators that the process we use to grow the fat cells results in a product that is safe to eat. We are very confident that this is the case and that it is significantly safer than meat produced using traditional animal agriculture. The environment we work in is sterile, and that's through necessity. The cells we work with have no immune system so they have to grow in an extremely clean environment. A slaughter house is a completely different environment." Steele also points out that the antibiotics used in traditional meat production to deal with diseases are not needed in a sterile environment.

"On the safety side we [have no concerns] and on the health and nutritional side what we do gives you the same as if you were to eat traditional pork fat."

Maths and taxes

When we initially contacted Ed Steele it wasn't only because we were interested in meat alternatives, but also because we were on the look-out for companies that make use of a feature of UK taxation that perhaps isn't as well-known as it should be: research and development (R&D) in mathematics qualifies for tax relief, just as R&D in other subjects does.

The process that is being developed at Hoxton Farms can also be used to grow fat cells out of stem cells from cows, other mammals, and even fish.

For Hoxton Farms mathematical tax relief has been extremely useful. "It is hugely important; every year we submit an R&D tax credit claim," says Steele. "It's one of the amazing things about doing research in the UK and has been a huge help."

And although the UK government has recently slightly reduced the amount of R&D tax credits available, it has widened the scope on another front: as of April 2023 pure mathematics also counts as part of science for the purposes of R&D tax relief. This means that companies wanting to claim tax relief for R&D can now do so even if the R&D involves pure maths with no obvious, direct applications.

"As a mathematician I [completely agree with this]," says Steele. "[For example] basic research in biology has always counted as something you can claim R&D tax credits for, even though when you are doing basic research you are not trying to find out something in particular, you're just trying to learn about [biological] systems. I don't see why there would ever be a difference between pure maths research and basic research in biology. The maths that we use, while it is very applied, of course is built from pure maths. Unless you have the research going on in pure maths you will miss out on the future benefits in the applied fields."

Exactly when cultivated fat will appear routinely on the food labels we study in the supermarket isn't clear yet, but Steele thinks it will be a matter of a few years, rather than decades. Pork fat isn't the only animal fat that can be produced in this way. Beef and other mammalian fats, and even fish fat, are also options. Perhaps we will one day see an end to the cruelty and environmental damage that comes with the meat industry and fisheries. And if we do, then this will be in some small part down to mathematics.

About this article

Marianne Freiberger, Editor of Plus,interviewed Ed Steele in July 2023.

This article was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) - you can find all the content from the collaboration here.
The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.

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Veganism, vegetarianism, and flexitarianism have been booming. There are three main reasons for people turning away from meat and dairy: concerns about their own health, about the environmental impact of the meat and dairy industries, and about animal cruelty.

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using inverse mathematical mechanics to solve crime

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https://plus.maths.org/content/can-data-science-help-rebuild-our-trauma-networks

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Ambulance and A&E waiting times have been much in the news lately. It seems that many places could benefit from an increase in capacity, and here is where mathematics can help.

If we were to suffer a terrible injury here in Cambridge, the East of England Trauma Network would spring into action and an ambulance would take us to Addenbrooke's Hospital in Cambridge. But if we were elsewhere in the region, more than a 45 minute ambulance ride away from Addenbrooke's, we would be rushed to one of the twelve other hospitals in the region that complete the network.

The East of England Trauma Network ( Image used with permission)

Addenbrooke's is the only Major Trauma Centre in the region, meaning that it is the only hospital that can provide all the services that might be required. This means that patients initially cared for and assessed at other hospitals may have to transfer to Addenbrooke's to have further treatment if required. But a lot has changed over the ten years since the East of England Trauma Network was first established. "The number of beds for major trauma patients in Addenbrooke's is the same, but the number of patients has increased," says Houyuan Jiang, Professor of Management Science at the Judge Business School. "Demand is increasing because the size of the population is getting bigger and the population is getting older. It's time to reassess the service needs and supply."

Improving a trauma network

Jiang, along with colleague Feryal Erhun, Professor of Operations and Technology Management at the Judge Business School, are members of the Cambridge Mathematics of Information in Healthcare Hub (CMIH), a collaboration of 19 co-investigators from different disciplines across the University of Cambridge who use mathematics to meet the data challenges across all types of medical treatments. (You can read more about CMIH here.) Jiang was speaking at a CMIH event in June 2023, organised by the Newton Gateway to Mathematics, about his work supporting the development of the East of England Trauma Network to meet the future healthcare needs of the people in the region.

The proposal is to build a second Major Trauma Centre by upgrading the Trauma Unit at one of the other hospitals in the region. But where should it be? How many beds for major trauma patients should it have? And will this meet the demand across the network? Like many of the researchers in CMIH, Jiang used a machine learning approach to draw on the extensive amount of data available to derive possible answers to these questions. (You can read an easy introduction to machine learning here.) Jiang and his colleagues used data from the past ten years relating to the use of the trauma hospitals in the region, as well as projections (from the Office for National Statistics) for how the population will change into the future and the impact this would have on demand. They also drew on information from Google Maps to judge distances of potential patients to hospitals.

Once Jiang and his colleagues had built machine learning models based on these extensive data sets, they used counterfactual analysis to explore how much you have to change the inputs into the models (say, the location of the Major Trauma Centre, or the number of major trauma beds) to affect the output of the model (say, that the network can meet demands in the required amount of time). Machine learning models can often appear to be mysterious black boxes and counterfactual analysis can help researchers explain and interpret what their models are doing.

Bringing the CMIH family together

Jiang's work on using a mathematical data driven approach to infrastructure and planning decisions in health care was just one example of the wide range of work done by members of CMIH at the event. Also speaking was neurosurgeon Chao Li, who explained how image analysis algorithms can help with the diagnosis and treatment of brain tumours. And biomedical physicist Sarah Bohndiek discussed some of the promising medical applications of using multispectral imaging, where objects are imaged using many different wavelengths of light. One potential application would be new medical equipment that could help spot, and so treat, precancerous regions in the oesophagus, stopping them developing into cancer.

A beautiful example of multispectral imaging

The breadth of speakers at this event and the links between their work really demonstrated the range and strength of the multidisciplinary collaboration supported by CMIH. "The CMIH is quite diverse in terms of research topics, with early career researchers, and with people coming from different disciplines and from industry," says Carola-Bibiane Schönlieb, Professor of Applied Mathematics at the University of Cambridge and Co-Director of CMIH. "We all have in common a passion for bringing data analytics into clinical research."

James Rudd, Professor of Cardiovascular Medicine and Clinical lead of CMIH, said the work produced by the CMIH had already resulted in real impacts on the healthcare received by patients: "Quite a few [of the nearly 100 papers produced by CMIH so far] have impacted policy. For example, a model [developed by CMIH] for waiting lists for cardiovascular disease is now being used by NHS England."

And although the event marks the end of this phase of the CMIH, Schönlieb says that there are major developments in healthcare analytics that this group of researchers can play an important role in. One area of potential future work, which she suggested could become CMIH3, is looking at interfacing the mathematics and data science behind the health of our population, with that behind the health of our environment. Looking at these two societal challenges in a unified way may be the next frontier.

About this article

This article is based on a talk by Houyuan Jiang at the CMIH Academic Engagement Event which took place on 21 June 2023, hosted by the Newton Gateway to Mathematics.

The work Jiang spoke about is led by Dr Esther Kwong, Specialised Commissioning Public Health Consultant from the Healthcare team at NHS England. The collaboration with the Judge Business School at the University of Cambridge was established for capacity modelling and forecasting. Esther Kwong, Lauren Rixen, Zidong Liu, Feryal Erhun, and Houyuan Jiang have authored a report on this work, Public Health Needs Assessment for Major Trauma, which will be publicly available later in the year.

Rachel Thomas is Editor of Plus.

This article was produced as part of our collaborations with the Cambridge Mathematics of Information in Healthcare Hub (CMIH), the Newton Gateway to Mathematics and the Isaac Newton Institute for Mathematical Sciences (INI).

The INI is an international research centre in Cambridge which attracts leading mathematicians from all over the world. The Newton Gateway is the impact initiative of the INI, which engages with users of mathematics. You can find all the content from the collaboration here.

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Can data science help rebuild our trauma networks?

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Ambulance and A&E waiting times have been much in the news lately. It seems that many places could benefit from an increase in capacity, and here is where mathematics can help.

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Quantum mechanics is the physics of the extremely small. With something so far outside our everyday experience it’s not surprising to find mathematics at the heart of it all. But at the quantum scale nothing in life is certain... Peter Landshoff explains.

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Think of a right-angled triangle. Then you'll probably remember from school that you can use the sine and cosine functions to find out more about the triangle. If is one of the angles that isn't the right angle then you have

and

For the angle α, the sine gives the ratio of the length of the opposite side to the length of the hypotenuse. The cosine gives the ratio of the length of the adjacent side to the length of the hypotenuse. (Image by Dnu72 - CC BY-SA 3.0.)

The sine and the cosine functions can do a lot more than help you solve geometry problems. They can be used to build any waveform - the music you are listening to, the digital signal you are sending over wifi, even the swell on the sea - no matter how complicated these natural or human made oscillations might be.

Going round and round

The first step to constructing a wave is to imagine a circle of radius 1 drawn in Cartesian coordinates, with the centre of the circle lying at the point Imagine moving around the circle in an anticlockwise direction, starting at the right-most point, . After you have rotated through an angle of that is less than 90 degrees (corresponding to less than in radians) the point you are at defines a right-angled triangle, with corners and The hypotenuse of this triangle has length 1 because the point lies on our circle of radius is 1. For this triangle we have and

A right-angled triangle formed from a point on the unit circle.

You can continue to move around the circle anti-clockwise to make the angle bigger than When you do this, the triangle with corners and no longer has as one of its angles.

When α is greater than π/2, then the right-angled triangle formed from a point on the unit circle no longer contains the angle α.

However, there is nothing to stop you from extending the definitions of the sine and cosine of in analogy with what we had before:

where are the coordinates of the point you are at.

What happens to the sine and cosine as you move once around the circle? In one circuit of the circle you will have turned through an angle of . As you moved around the circle your vertical coordinate (the sine) started at and increased steadily until it reached a maximum of when you were on top of the circle. As you kept moving it then dropped down to again in a symmetrical fashion, before reaching a minimum of , and finally coming back up again to 0.

If you plot how the vertical coordinate (in red in the figure below) varies with the angle turned through (from 0 to ) you get a regular wave shape. It starts at 0, goes up to the maximum of 1, then down to 0 again, before dropping to the minimum of -1, and up again to 0.

The red wave is the sine of the angle plotted against the angle (coming from the vertical coordinate) and the blue wave is the cosine of the angle plotted against the angle (coming from the horizontal coordinate).

This wave pattern repeats as you carry on going around your circle a second time, increasing the angle you turned through from to and a third time travelling from to , and so on. You end up with an infinitely long, perfectly regular wave. The horizontal coordinates (the cosine) give a similarly perfect wave (plotted in blue above), offset against the first one by a distance of along the horizontal axis. This extends our definition of the sine and cosine to angles greater than . (You can also define the sine and cosine for negative numbers by going round the circle in a clockwise direction.)

The red wave is the sine of the angle plotted against the angle (coming from the vertical coordinate) and the blue wave is the cosine of the angle plotted against the angle (coming from the horizontal coordinate).

Squeezing and stretching

We've now seen how moving around the unit circle can give us two functions, the sine function and the cosine function, each of which comes with a graph that describes a regular wave. When you encounter these functions in text books the variable is usually called rather than , so you will see something like

We will switch to this notation now. This means that now represents what used to be our angle and that it no longer represents a coordinate of a point on the unit circle. That's potentially a little confusing, but hang in there and you will get used to it.

The wavelength is the distance between two peaks of a wave, and in our example so far this is . That's because to go through one complete cycle of our wave we had to turn through an angle of .

It's also possible to create waves with different wavelengths. To create a wave with wavelength you multiply the variable by to get the functions

By making the wavelength shorter you essentially squeeze the wave and by making it longer you stretch it.

In the Geogebra applet below, use the slider to change the wavelength of the sine (red) and cosine (blue) waves.

Going higher and going lower

The peaks of the waves we have created so far have the value 1 and the troughs have the value -1. To create a wave that has higher peaks and lower troughs, you simply multiply the entire function by a constant to get

This constant is called the amplitude of the wave.

In the Geogebra applet below use the sliders to vary the amplitude and wavelength for the sine wave (red) and the cosine wave (blue).

Speeding up and slowing down

So far the waves we have created are stationary: they don't change over time. However, it is also possible to create waves that travel along. Our wave function will now be a function of two variables and As before represents the position on the horizontal axis, while denotes time. Suppose you want your wave to be moving along at a speed of The functions

produce such moving waves. If you are keeping the time variable fixed then you are essentially seeing a snapshot in time, giving you a stationary wave just like the ones we had above. If you are keeping your location on the -axis fixed, then you are seeing the corresponding -variable as it moves up and down over time. This is a bit like watching a particular point on the surface of a lake move up and down as waves ripple through.

In the Geogebra applet below use the slider to change the speed. Setting the speed to 0 (or pressing the pause button) corresponds to stopping time, so you're keeping the time variable fixed. In this case you see a snapshot in time which looks like and ordinary sine (or cosine) wave, only perhaps shifted along the horizontal axis by some distance. Focusing on the blue dot corresponds to keeping the variable fixed. (We are not giving you the option of changing the wavelength and amplitude here as too much choice can be confusing!)

The reason the argument of our functions now is is that in a time period of length a wave travelling at speed will have travelled a distance of The height of the wave at point along the -axis at time will therefore be the same as the height was at time 0 at the point because that's how far the wave has travelled in time .

Decoding messages

At first sight these sine and cosine waves we have created appear too perfect to tell you much about the waves we encounter in real life. But you can hear one in action if you strike a tuning fork. The sound you hear is the result of the vibrations of your ear drum, stimulated by the sound wave from the tuning fork travelling through the air. For a tuning fork, if you plotted the intensity, or pressure, of this vibration over time, you would see a perfect sine wave in action.

As you can see the sound wave of something like speech is more complicated. But any sound wave, indeed any repeating function, can be broken up into a number of sine waves of various frequencies and amplitudes (intensities). This is the result of work that started with the French mathematician Joseph Fourier, who lived through the French revolution in the eighteenth century. The expression of a sound wave, or any signal varying over time, as the sum of its constituent sine waves, is known as the Fourier transform of that signal. (You can read a more detailed explanation of the maths involved here - the maths is quite complicated but the mathematical ideas involved are lovely!)

The function f varies in time - representing a sound wave. The Fourier transform process takes f and decomposes it into its constituent sine waves, with particular frequencies and amplitudes. The Fourier transform is represented as spikes in the frequency domain, the height of the spike showing the amplitude of the wave of that frequency.

About this article

This article was produced as part of our coverage of the Dispersive hydrodynamics: mathematics, simulation and experiments, with applications in nonlinear waves programme hosted by the Isaac Newton Institute for Mathematical Sciences. You can find more content about the programme here.

Rachel Thomas and Marianne Freiberger are the Editors of Plus.

This article is based on the book Numericon: A journey through the hidden lives of numbers by Marianne Freiberger and Rachel Thomas and our article Fourier transforms of images.

This article was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) - you can find all the content from the collaboration here.
The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.

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Think of a right-angled triangle. Then you'll probably remember from school that you can use the sine and cosine functions to find out more about the triangle. If is one of the angles that isn't the right angle then you have