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https://venngage.com/blog/misleading-graphs/

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In this post-truth era, graphs are being used to skew data and spin narrative like never before. Especially with the velocity at which some of these topics spread across social media.

All it takes is a single graph from a less-than-reputable source, blasted out to a list of followers, to spread a false narrative around the world.

We have already seen this happen many times during the COVID-19 response, which is why we added a new section featuring a few of those misleading graphs!

Now the data doesn't even have to be bad-it could just be presented in a misleading way. I mean, there is a whole Wikipedia page , Reddit community , and hundreds of articles about how graphs can be used to misinform readers.

Now, I can't make these data-skewing creators stop, but I can help you spot these misleading graphs when they crop up. I've included examples of how to graph the data properly, too.
Not a designer? No problem. Create better graphs in minutes for free with Venngage's Graph Maker.
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In the meantime, following data visualization best practices ensures that your graphs are always clear and understandable, like in this infographic (the accurate part, at least!).
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Use the links below to jump to each section:

1. Omitting the baseline

Omitting baselines, or the axis of a graph, is one of the most common ways data is manipulated in graphs. This misleading tactic is frequently used to make one group look better than another.

Truncating graphs can make something that is not very significant look like a massive difference.

Just take a look at the first example below that compared how Democrats, Republicans, and Independents felt about a certain issue.

If you took a cursory glance at this graph you would probably think that Democrats agreed almost three times more than Republicans and Independents. But a closer look shows that difference was only about a 14% difference. This graph was obviously created to push an incorrect idea about a certain group.

If they wanted to properly show the differences, or just report the facts objectively, they could have included a more accurate graph like this:
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Whoa, that is much better!

In this graph, they are trying to do the same thing as the previous example: deceive their audience. But the margin is a lot closer, with only an 11% increase.

That didn't stop Fox from publishing a graph that makes one tax rate seem almost 4x larger than the other.

Oh, and the small numbers that they used not only on the axis but also as labels almost confirm they wanted to mislead their audience.

Here is how it should have looked (as you can see, the difference is not that impressive):

Writers also get themselves into trouble when they try too hard to be unique or creative with a graph.

One of the first things they remove is the baselines or axis. I believe this is because the writers think it actually distracts from the data. But it only makes the data harder to read!

As you can see in this graph, there are no baselines at all! Now, this misleading graph makes it seem like our current president has nominated judges at double or triple the rate of his predecessors.

When the graph is adjusted and the axis are added you can see the real story. He definitely is nominating at a high rate, but it's not as crazy as the original draft made it seem:
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It is important to remember that axis manipulation like this is not limited to just politics or hot-button issues.

Even music journalists sometimes find themselves on the wrong side of a misleading or bad graph. Just check out this bad graph that I found from :

I'm guessing Drake was pretty happy to see this chart, but it really should have looked like this:
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Sorry Drake.

You can also see these kinds of misleading graphs it in tech:

In business meetings:

And even in nutrition information:

Honestly, wherever people are using data visualizations to backup their claims, you will probably find someone misleading graphs like the examples above.

Related: What Is Data Visualization? (Definition, Examples, Best Practices)

2. Manipulating the Y-Axis

Axis manipulation is almost the opposite of truncating data, because they include the axis and baselines but change them so much that they lose meaning.

This is a very powerful tool on social media and can be used to push a false narrative.

The saddest thing is that these organizations know exactly what they are doing.

What should frustrate you, the reader, is that both of these organizations had access to the same data and tools. But only one decided to present it in a trustworthy manner.

I mean, I could do the exact same thing using data about my beloved Arkansas Razorbacks.

In this example, I looked at their win totals over the last 15 or so years and put them on a normal line graph. But I then took the same approach as the previous graph and manipulated the y-axis:

With those changes, I can make it looks like the Razorbacks have been winning fairly consistently over the past few years.

Awesome!

But if you follow college football you will know that we have not been winners most of the time, and the graph should look a little closer to this:
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This one is not so inspiring...

Again, this is the exact same data, presented in the same type of data visualization, but each graph tells a completely different story.

It only took one tiny change to completely flip the story. That should make you uncomfortable.
Create your own line graph for free with Venngage's Line Graph Maker.
Here's another example. This graph makes it seem like you have a terrible credit rating, causing you to freak out:

When it really isn't that bad of a rating, using 700 as the max:
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And even though this seems rather simple in the grand scheme of things, this approach can be used to mislead people about their situation. Or try to sell them something they don't need.

I wanted to include this example to show that not all misleading graphs are line and bar charts. Some can be innocent pie charts that just got caught up with some shady people!

As you can see, this misleading tactic is being used in politics, on social media, and in businesses to push an agenda or idea.

3. Cherry Picking Data

Another way to skew data is by only including certain parts of the data in your misleading charts or graphs. Usually, it is only the data that puts your viewpoint in a positive light or your opponent's in a negative light.

For example, only including a month where there was a sales spike and not the rest of the yearly data.

Or only sharing misleading poll results that make a certain person look favorable to everyone. But when you take a harder look at this graph, it only includes people from his own party:

And that party is shrinking by the day . This is definitely misleading, especially when the rest of the results looked like this:

If someone were to quickly look at the original bad graph, they would probably think everyone loves the president. When actually only about 35% of the country approves of his administration.

This is more common in graphs that have time as one of their axes. I mean it is pretty easy to start with a year that confirms what you are trying to say.

Both improper extraction and omitting data are things that you want to avoid!

Let's start with an example of improper extraction because I don't think I need to show you what omitting data looks like. In this case, I struggled to find some real-world examples, because who is really going to admit they left out data.

(Not many people!)

In the first graph below, a reader could obviously be mislead into thinking that the UK National debt has never been higher! This graph could be used to justify a politician voting on some piece of legislation that would lower the debt.

However, when you take a look at the full time series, you can see that national debt is actually pretty low in comparison.

This fictitious creator decided to also start the graph right at a low point and falsely illustrate that it could have been rising from zero to these rates. They also stuffed the graph with a bunch of random points to make it seem like the data set was much larger, when it only covered 10 years!

If you want another example of improper extraction, look no further than the stock market.

There are thousands of data points that stock analysts look at before they make trades or recommend people buy something. So there are a lot of things that they can omit to make certain company stock look better or worse overall.

But I think something that is very easy to mislead readers with is stock price. For example, take a look at the graph below, which shows that Twitter has been on an large upward swing.

As an outsider looking in, with just this graph at my disposal, I would think that they have been doing something right lately!

But I would be mistaken...

They have been on an unprecedented slide for the past year or so, and that increase is just a tiny blip on the long term graph.

Now, if I were a less than honest stock trader, I could try to unload a ton of Twitter stock just by using that graph.

And this type of misinformation could be used to manipulate about any piece of data you want to fit your goals.

Like this example, which tried to justify climate change not being real:

Mostly because people do not want to take a look at the raw data and they see graphs as a beacon of honesty.

I mean, why would someone lie on the internet, ?

4. Using The Wrong Graph

So far, I have talked about intentional misinformation tactics that writers use to push their agendas.

Now I think we should take a look at types of misinformation that can happen through sheer incompetence.

This usually involves picking a type of graph or chart that does not fit the data you are trying to present. And more often than not, the misunderstood pie chart is to blame for this.

It is not the pie chart's fault that it can be so misleading-it's the people who think it can be used for any and every type of data.

For example, take a look at this pie chart from the NFL Draft:

I am not sure what they were trying to do with this chart but as a multibillion-dollar company, they should have a competent graphics person.

First, in what world is 64 prospects half of 69 prospects? And second, why did they not use a bar graph for this data?

If you were scrolling through your Twitter feed and saw misleading graphs like this, it would make sense that you thought USC blew the others out of the water.

But if they wanted to share a more accurate graph, they should have created a column chart like this:
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It may not be as flashy as the first one but at least it is accurate.

Here is another questionable graph from the world of college football. This time, they attempted to graph projected win totals:

They ranked each team correctly from highest to lowest, but the inclusion of the bar graph made no sense to most people.
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If your school had a longer name it looked like they would win more in this graph. And if you were quickly scanning a social media feed, that would be a fair conclusion.

That's much better, but it still hurts seeing my school, Arkansas, at the bottom of both graphs.

Another way brands can be unwittingly deceptive is by trying to be too "innovative" with their graphs or charts.

In the example below, they used a pie chart and I really can't understand why!

When it really should have been a timeline or even a simple table:

I mean, what are they even trying to show with that terrible graph? It's not even labeled correctly to help the reader figure out what is going on!

It may look cool, but functionality should always come first!

In this example from Microsoft, by trying to be conceptual, they created a misleading data visualization:

Even if Microsoft Edge is faster than Chrome or Firefox, it is just by a slight margin. Not around 25% faster than Chrome or 50% faster than Firefox, as the visualization would lead you to believe.
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They should have used a column chart if they wanted to be accurate with their data:

Or if they still wanted to use something a little less boring, they could have gone with a bubble chart like this:

Since I gave the pie chart a bit of grief above, I'm going to show a place that it definitely should have been used! You can also use Venngage's free Pie Chart Maker to create pie charts that accurately depict your data.

In the example below, The Intercept was trying to show how the Russia issues have taken over the news lately:

It fell just a little bit short, mainly because the labels they chose are not very descriptive.
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And unless you calculated it yourself, you were left guessing what the actual split was between the two.

If I was creating this visualization, I would have gone straight to the pie chart:

Not only does it include the same information, it makes it easy for someone to quickly spot the difference.

Helping readers quickly understand the data should be the goal of any data visualization.

Not a designer? Create better charts for free Venngage's Chart Maker.

5. Going Against Conventions

To conclude our list of misleading data visualization tactics, I thought it would be a good idea to look at misleading graphs and charts that alter long-held conventions or associations.

If you are a little confused with what I am talking about, think about a graph where red represents Democrats and blue represents Republicans.

It would be pandemonium!

Or a simpler example is using green for losses, and red for profits.

That would make no sense to a competent graph maker but would be a great tool to manipulate an audience.

In this map about STI rates across the country, they choose to use a dark color to denote low levels and a light color for high levels:

This use of color goes against almost every map data visualization I have even seen. So I do think it was created to intentionally mislead the reader.

And to make things even more mind-boggling, the higher the number is, the lower the rates are, supposedly. So the map is confusing all around, which could make someone rightly think that Idaho is a hotbed for STIs, when it is really the southern states.

Here is an example of how a map like that should have looked (we didn't have access to the raw data to recreate the map, so we found something similar):

The dark colors are used to denote high values and the light ones are low ones.

Here is another example of a map using an insane color palette:

Could you even follow what they were trying to say? I couldn't!

Just like I pointed out above, these map makers should have used a single color palette with shades and tints. Almost everyone knows how to decipher those types of maps.

Next, we have a graph where the writer wanted to outright push a false idea to its audience. Honestly, it is one of the most egregious graph manipulations I have ever seen because of how blatant their intent was.

They actually flipped a graph upside down. This made it look like gun deaths were going down when in reality, gun deaths were spiking after the Stand Your Ground law was enacted:

See what I mean? This was done to push an agenda.

A simple rotation and mirroring of the graph will show you what it should have looked like:

There! Good as new, well...kinda.

And for our final example, we have one from our favorite data manipulators, Fox News.

Why don't you take a look at why this graph made this list?

Did you spot it? If not, I don't blame you because I am guessing many of their viewers missed it too. And they are counting on that.

If you take a look at the x-axis, you will see that they choose to include a bunch of random time values for their graph. It's not like they just left a few months or quarters out of the graph at random-they purposefully chose those quarters to fit their narrative. You could also consider this an example of omitting data.

And the worst part about this example is not that it is a bad chart, but that they thought they could dupe their loyal viewers.

If a brand thinks so little of your intelligence that they push bad graphs on you, I would recommend finding another source.

There's a ton of data out there about the COVID-19 pandemic right now. And as time goes on, there's going to be an overwhelming amount of data shared across the world.

That means that there are probably going to be some misleading graphs shared as well.

You will probably see a lot of these on social media, in news articles and on network TV.

Some of these are going to be intentionally misleading to shift the blame around, but others are misleading because the creator doesn't know how to visualize data correctly.

So let's take a look at some of the most recent examples of those bad graphs, and see if we can fix a few of them with Venngage.

Related: Bad Infographics: The Worst Infographics of 2020 (With Lessons for 2021)

1. Overloading readers with data

One of the most common mistakes I have seen people make when creating graphs and visualization is that they include way too much data.

Trying to include all the data on one visual usually confuses more people than helps them.

For example, this graph that was shared by al White House official a few weeks ago tells us almost nothing about the rise in cases.

After you look at the first two trend lines, the rest of the graph is basically worthless, unless you have a magnifying glass handy.

Not only is there way too many trend lines on that graph, the legend doesn't help you decipher anything.

According to this graph, the state with the most cases could be either New York, Rhode Island, New Hampshire, Oregon, or Minnesota.

A better way to present this data might have been an interactive graph that you can isolate your state from the others, or just create single graphs for each state.

Honestly, I don't think it was their intention to mislead people with this graph but it still could have hurt a lot of people who are looking for guidance in this troubling time.

That's truly why bad graphs are so harmful in this pandemic. If cold hard data isn't presented in a truthful and easy to decipher way, it may cause people to not take it seriously or panic.

2. Presenting the wrong data

Even when people don't have too much data on your visualization, sometimes they just present the wrong data.

The main thing that sticks out to me in this map visualization is that they used absolute numbers as their data source.

Instead, they should have compared the number of cases to the population of the country, city or state. This would have made comparing the different locations a lot easier. It also would have made the chart actually useful.

For example, if Kansas City and Los Angeles had the exact same number of cases, Kansas City would be in much worse shape than Los Angeles.

But according to this map, someone could conclude that they both are in the same predicament.

Like the last example this map literally tells us nothing useful, except where the biggest cities in the country are.

Additionally, stopping the range at 101+ is very misleading, there's a massive difference between 101 cases and 500 or 1000.

Another example of using the wrong data can be seen in this pie chart from the Georgia Dept. of Health.

In this pie chart, they put every adult under 60 years old in one group, and then anyone 60 years or older in another:

If you were to take a quick look at this, you would probably say that COVID-19 really isn't that dangerous to people over 60. The other group makes up more than half of the pie chart!

But according to the experts, that's the exact opposite or reality. Those over 60 are the most impacted by the pandemic.

3. Featuring the wrong graph

Now that we looked at a few pretty terrible data visualizations, let's see if we can improve some that just used the wrong graphs.

Like I said earlier, using the wrong graph isn't a great excuse but it's better than being intentionally misleading or lazy with your data.

As you can see in this pie chart, the percentages from each section add up to more than 100%. This is a common problem when people try to visualize survey data that has multiple answers.

The creator of the original pie chart should have presented the data kinda like this:

presents very similar data in an effective way in this simple bar chart:

4. Manipulating the Y & X-axis

Let's get back to some actually misleading graphs, starting with this example from Fox News:

As you can see, both the x-axis and y-axis have been manipulated, which is very common on graphs like this.

Instead of starting using a consistent scale on the y-axis of this graph, they jump from 0 to 5,000 and the to 20,000.

On the x-axis the number of days between each important date are inconsistent as well.

I'm not sure if their goal was to minimize how fast it was spreading, but I think they made the opposite case. In only a handful of days, the number of cases has basically tripled or more.

Here is a rough estimate of what that graph should have looked like:

By using accurate axes in this revised example shows truly how fast the virus was spreading in the later days of this chart.

It is a lot easier to see that the number of cases tripled in about 5 days between 3/15 and 3/20, and then tripled again in the next 4 days. The other graph is a lot less clear, which could lead some viewers to believe that we have this thing under control.

5. Going against normal conventions

For our final example, I found another visualization that attempted to map the number of cases throughout the world.

This one goes against normal color conventions that you would see on a similar map:

First, they really should have used a ratio of cases to the population of each country instead of just a number. That would have painted a more accurate picture.

The scale that they used on this map is also pretty terrible, and like I said with the other map there's a big difference between 10,001 and 100,000. Especially in smaller countries across the world.

But the worst part of this graph is the colors used to show that scale-I really have no idea what they were thinking when they picked that palette.
Why was a shade of purple included? Why are half of the colors muted, except for one bright color?
The red that denotes 100k+ or more cases is the darkest, but it's not the most eye catching of the colors. Your attention definitely isn't drawn to that color when you first look at the visual.

Then it just goes downhill from there with the color selection once you take a hard look at each.

Why does the scale go from a subdued color to a very vivid color and then back to a subdued color?

I really don't have any answer to those questions at this moment. But if you have any ideas, please let me know in the comments!

What blows my mind is that they could have just picked lighter shades and tones of a single that dark red to build an effective scale of colors!

Conclusion

Would you have guessed there were that many brands that play loose and fast with graphs?

As with any type of news story, I would recommend first checking where the graph is coming from and then taking a look at the data.

Like I said in the intro, most people sharing misleading graphs do not have your best interests in mind.

For example, if a graph that shows the benefits of coconut oil is being shared by a company that just happens to sell coconut oil, that graph may be skewed. Maybe look at some other sources before you order a case of it.
Do you want to learn more about picking the right charts for your data? Read this guide guide:
And if it is just one person or group sharing this particular graph, that is another red flag.

So be vigilant by always checking your sources, stay skeptical and if you feel like a writer is being misleading, call them out on it!

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5 Ways Writers Use Misleading Graphs To Manipulate You [INFOGRAPHIC] - Venngage

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Do some graphs look fishy to you? There's probably a good reason. Here are 5 ways writers use misleading graphs to manipulate you.

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As the New York literary journal The Knickerbocker extolled at the start of the American Civil War, "ARTISTS! ... remember that your elegant brushes are recording the history of a nation."

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Stalin's Manipulation of Visual Representations

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Stalin's Manipulation of Visual Representations

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https://www.history.com/news/josef-stalin-great-purge-photo-retouching

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Now you see him-now you don't. Compare a photo taken in the 1930s of five Communist Party officials in the USSR and you'll see Avel Enukidze, photographed next to Soviet premier Vyacheslav Molotov and others. But during Josef Stalin's Great Purge, the onetime member of the Communist party's highest governing body was deemed an enemy of the state and executed by firing squad.

Then, he disappeared from Soviet photographs, too, his existence blotted out by a retouched suit on another official from the original photo.

Enukidze's erasure was the product of a real conspiracy to change public perception in the USSR during Joseph Stalin's dictatorship. Stalin's commitment to censorship and photo doctoring was so strong that, at the height of the Soviet Union 's international power, he rewrote history using photo alteration. The stakes weren't just historical: Each erasure meant a swing of Stalin's loyalties, and most disappeared subjects also disappeared (or were killed) in real life, too.

After consolidating his power in 1929, Stalin declared war on Soviets he considered tainted by their connections to the political movements that had come before him. Beginning in 1934 he wiped out an ever-changing group of political "enemies." An estimated 750,000 people died during the Great Purge, as it is now known, and more than a million others were banished to remote areas to do hard labor in gulags.

During the purges, many of Stalin's enemies simply vanished from their homes. Others were executed in public after show trials. And since Stalin knew the value of photographs in both the historical record and his use of mass media to influence the Soviet Union, they often disappeared from photos, too.

Stalin used a large group of photo retouchers to cut his enemies out of supposedly documentary photographs. One such erasure was Nikola Yezhov, a secret police official who oversaw Stalin's purges. For a while Yezhov worked at Stalin's right hand, interrogating, falsely accusing and ordering the execution of thousands of Communist Party officials. But in 1938, Yezhov fell from Stalin's favor after being usurped by one of his own deputies. He was denounced, secretly arrested, tried in a secret court, and executed.

Stalin's censors then removed Yezhov from the photographic record, including cutting him from a photograph in which he smiled next to his former boss, Stalin, next to a waterway. The photo retouchers removed Yezhov from the photo and inserted new water to cover up the space where Yezhov would have been.

Stalin did the same with scores of party officials who had been photographed next to him at various events. Sometimes, official censors had to retouch photos over and over again as the list of political enemies grew longer. In one photograph, Stalin is shown with a group of three of his deputies. As each deputy fell out his favor, they were snipped out of the photo until only Stalin remained.

It's thought that Stalin's obsession with photo doctoring constituted a mini industry in the USSR. Publishers were contacted by Stalin's minions and told to eliminate the enemy du jour from upcoming photos-and they did. According to design historian David King, who uncovered thousands of doctored photos and their original versions, the work was not performed in one location or even through an official ministry.

Rather," King writes, "photographic manipulation worked very much on an ad hoc basis. Orders were followed, quietly. A word in an editor's ear or a discreet telephone conversation from a 'higher authority' was sufficient to eliminate all further reference-visual or literal-to a victim, no matter how famous she or he had been."

Sometimes, photo doctoring meant going back to the past to change the historical record, as when Stalin ordered Leon Trotsky, once a leading figure in the Communist Party, eliminated from all photos. After Trotsky was exiled by Stalin for mounting a failed opposition to his leadership, the revolutionary was snipped, airbrushed and covered up in countless photographs. Sometimes, Stalin inserted himself in photos at key moments in history, or had photo technicians make him look taller or more handsome.

Even citizens had to get in on the act. As Stalin's purges became more and more widespread, civilians who feared being branded as his political enemies began to realize that owning photos of Stalin's political enemies-even photos in books or magazines-was dangerous. They learned to deface their own materials with scissors or ink. "Such was the atmosphere of fear that families of those arrested and condemned were compelled to destroy even the image of their loved ones in their own personal records," writes biographer Helen Rappaport.

Stalin's obsession with image manipulation didn't stop with photos. As historian Jan Plamper notes, the omnipresent portraits of Stalin that were in every home and business were subject to maniacal oversight. The dictator commissioned an army of painters to create his official portraits, offering some artists massive amounts of money to paint him. Then, the official portrait was reproduced and retouched over and over until it met with Stalin's liking.

"The amount and detail of documentation on retouching (and the entire reproduction process) is astounding," writes Plamper. "This reflects a heightened concern to fix upon paper clear responsibilities-and tremendous anxiety, lest something go awry."

As photo doctoring became more and more common in the USSR's propaganda effort, it also became a way to evade Stalin's wrath. Take the famous photo of Soviet soldiers raising their flag over the bombed-out Reichstag during the Battle of Berlin at the end of World War II. This now iconic photo was staged (it was inspired by the flag-raising at Iwo Jima). It was also altered specifically to sidestep Stalin's anger: The photographer concealed the wrists of the soldiers, which were covered in stolen wristwatches they had looted from German citizens on their way to the Reichstag. Stalin had ordered his soldiers not to loot-so the watches would have caused the soldiers to be disciplined and, perhaps, killed.

Stalin wasn't the only dictator who loved to doctor photos. Adolf Hitler removed his propaganda minister, Joseph Goebbels, from a photo of him and director Leni Riefenstahl in 1937, though his motivations for doing so are uncertain. Benito Mussolini circulated a famous photograph of himself riding victorious atop a horse-after cropping out the handler holding the horse. And Kim Jong-Un apparently uses Photoshop to make his ears look smaller.

But as Stalin shows, manipulating photos isn't always about the size of one's ears. It can be a way of literally erasing today's political enemies from tomorrow's picture of history-and making the future as unreliable as a present filled with propaganda and lies.

Fact Check

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How Photos Became a Weapon in Stalin's Great Purge | HISTORY

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Stalin didn't have Photoshop-but that didn't keep him from wiping the traces of his enemies from the history books. Even the famous photo of Soviet soldiers raising their flag after the Battle of Berlin was altered.

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https://www.britannica.com/science/mathematics/Mathematics-in-ancient-Egypt

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The introduction of writing in Egypt in the predynastic period ( c. 3000 bce) brought with it the formation of a special class of literate professionals, the scribes. By virtue of their writing skills, the scribes took on all the duties of a civil service: record keeping, tax accounting, the management of public works (building projects and the like), even the prosecution of war through overseeing military supplies and payrolls. Young men enrolled in scribal schools to learn the essentials of the trade, which included not only reading and writing but also the basics of mathematics.

One of the texts popular as a copy exercise in the schools of the New Kingdom (13th century bce) was a satiric letter in which one scribe, Hori, taunts his rival, Amen-em-opet, for his incompetence as an adviser and manager. "You are the clever scribe at the head of the troops," Hori chides at one point,

a ramp is to be built, 730 cubits long, 55 cubits wide, with 120 compartments-it is 60 cubits high, 30 cubits in the middle...and the generals and the scribes turn to you and say, "You are a clever scribe, your name is famous. Is there anything you don't know? Answer us, how many bricks are needed?" Let each compartment be 30 cubits by 7 cubits.

This problem, and three others like it in the same letter, cannot be solved without further data. But the point of the humour is clear, as Hori challenges his rival with these hard, but typical, tasks.

What is known of Egyptian mathematics tallies well with the tests posed by the scribe Hori. The information comes primarily from two long papyrus documents that once served as textbooks within scribal schools. The Rhind papyrus (in the British Museum) is a copy made in the 17th century bce of a text two centuries older still. In it is found a long table of fractional parts to help with division, followed by the solutions of 84 specific problems in arithmetic and geometry. The Golenishchev papyrus (in the Moscow Museum of Fine Arts), dating from the 19th century bce, presents 25 problems of a similar type. These problems reflect well the functions the scribes would perform, for they deal with how to distribute beer and bread as wages, for example, and how to measure the areas of fields as well as the volumes of pyramids and other solids.

The Egyptians, like the Romans after them, expressed numbers according to a decimal scheme, using separate symbols for 1, 10, 100, 1,000, and so on; each symbol appeared in the expression for a number as many times as the value it represented occurred in the number itself. For example, stood for 24. This rather cumbersome notation was used within the hieroglyphic writing found in stone inscriptions and other formal texts, but in the papyrus documents the scribes employed a more convenient abbreviated script, called hieratic writing, where, for example, 24 was written .

In such a system, addition and subtraction amount to counting how many symbols of each kind there are in the numerical expressions and then rewriting with the resulting number of symbols. The texts that survive do not reveal what, if any, special procedures the scribes used to assist in this. But for multiplication they introduced a method of successive doubling. For example, to multiply 28 by 11, one constructs a table of multiples of 28 like the following:

The several entries in the first column that together sum to 11 (i.e., 8, 2, and 1) are checked off. The product is then found by adding up the multiples corresponding to these entries; thus, 224 + 56 + 28 = 308, the desired product.

To divide 308 by 28, the Egyptians applied the same procedure in reverse. Using the same table as in the multiplication problem, one can see that 8 produces the largest multiple of 28 that is less then 308 (for the entry at 16 is already 448), and 8 is checked off. The process is then repeated, this time for the remainder (84) obtained by subtracting the entry at 8 (224) from the original number (308). This, however, is already smaller than the entry at 4, which consequently is ignored, but it is greater than the entry at 2 (56), which is then checked off. The process is repeated again for the remainder obtained by subtracting 56 from the previous remainder of 84, or 28, which also happens to exactly equal the entry at 1 and which is then checked off. The entries that have been checked off are added up, yielding the quotient: 8 + 2 + 1 = 11. (In most cases, of course, there is a remainder that is less than the divisor.)

For larger numbers this procedure can be improved by considering multiples of one of the factors by 10, 20,...or even by higher orders of magnitude (100, 1,000,...), as necessary (in the Egyptian decimal notation, these multiples are easy to work out). Thus, one can find the product of 28 by 27 by setting out the multiples of 28 by 1, 2, 4, 8, 10, and 20. Since the entries 1, 2, 4, and 20 add up to 27, one has only to add up the corresponding multiples to find the answer.

Computations involving fractions are carried out under the restriction to unit parts (that is, fractions that in modern notation are written with 1 as the numerator). To express the result of dividing 4 by 7, for instance, which in modern notation is simply 4/7, the scribe wrote 1/2 + 1/14. The procedure for finding quotients in this form merely extends the usual method for the division of integers, where one now inspects the entries for 2/3, 1/3, 1/6, etc., and 1/2, 1/4, 1/8, etc., until the corresponding multiples of the divisor sum to the dividend. (The scribes included 2/3, one may observe, even though it is not a unit fraction.) In practice the procedure can sometimes become quite complicated (for example, the value for 2/29 is given in the Rhind papyrus as 1/24 + 1/58 + 1/174 + 1/232) and can be worked out in different ways (for example, the same 2/29 might be found as 1/15 + 1/435 or as 1/16 + 1/232 + 1/464, etc.). A considerable portion of the papyrus texts is devoted to tables to facilitate the finding of such unit-fraction values.

These elementary operations are all that one needs for solving the arithmetic problems in the papyri. For example, "to divide 6 loaves among 10 men" (Rhind papyrus, problem 3), one merely divides to get the answer 1/2 + 1/10. In one group of problems an interesting trick is used: "A quantity ( a h a) and its 7th together make 19-what is it?" (Rhind papyrus, problem 24). Here one first supposes the quantity to be 7: since 1 ¹/ ₇ of it becomes 8, not 19, one takes 19/8 (that is, 2 + 1/4 + 1/8), and its multiple by 7 (16 + 1/2 + 1/8) becomes the required answer. This type of procedure (sometimes called the method of "false position" or "false assumption") is familiar in many other arithmetic traditions (e.g., the Chinese, Hindu, Muslim, and Renaissance European), although they appear to have no direct link to the Egyptian.

The geometric problems in the papyri seek measurements of figures, like rectangles and triangles of given base and height, by means of suitable arithmetic operations. In a more complicated problem, a rectangle is sought whose area is 12 and whose height is 1/2 + 1/4 times its base (Golenishchev papyrus, problem 6). To solve the problem, the ratio is inverted and multiplied by the area, yielding 16; the square root of the result (4) is the base of the rectangle, and 1/2 + 1/4 times 4, or 3, is the height. The entire process is analogous to the process of solving the algebraic equation for the problem ( x × ³/ ₄ x = 12), though without the use of a letter for the unknown. An interesting procedure is used to find the area of the circle (Rhind papyrus, problem 50): 1/9 of the diameter is discarded, and the result is squared. For example, if the diameter is 9, the area is set equal to 64. The scribe recognized that the area of a circle is proportional to the square of the diameter and assumed for the constant of ¹/ ₇ , first proposed by proportionality (that is, π/4) the value 64/81. This is a rather good estimate, being about 0.6 percent too large. (It is not as close, however, as the now common estimate of 3 Archimedes, which is only about 0.04 percent too large.) But there is nothing in the papyri indicating that the scribes were aware that this rule was only approximate rather than exact.

A remarkable result is the rule for the volume of the truncated pyramid (Golenishchev papyrus, problem 14). The scribe assumes the height to be 6, the base to be a square of side 4, and the top a square of side 2. He multiplies one-third the height times 28, finding the volume to be 56; here 28 is computed from 2 × 2 + 2 × 4 + 4 × 4. Since this is correct, it can be assumed that the scribe also knew the general rule: A = ( h/3)( a ² + a b + b ²). How the scribes actually derived the rule is a matter for debate, but it is reasonable to suppose that they were aware of related rules, such as that for the volume of a pyramid: one-third the height times the area of the base.

The Egyptians employed the equivalent of similar triangles to measure distances. For instance, the seked of a pyramid is stated as the number of palms in the horizontal corresponding to a rise of one cubit (seven palms). Thus, if the seked is 5 ¹/ ¹/ ₃ cubits (Rhind papyrus, problem 57). The Greek sage ₄ and the base is 140 cubits, the height becomes 93 Thales of Miletus (6th century bce) is said to have measured the height of pyramids by means of their shadows (the report derives from Hieronymus, a disciple of Aristotle in the 4th century bce). In light of the seked computations, however, this report must indicate an aspect of Egyptian surveying that extended back at least 1,000 years before the time of Thales.

Egyptian seked Encyclopædia Britannica, Inc.

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Mathematics - Mathematics in ancient Egypt

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Mathematics - Mathematics - Mathematics in ancient Egypt: The introduction of writing in Egypt in the predynastic period (c. 3000 bce) brought with it the formation of a special class of literate professionals, the scribes.