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Matemáticas cercanas

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El número π es seguramente el número más famoso de las matemáticas.

Como todo el mundo sabrá su valor es 3 " y algo más ".

Sobre ese "y algo más" la gran mayoría recuerda que es 3,14... (aproximación con dos decimales que habitualmente se utiliza en la escuela), o con algún decimal más 3,1415926... o, en un alarde de capacidad memorística, puede que 3,14159265358979323846264 ...
Pared del Mathematikum de Giessen con algunos de los decimales de Pi (Imagen de Dontworry bajo Licencia CC BY-SA 4.0 via Wikimedia Commons)
Incluso se puede llegar al extremo del joven estudiante Rajveer Meena, que fue capaz de decir de memoria 70.000 decimales el 21 de marzo de 2015 en un tiempo de 9 horas y 7 minutos.

Sí, no me he equivocado... ¡70.000!... conmigo no contéis para algo así porque lo mío es razonar, no memorizar.

Pero ¿cómo podemos calcular decimales de π?

Ya en el Papiro de Ahmes, conocido también como Papiro Rhind, escrito por el escriba Ahmes ( A'h-mosè) a mediados del siglo XVI a. C. se hacía una aproximación de π considerando que un cuadrado de lado 8 equivalía en superficie a un círculo de diámetro 9.

Parte de la primera sección del Papiro de Ahmes o Papiro Rhind (Imagen de dominio público).

A lo largo de la historia se han ido utilizando nuevos métodos que han permitido obtener mejores aproximaciones de este tan popular número.

En el siguiente vídeo de Quantum Fracture se muestran, de manera bastante didáctica y amena, tres métodos que permiten ir obteniendo decimales de π, unos más eficientes que otros, pero que al menos podemos emplear para obtener los primeros decimales: El método de Arquímedes o de los polígonos regulares, el método de Montecarlo y el método empleado por Euler de las series infinitas ( problema de Basilea).

Además de éstos, hay muchos más métodos como, por ejemplo, el utilizado por Yasumasa Kanada (Universidad de Tokio) para obtener en diciembre de 2002 nada menos que 1.241.100.000.000 dígitos, empleando las fórmulas modificadas de Machin:

Pero si hay un método que destaca es, desde luego, el algoritmo de Chudnovsky, cuya expresión es:

Este algoritmo de cálculo fue el utilizado por los informáticos Alexander Yee y Shigeru Kondo que, el 28 de Diciembre de 2013, consiguieron calcular 12,1 billones de decimales de π (12.100.000.000.000 decimales), bueno exactamente 12,1 billones cincuenta decimales (12.100.000.000.050 decimales).

Para que os hagáis una idea, si escribiéramos seguidos los 12,1 billones de decimales calculados por Yee y Kondo con una fuente típica, por ejemplo Times New Roman tamaño 12, el papel necesario para ello daría aproximadamente 60 vueltas a la Tierra.

Por cierto, que si queréis saber más sobre el algoritmo de Chudnovsky os recomiendo la entrada de Gaussianos donde se explica perfectamente y con más detalle, y de la que he tomado prestada la expresión anterior.

Y para terminar, ya que se va acercando el día de π, os invito a visitar estas otras entradas del blog que tratan sobre este número (alguna la he enlazado anteriormente)...
¿Quieres buscar un número entre los decimales de π? ¿Magia? ¿Casualidad? ... el dígito un millón de π ¿Sabías que...? Curiosidades de π ¿Sabías que...? Curiosidades de π (II) ¿Sabías que...? Curiosidades de π (III) ¿Sabías que...? Curiosidades de π (IV) ¿Sabías que...? Sobre el día de PI 22/7 Día de aproximación de Pi o Casual Pi Day ¡Su historia en 15 viñetas!
Por cierto que...

Este post participa en la Edición 7.X del Carnaval de Matemáticas cuyo anfitrión es el Blog del IMUS.

Referencias utilizadas: El día del número PI de matematicascercanas.com (14 de marzo de 2015) El algoritmo de Chudnovsky, o cómo se calculan los decimales de Pi en el siglo XXI de Gaussianos (14 de marzo de 2013) 3 Maneras de Saber que π = 3.14159... de Quantum Fracture (26 de enero de 2017) 12.1 Trillion Digits of Pi de numberworld.org (página de Alexander Yee) Número π de Wikipedia, la enciclopedia libre

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Algunas maneras de obtener decimales de π - matematicascercanas

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El número π es seguramente el número más famoso de las matemáticas. Como todo el mundo sabrá su valor es 3 " y algo más ". Sobre ese "y algo más" la gran mayoría recuerda que es 3,14... (aproximación con dos decimales que habitualmente se utiliza en la escuela), o con algún decimal más 3,1415926...

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The infinite life of pi - Reynaldo Lopes

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Grandes temas de la matemática: Capítulo 1: El número PI

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Alphabetical Index Interactive Entries Random Entry New in MathWorld

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e has always bothered me - not the letter, but the mathematical constant. What does it really mean?

Math books and even my beloved Wikipedia describe e using obtuse jargon:

The mathematical constant e is the base of the natural logarithm.

And when you look up the natural logarithm you get:

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.

Nice circular reference there. It's like a dictionary that defines labyrinthine with Byzantine: it's correct but not helpful. What's wrong with everyday words like "complicated"?

I'm not picking on Wikipedia - many math explanations are dry and formal in their quest for rigor. But this doesn't help beginners trying to get a handle on a subject (and we were all a beginner at one point).

No more! Today I'm sharing my intuitive, high-level insights about what e is and why it rocks. Save your rigorous math book for another time. Here's a quick video overview of the insights:

e is NOT Just a Number

Describing e as "a constant approximately 2.71828..." is like calling pi "an irrational number, approximately equal to 3.1415...". Sure, it's true, but you completely missed the point.

Pi is the ratio between circumference and diameter shared by all circles. It is a fundamental ratio inherent in all circles and therefore impacts any calculation of circumference, area, volume, and surface area for circles, spheres, cylinders, and so on. Pi is important and shows all circles are related, not to mention the trigonometric functions derived from circles (sin, cos, tan).

e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.

e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more. Even jagged systems that don't grow smoothly can be approximated by e.

Just like every number can be considered a scaled version of 1 (the base unit), every circle can be considered a scaled version of the unit circle (radius 1), and every rate of growth can be considered a scaled version of e (unit growth, perfectly compounded).

So e is not an obscure, seemingly random number. e represents the idea that all continually growing systems are scaled versions of a common rate.

Understanding Exponential Growth

Let's start by looking at a basic system that doubles after an amount of time. For example,

Bacteria can split and "doubles" every 24 hours

We get twice as many noodles when we fold them in half.

Your money doubles every year if you get 100% return (lucky!)

And it looks like this:

Splitting in two or doubling is a very common progression. Sure, we can triple or quadruple, but doubling is convenient, so hang with me here.

Mathematically, if we have x splits then we get times as much stuff than when we started. With 1 split we have or 2 times as much. With 4 splits we have times as much. As a general formula:

Said another way, doubling is 100% growth. We can rewrite our formula like this:

It's the same equation, but we separate 2 into what it really is: the original value (1) plus 100%. Clever, eh?

Of course, we can substitute any number (50%, 25%, 200%) for 100% and get the growth formula for that new rate. So the general formula for x periods of return is:

This just means we use our rate of return, (1 + return), "x" times.

A Closer Look

Our formula assumes growth happens in discrete steps. Our bacteria are waiting, waiting, and then boom, they double at the very last minute. Our interest earnings magically appear at the 1 year mark. Based on the formula above, growth is punctuated and happens instantly. The green dots suddenly appear.

The world isn't always like this. If we zoom in, we see that our bacterial friends split over time:

Mr. Green doesn't just show up: he slowly grows out of Mr. Blue. After 1 unit of time (24 hours in our case), Mr. Green is complete. He then becomes a mature blue cell and can create new green cells of his own.

Does this information change our equation?

Nope. In the bacteria case, the half-formed green cells still can't do anything until they are fully grown and separated from their blue parents. The equation still holds.

Money Changes Everything

But money is different. As soon as we earn a penny of interest, that penny can start earning micro-pennies of its own. We don't need to wait until we earn a complete dollar in interest - fresh money doesn't need to mature.

Based on our old formula, interest growth looks like this:

But again, this isn't quite right: all the interest appears on the last day. Let's zoom in and split the year into two chunks. We earn 100% interest every year, or 50% every 6 months. So, we earn 50 cents the first 6 months and another 50 cents in the last half of the year:

But this still isn't right! Sure, our original dollar (Mr. Blue) earns a dollar over the course of a year. But after 6 months we had a 50-cent piece, ready to go, that we neglected! That 50 cents could have earned money on its own:

Because our rate is 50% per half year, that 50 cents would have earned 25 cents (50% times 50 cents). At the end of 1 year we'd have

Our original dollar (Mr. Blue)

The dollar Mr. Blue made (Mr. Green)

The 25 cents Mr. Green made (Mr. Red)

Giving us a total of $2.25. We gained $1.25 from our initial dollar, even better than doubling!

Let's turn our return into a formula. The growth of two half-periods of 50% is:

Diving into Compound Growth

It's time to step it up a notch. Instead of splitting growth into two periods of 50% increase, let's split it into 3 segments of 33% growth. Who says we have to wait for 6 months before we start getting interest? Let's get more granular in our counting.

Charting our growth for 3 compounded periods gives a funny picture:

Think of each color as shoveling money upwards towards the other colors (its children), at 33% per period:

Month 0: We start with Mr. Blue at $1.

Month 4: Mr. Blue has earned 1/3 dollar on himself, and creates Mr. Green, shoveling along 33 cents.

Month 8: Mr. Blue earns another 33 cents and gives it to Mr. Green, bringing Mr. Green up to 66 cents. Mr. Green has actually earned 33% on his previous value, creating 11 cents (33% * 33 cents). This 11 cents becomes Mr. Red.

Month 12: Things get a bit crazy. Mr. Blue earns another 33 cents and shovels it to Mr. Green, bringing Mr. Green to a full dollar. Mr. Green earns 33% return on his Month 8 value (66 cents), earning 22 cents. This 22 cents gets added to Mr. Red, who now totals 33 cents. And Mr. Red, who started at 11 cents, has earned 4 cents (33% * .11) on his own, creating Mr. Purple.

Phew! The final value after 12 months is: 1 + 1 + .33 + .04 or about 2.37.

Take some time to really understand what's happening with this growth:

Each color earns interest on itself and hands it off to another color. The newly-created money can earn money of its own, and on the cycle goes.

I like to think of the original amount (Mr. Blue) as never changing. Mr. Blue shovels money to create Mr. Green, a steady 33 every 4 months since Mr. Blue does not change. In the diagram, Mr. Blue has a blue arrow showing how he feeds Mr. Green.

Mr. Green just happens to create and feed Mr. Red (green arrow), but Mr. Blue isn't aware of this.

As Mr. Green grows over time (being constantly fed by Mr. Blue), he contributes more and more to Mr. Red. Between months 4-8 Mr. Green gives 11 cents to Mr. Red. Between months 8-12 Mr. Green gives 22 cents to Mr. Red, since Mr. Green was at 66 cents during Month 8. If we expanded the chart, Mr. Green would give 33 cents to Mr. Red, since Mr. Green reached a full dollar by Month 12.

Make sense? It's tough at first - I even confused myself a bit while putting the charts together. But see that each dollar creates little helpers, who in turn create helpers, and so on.

We get a formula by using 3 periods in our growth equation:

We earned $1.37, even better than the $1.25 we got last time!

Can We Get Infinite Money?

Why not take even shorter time periods? How about every month, day, hour, or even nanosecond? Will our returns skyrocket?

Our return gets better, but only to a point. Try using different numbers of n in our magic formula to see our total return:

n (1 + 1/n)^n ----------------------- 1 2 2 2.25 3 2.37 5 2.488 10 2.5937 100 2.7048 1,000 2.7169 10,000 2.71814 100,000 2.718268 1,000,000 2.7182804 ...

The numbers get bigger and converge around 2.718. Hey... wait a minute... that looks like e!

Yowza. In geeky math terms, e is defined to be that rate of growth if we continually compound 100% return on smaller and smaller time periods:

This limit appears to converge, and there are proofs to that effect. But as you can see, as we take finer time periods the total return stays around 2.718.

But what does it all mean?

The number e (2.718...) is the maximum possible result when compounding 100% growth for one time period. Sure, you started out expecting to grow from 1 to 2 (that's a 100% increase, right?). But with each tiny step forward you create a little dividend that starts growing on its own. When all is said and done, you end up with e (2.718...) at the end of 1 time period, not 2. e is the maximum, what happens when we compound 100% as much as possible.

So, if we start with $1.00 and compound continuously at 100% return we get 1e. If we start with $2.00, we get 2e. If we start with $11.79, we get 11.79e.

e is like a speed limit (like c, the speed of light) saying how fast you can possibly grow using a continuous process. You might not always reach the speed limit, but it's a reference point: you can write every rate of growth in terms of this universal constant.

(Aside: Be careful about separating the increase from the final result. 1 becoming e (2.718...) is an increase (growth rate) of 171.8%. e, by itself, is the final result you observe after all growth is taken into account (original + increase)).

What about different rates?

Good question. What if we grow at 50% annually, instead of 100%? Can we still use e?

Let's see. The rate of 50% compound growth would look like this:

Hrm. What can we do here? Remember, 50% is the total return, and n is the number of periods to split the growth into for compounding. If we pick n=50, we can split our growth into 50 chunks of 1% interest:

Sure, it's not infinity, but it's pretty granular. Now imagine we also divided our regular rate of 100% into chunks of 1%:

Ah, something is emerging here. In our regular case, we have 100 cumulative changes of 1% each. In the 50% scenario, we have 50 cumulative changes of 1% each.

What is the difference between the two numbers? Well, it's just half the number of changes:

This is pretty interesting. 50 / 100 = .5, which is the exponent we raise e to. This works in general: if we had a 300% growth rate, we could break it into 300 chunks of 1% growth. This would be triple the normal amount for a net rate of .

Even though growth can look like addition (+1%), we need to remember that it's really a multiplication (x 1.01). This is why we use exponents (repeated multiplication) and square roots ( means "half" the number of changes, i.e. half the number of multiplications).

Although we picked 1%, we could have chosen any small unit of growth (.1%, .0001%, or even an infinitely small amount!). The key is that for any rate we pick, it's just a new exponent on e:

What about different times?

Suppose we have 300% growth for 2 years. We'd multiply one year's growth () by itself:

And in general:

Because of the magic of exponents, we can avoid having two powers and just multiply rate and time together in a single exponent.

The big secret: e merges rate and time.

This is wild! can mean two things:

x is the number of times we multiply a growth rate: 100% growth for 3 years is

x is the growth rate itself: 300% growth for one year is

Won't this overlap confuse things? Will our formulas break and the world come to an end?

It all works out. When we write:

the variable x is a combination of rate and time.

Let me explain. When dealing with continuous compound growth, 10 years of 3% growth has the same overall impact as 1 year of 30% growth (and no growth afterward).

10 years of 3% growth means 30 changes of 1%. These changes happen over 10 years, so you are growing continuously at 3% per year.

1 period of 30% growth means 30 changes of 1%, but happening in a single year. So you grow for 30% a year and stop.

The same "30 changes of 1%" happen in each case. The faster your rate (30%) the less time you need to grow for the same effect (1 year). The slower your rate (3%) the longer you need to grow (10 years).

But in both cases, the growth is in the end. We're impatient and prefer large, fast growth to slow, long growth but e shows they have the same net effect.

So, our general formula becomes:

If we have a return of r for t time periods, our net compound growth is . This even works for negative and fractional returns, by the way.

Example Time!

Examples make everything more fun. A quick note: We're so used to formulas like and regular, compound interest that it's easy to get confused (myself included). Read more about simple, compound and continuous growth.

These examples focus on smooth, continuous growth, not the jumpy growth that happens at yearly intervals. There are ways to convert between them, but we'll save that for another article.
Example 1: Growing crystals
Suppose I have 300kg of magic crystals. They're magic because they grow throughout the day: I watch a single crystal, and in the course of 24 hours it sheds off its own weight in crystals. (The baby crystals start growing immediately at the same rate, but I can't track that - I'm watching how much the original sheds). How much will I have after 10 days?

Well, since the crystals start growing immediately, we want 300 · e continuous growth. Our rate is 100% every 24 hours, so after 10 days we get: 1 · 10 = 6.6 million kg of our magic gem.

This can be tricky: notice the difference between the input rate and the total output rate. The "input" rate is how much a single crystal changes: 100% in 24 hours. The net output rate is e (2.718x) because the baby crystals grow on their own.

In this case we have the input rate (how fast one crystal grows) and want the total result after compounding (how fast the entire group grows because of the baby crystals). If we have the total growth rate and want the rate of a single crystal, we work backwards and use the natural log.
Example 2: Maximum interest rates
Suppose I have $120 in an account with 5% interest. My bank is generous and gives me the maximum possible compounding. How much will I have after 10 years?

Our rate is 5%, and we're lucky enough to compound continuously. After 10 years, we get $120 · e.05 · 10 = $197.85. Of course, most banks aren't nice enough to give you the best possible rate. The difference between your actual return and the continuous one is how much they don't like you.
Example 3: Radioactive decay
I have 10kg of a radioactive material, which appears to continuously decay at a rate of 100% per year. How much will I have after 3 years?

Zip? Zero? Nothing? Think again.

Decaying continuously at 100% per year is the trajectory we start off with. Yes, we do begin with 10kg and expect to "lose it all" by the end of the year, since we're decaying at 10 kg/year.

We go a few months and get to 5kg. Half a year left? Nope! Now we're losing at a rate of 5kg/year, so we have another full year from this moment!

We wait a few more months, and get to 2kg. And of course, now we're decaying at a rate of 2kg/year, so we have a full year (from this moment). We get 1 kg, have a full year, get to .5 kg, have a full year - see the pattern?

As time goes on, we lose material, but our rate of decay slows down. This constantly changing growth is the essence of continuous growth & decay.

After 3 years, we'll have 10 · e-1 · 3 = .498 kg. We use a negative exponent for decay - we want a fraction (1/) vs a growth multiplier (). [Decay is commonly given in terms of "half life" -- we'll talk about converting these rates in a future article.]
More Examples
If you want fancier examples, try the Black-Scholes option formula (notice e used for exponential decay in value) or radioactive decay. The goal is to see in a formula and understand why it's there: it's modeling a type of growth or decay.

And now you know why it's "e", and not pi or some other number: e raised to "r*t" gives you the growth impact of rate r and time t.

There's More To Learn

My goal was to:

Explain why e is important: It's a fundamental constant, like pi, that shows up in growth rates.

Give an intuitive explanation: e lets you see the impact of any growth rate. Every new "piece" (Mr. Green, Mr. Red, etc.) helps add to the total growth.

Show how it's used: lets you predict the impact of any growth rate and time period.

Get you hungry for more: In the upcoming articles, I'll dive into other properties of e.

This article is just the start - cramming everything into a single page would tire you and me both. Dust yourself off, take a break and learn about e's evil twin, the natural logarithm.

Other Posts In This Series

1. An Intuitive Guide To Exponential Functions & e

2. Demystifying the Natural Logarithm (ln)

3. A Visual Guide to Simple, Compound and Continuous Interest Rates

4. Understanding Exponents (Why does 0^0 = 1?)

5. Using Logarithms in the Real World

6. How To Think With Exponents And Logarithms

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An Intuitive Guide To Exponential Functions & e

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e has always bothered me - not the letter, but the mathematical constant. What does it really mean? Math books and even my beloved Wikipedia describe e using obtuse jargon: The mathematical constant e is the base of the natural logarithm.

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Las matemáticas que se estudiaban en México hacia el año de 1630 no eran otras que las del Renacimiento: geometría clásica, álgebra, aritmética y trigonometría. Lugar relevante ocupaba la ciencia de los números, a la que podemos llamar "aritmosofía", pues como heredera de la vieja tradición hermético-pitagórica intentaba determinar el papel de los números en la aprehensión de la idea de belleza basada en una armonía o un equilibrio. Por otra parte, desde el punto de vista matemático el mundo astronómico de las esferas y los círculos planetarios era tan armonioso que resultaba fiel retrato de su Creador, tal como lo afirmó fray Luis de Granada en su "Introducción al símbolo de la fé".

Cuando fray Diego Rodríguez resolvió un cierto caso de la ecuación de tercer grado encontró que existían raíces que no eran reales, es decir que estaban formadas por la raíz cuadrada de un número negativo. Como lo hiciera Descartes por esas misma fechas, el padre Rodríguez las declaró "falsas" y las rechazó por imposibles. Pero a medida que avanzaba en sus estudios algebraicos se dió cuenta de que los números no reales eran difíciles de evitar y -derrotado- terminó, como sus colegas del otro lado del Atlántico, por aceptar su existencia. Así entraron en México los números "imaginarios", es decir aquellos números que no eran reales pero que paradójicamente sí existían. Apenas podemos comprender la transformación mental que supuso este descubrimiento. Era la aceptación de la realidad de lo imaginario. Hacia el final de sus voluminosos manuscritos de álgebra, el padre Rodríguez capitulaba y aceptaba la existencia de lo que por definición no podía existir. Su solución personal y originalísima de las ecuaciones de cuarto grado ya incluye raices imaginarias. Pero no todos aceptaron tan fácilmente esta revolución en los números que corría paralela a la revolución en los círculos y las elipses planetarias. Leibniz rechazó su existencia cuando afirmó irónicamente que los números imaginarios eran un "excelente y maravilloso refugio del Espíritu Santo, una especie de anfibio entre ser y no ser"; y todavía en el siglo XVIII Euler los describió como "nada, o menos que nada".

El arte barroco y los números imaginarios en México en el siglo XVII

Elías Trabulse

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Así entraron en México los números "imaginarios"

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Las matemáticas que se estudiaban en México hacia el año de 1630 no eran otras que las del Renacimiento: geometría clásica, álgebra, aritmética y trigonometría. Lugar relevante ocupaba la ciencia de los números, a la que podemos llamar "aritmosofía", pues como heredera de la vieja tradición hermético-pitagórica intentaba determinar el papel de los números en la aprehensión de la idea de belleza basada en una armonía o un equilibrio.

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Imaginary numbers always confused me. Like understanding e, most explanations fell into one of two categories:

It's a mathematical abstraction, and the equations work out. Deal with it.

It's used in advanced physics, trust us. Just wait until college.

Gee, what a great way to encourage math in kids! Today we'll assault this topic with our favorite tools:

And our secret weapon: learning by analogy. We'll approach imaginary numbers by observing its ancestor, the negatives. Here's your guidebook:

It doesn't make sense yet, but hang in there. By the end we'll hunt down i and put it in a headlock, instead of the reverse.

Video Walkthrough:

Really Understanding Negative Numbers

Negative numbers aren't easy. Imagine you're a European mathematician in the 1700s. You have 3 and 4, and know you can write 4 - 3 = 1. Simple.

But what about 3-4? What, exactly, does that mean? How can you take 4 cows from 3? How could you have less than nothing?

Negatives were considered absurd, something that "darkened the very whole doctrines of the equations" ( Francis Maseres, 1759). Yet today, it'd be absurd to think negatives aren't logical or useful. Try asking your teacher whether negatives corrupt the very foundations of math.

What happened? We invented a theoretical number that had useful properties. Negatives aren't something we can touch or hold, but they describe certain relationships well (like debt). It was a useful fiction.

Rather than saying "I owe you 30" and reading words to see if I'm up or down, I can write "-30" and know it means I'm in the hole. If I earn money and pay my debts (-30 + 100 = 70), I can record the transaction easily. I have +70 afterwards, which means I'm in the clear.

The positive and negative signs automatically keep track of the direction - you don't need a sentence to describe the impact of each transaction. Math became easier, more elegant. It didn't matter if negatives were "tangible" - they had useful properties, and we used them until they became everyday items. Today you'd call someone obscene names if they didn't "get" negatives.

But let's not be smug about the struggle: negative numbers were a huge mental shift. Even Euler, the genius who discovered e and much more, didn't understand negatives as we do today. They were considered "meaningless" results (he later made up for this in style).

It's a testament to our mental potential that today's children are expected to understand ideas that once confounded ancient mathematicians.

Enter Imaginary Numbers

Imaginary numbers have a similar story. We can solve equations like this all day long:

The answers are 3 and -3. But suppose some wiseguy puts in a teensy, tiny minus sign:

Uh oh. This question makes most people cringe the first time they see it. You want the square root of a number less than zero? That's absurd! (Historically, there were real questions to answer, but I like to imagine a wiseguy.)

It seems crazy, just like negatives, zero, and irrationals (non-repeating numbers) must have seemed crazy at first. There's no "real" meaning to this question, right?

Wrong. So-called "imaginary numbers" are as normal as every other number (or just as fake): they're a tool to describe the world. In the same spirit of assuming -1, .3, and 0 "exist", let's assume some number i exists where:

That is, you multiply i by itself to get -1. What happens now?

Well, first we get a headache. But playing the "Let's pretend i exists" game actually makes math easier and more elegant. New relationships emerge that we can describe with ease.

You may not believe in i, just like those fuddy old mathematicians didn't believe in -1. New, brain-twisting concepts are hard and they don't make sense immediately, even for Euler. But as the negatives showed us, strange concepts can still be useful.

I dislike the term "imaginary number" - it was considered an insult, a slur, designed to hurt i 's feelings. The number i is just as normal as other numbers, but the name "imaginary" stuck so we'll use it.

Visual Understanding of Negative and Complex Numbers

As we saw last time, the equation $x^2 = 9$ really means:

or
What transformation x, when applied twice, turns 1 to 9?
The two answers are "x = 3" and "x = -3": That is, you can "scale by" 3 or "scale by 3 and flip" (flipping or taking the opposite is one interpretation of multiplying by a negative).

Now let's think about $x^2 = -1$, which is really

What transformation x, when applied twice, turns 1 into -1? Hrm.

We can't multiply by a positive twice, because the result stays positive

We can't multiply by a negative twice, because the result will flip back to positive on the second multiplication

But what about... a rotation! It sounds crazy, but if we imagine x being a "rotation of 90 degrees", then applying x twice will be a 180 degree rotation, or a flip from 1 to -1!

Yowza! And if we think about it more, we could rotate twice in the other direction (clockwise) to turn 1 into -1. This is "negative" rotation or a multiplication by -i:

If we multiply by -i twice, the first multiplication would turn 1 into -i, and the second turns -i into -1. So there's really two square roots of -1: i and -i.

This is pretty cool. We have some sort of answer, but what does it mean?

i is a "new imaginary dimension" to measure a number

i (or -i) is what numbers "become" when rotated

Multiplying i is a rotation by 90 degrees counter-clockwise

Multiplying by -i is a rotation of 90 degrees clockwise

Two rotations in either direction is -1: it brings us back into the "regular" dimensions of positive and negative numbers.

Numbers are 2-dimensional. Yes, it's mind bending, just like decimals or long division would be mind-bending to an ancient Roman. ( What do you mean there's a number between 1 and 2?). It's a strange, new way to think about math.

We asked "How do we turn 1 into -1 in two steps?" and found an answer: rotate it 90 degrees. It's a strange, new way to think about math. But it's useful. (By the way, this geometric interpretation of complex numbers didn't arrive until decades after i was discovered).

Also, keep in mind that having counter-clockwise be positive is a human convention - it easily could have been the other way.

Finding Patterns

Let's dive into the details a bit. When multiplying negative numbers (like -1), you get a pattern:

Since -1 doesn't change the size of a number, just the sign, you flip back and forth. For some number "x", you'd get:

This idea is useful. The number "x" can represent a good or bad hair week. Suppose weeks alternate between good and bad; this is a good week; what will it be like in 47 weeks?

So -x means a bad hair week. Notice how negative numbers "keep track of the sign": we can throw $(-1)^{47}$ into a calculator without having to count (" Week 1 is good, week 2 is bad... week 3 is good... "). Things that flip back and forth can be modeled well with negative numbers.

Ok. Now what happens if we keep multiplying by $i$?

Very funny. Let's reduce this a bit:

$1 = 1$ (No questions here)

$i = i$ (Can't do much)

$i^2 = -1$ (That's what i is all about)

$i^3 = (i \cdot i) \cdot i = -1 \cdot i = -i$ (Ah, 3 rotations counter-clockwise = 1 rotation clockwise. Neat.)

$i^4 = (i \cdot i) \cdot (i \cdot i) = -1 \cdot -1 = 1$ (4 rotations bring us "full circle")

$i^5 = i^4 \cdot i = 1 \cdot i = i$ (Here we go again...)

Represented visually:

We cycle every 4th rotation. This makes sense, right? Any kid can tell you that 4 left turns is the same as no turns at all. Now rather than focusing on imaginary numbers ($i$, $i^2$), look at the general pattern:

Like negative numbers modeling flipping, imaginary numbers can model anything that rotates between two dimensions "X" and "Y". Or anything with a cyclic, circular relationship - have anything in mind?

'Cos it'd be a sin if you didn't. There'll be more in future articles. [Editor's note: Kalid is in electroshock therapy to treat his pun addiction.]

Understanding Complex Numbers

There's another detail to cover: can a number be both "real" and "imaginary"?

You bet. Who says we have to rotate the entire 90 degrees? If we keep 1 foot in the "real" dimension and another in the imaginary one, it looks like this:

We're at a 45 degree angle, with equal parts in the real and imaginary (1 + i). It's like a hotdog with both mustard and ketchup - who says you need to choose?

In fact, we can pick any combination of real and imaginary numbers and make a triangle. The angle becomes the "angle of rotation". A complex number is the fancy name for numbers with both real and imaginary parts. They're written a + bi, where

a is the real part

b is the imaginary part

Not too bad. But there's one last question: how "big" is a complex number? We can't measure the real part or imaginary parts in isolation, because that would miss the big picture.

Let's step back. The size of a negative number is not whether you can count it - it's the distance from zero. In the case of negatives this is:

Which is another way to find the absolute value. But for complex numbers, how do we measure two components at 90 degree angles?
It's a bird... it's a plane... it's Pythagoras!
Geez, his theorem shows up everywhere, even in numbers invented 2000 years after his time. Yes, we are making a triangle of sorts, and the hypotenuse is the distance from zero:

Neat. While measuring the size isn't as easy as "dropping the negative sign", complex numbers do have their uses. Let's take a look.

A Real Example: Rotations

We're not going to wait until college physics to use imaginary numbers. Let's try them out today. There's much more to say about complex multiplication, but keep this in mind:

Let's take a look. Suppose I'm on a boat, with a heading of 3 units East for every 4 units North. I want to change my heading 45 degrees counter-clockwise. What's the new heading?

Some hotshot will say " That's simple! Just take the sine, cosine, gobbledegook by the tangent... fluxsom the foobar... and... ". . Sorry, did I break your calculator? Care to answer that question again?

Let's try a simpler approach: we're on a heading of 3 + 4i (whatever that angle is; we don't really care), and want to rotate by 45 degrees. Well, 45 degrees is 1 + i (perfect diagonal), so we can multiply by that amount!

Here's the idea:

Original heading: 3 units East, 4 units North = 3 + 4i

Rotate counter-clockwise by 45 degrees = multiply by 1 + i. (Here's why multiplication, not addition, performs the rotation.)

If we multiply them together we get:

So our new orientation is 1 unit West (-1 East), and 7 units North, which you could draw out and follow.

But yowza! We found that out in 10 seconds, without touching sine or cosine. There were no vectors, matrices, or keeping track what quadrant we are in. It was just arithmetic with a touch of algebra to cross-multiply. Imaginary numbers have the rotation rules baked in: it just works.

Even better, the result is useful. We have a heading (-1, 7) instead of an angle (atan(7/-1) = 98.13, keeping in mind we're in quadrant 2). How, exactly, were you planning on drawing and following that angle? With the protractor you keep around?

No, you'd convert it into cosine and sine (-.14 and .99), find a reasonable ratio between them (about 1 to 7), and sketch out the triangle. Complex numbers beat you to it, instantly, accurately, and without a calculator.

If you're like me, you'll find this use mind-blowing. And if you don't, well, I'm afraid math doesn't toot your horn. Sorry.

Trigonometry is great, but complex numbers can make ugly calculations simple (like calculating cosine(a+b) ). This is just a preview; later articles will give you the full meal.

Aside: Some people think "Hey, it's not useful to have North/East headings instead of a degree angle to follow!"

Really? Ok, look at your right hand. What's the angle from the bottom of your pinky to the top of your index finger? Good luck figuring that out on your own.

With a heading, you can at least say "Oh, it's X inches across and Y inches up" and have some chance of working with that bearing.

Complex Numbers Aren't

That was a whirlwind tour of my basic insights. Take a look at the first chart - it should make sense now.

There's so much more to these beautiful, zany numbers, but my brain is tired. My goals were simple:

Convince you that complex numbers were considered "crazy" but can be useful (just like negative numbers were)

Show how complex numbers can make certain problems easier, like rotations

If I seem hot and bothered about this topic, there's a reason. Imaginary numbers have been a bee in my bonnet for years - the lack of an intuitive insight frustrated me.

Now that I've finally had insights, I'm bursting to share them. But it frustrates me that you're reading this on the blog of a wild-eyed lunatic, and not in a classroom. We suffocate our questions and "chug through" - because we don't search for and share clean, intuitive insights. Egad.

But better to light a candle than curse the darkness: here's my thoughts, and one of you will shine a spotlight. Thinking we've "figured out" a topic like numbers is what keeps us in Roman Numeral land.

There's much more complex numbers: check out the details of complex arithmetic. Happy math.

Epilogue: But they're still strange!

I know, they're still strange to me too. I try to put myself in the mind of the first person to discover zero.

Zero is such a weird idea, having "something" represent "nothing", and it eluded the Romans. Complex numbers are similar - it's a new way of thinking. But both zero and complex numbers make math much easier. If we never adopted strange, new number systems, we'd still be counting on our fingers.

I repeat this analogy because it's so easy to start thinking that complex numbers aren't "normal". Let's keep our mind open: in the future they'll chuckle that complex numbers were once distrusted, even until the 2000's.

If you want more nitty-gritty, check out wikipedia, the Dr. Math discussion, or another argument on why imaginary numbers exist.

Other Posts In This Series

1. A Visual, Intuitive Guide to Imaginary Numbers

2. Intuitive Arithmetic With Complex Numbers

3. Understanding Why Complex Multiplication Works

4. Intuitive Guide to Angles, Degrees and Radians

5. Intuitive Understanding Of Euler's Formula

6. An Interactive Guide To The Fourier Transform

7. Intuitive Guide to Convolution

8. Intuitive Understanding of Sine Waves

9. An Intuitive Guide to Linear Algebra

10. A Programmer's Intuition for Matrix Multiplication

11. Imaginary Multiplication vs. Imaginary Exponents

12. Intuitive Guide to Hyperbolic Functions

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