Ancient Egyptians and Goofy Numbers

My friend Star asked me to explain an outmoded numerological interpretation of the Eye of Horus. This led into a discussion of some elementary number theory. Enjoy.

I’m not an Egyptologist, but I do know some aspects of the history of mathematics (at least as they were communicated to me throughout my undergraduate degree in physics). Let’s see what we can do. We can call this “Star Learns Elementary Number Theory”.
 
I know that a lot of ancient peoples thought there were only whole, positive numbers (1, 2, 3, 4, …); these are what 21st century mathematicians called ‘positive integers’. Instead of thinking of fractions as numbers between the integers (so that 0.5 is between 0 and 1), they thought of them as ratios of two integers (1:2). This avoids ever talking about a number existing between two numbers. At least this is how the Greeks thought of things.
 
It looks like the Egyptians thought something similar, but took this idea a step further. You can imagine writing all of the positive integers as the sum of two other numbers. For example, we can represent 2 as 1+1 and we can represent 3 as 1+2. You can do the same with fractions; 3/4 can be represented as 1/2+1/4. This is useful if your written language does not allow you to directly represent 3/4; apparently, ancient Egyptian was limited in that way.
 
The Eye of Horus stuff concerns an outmoded theory about how fractions were represented by the Egyptians. Apparently, Egyptologists used to think that each part of the eye represented a different base fraction and by adding together different parts of the eye you could get different numbers. It looks like Egyptologists have since abandoned that theory.
 
Unfortunately for the Egyptians, no matter how many base fractions one has, one can never represent all fractions. For one thing, there will always be fractions smaller than the base. The other problem is that there will be numbers one cannot represent using any fraction at all, let alone using the sum of two fractions. Let’s see why; this will take some algebra to work out, but bare with me. This is one of the most important discoveries of the ancient world.
 
Suppose that Harry, the Egyptian, is building a pyramid. He knows that the pyramid is going to be 1 foot tall and 2 feet wide. He wants to know the distance from the base of the pyramid to the top.
 
Imagine chopping the pyramid in half, so that we get a triangle 1 foot wide and 1 foot tall that looks like this.
 
So the question Harry wants to answer is what distance c is in the diagram. Luckily, Harry knows a Greek named Pythagoras who gives him a formula: multiply the width by itself, the height by itself, and then add the two numbers together; the result will equal the distance c multiplied by itself (c is the distance Harry wants).
 
Okay, so:
 
1*1=1
1*1=1
Add them together, we get 2.
 
What two numbers, when multiplied together, will equal 2?
 
Apparently, not 1, since 1 multiplied by itself is equal to 1. 2 won’t work either, since 2 multiplied by itself is equal to 4. Blowing Harry’s mind, you suggest that the number which, when multiplied by itself, gives 2 is somehow between 1 and 2. But which number between 1 and 2?
 
You decide to be clever (good things happen when you are clever) and you decide to represent the problem you are facing as an equation. You know that you want a fraction which, when multiplied by itself, produces 2. A fraction is one number divided by another; so let’s call those two numbers a and b. Furthermore, you know that this fraction, when multiplied by itself, produces 2:
 
2 = (a/b)*(a/b)
 
Lets stipulate that a/b has already been fully reduced. In other words, a and b are already as small as possible to represent that fraction.
 
You realize that you can re-write this as:
 
2 = (a*a)/(b*b)
 
Or, in other words:
 
2 = a^2 / b^2
 
But you want to know what a and b are. So let’s move the b^2 over to the left hand side:
 
2 * b^2 = a^2
 
This tells us that a^2 is divisible by 2 (do you see why?). But if a^2 is divisible by 2, then a^2 is even. But the only way for a^2 to be even is for a to be even; this is because your friend Gus has already proven that all integers are either even or odd and you know that the only way that a number, when multiplied by itself, can produce an even is if the original number was even. So a is even.
 
But if a is even, we can write a as 2k, where k is some new mystery number. So let’s do that:
 
2 * b^2 = (2k)^2
2 * b^2 = 4 * k^2
 
We can cancel out 2 on either side:
 
b^2 = 2 * k^2
 
But now we know that b^2 is divisible by 2. This implies, as before, that b is even.
 
So both a and b are even. Well, that can’t be right — we started off with a fraction consisting of two numbers that had no denominators in common. Yet we ended up with a fraction consisting of two numbers which were both multiples of 2, since they were both even. Did we mess up somewhere?
 
“WHAT THE HELL, STAR??” Both Harry and Pythagoras — our two new friends — are getting quite peeved with us. For one thing, we just threw doubt on the new religion Pythagoras has been developing, which declared as holy doctrine that all numbers were integers or ratios of two integers.
 
The problem is with the assumption we started with. There is no fraction consisting of two integers that will be equal to the square root of 2. Harry’s language is incapable of ever expressing the length he wants to calculate because it can only express fractions; but there is no fraction for that length. No matter how well he approximates it, using all sorts of tinier and tinier fractions, there will always be a little bit left over (or he will always go a little bit over). There is something his language leaves out, something which needs to be added in.
 
What’s missing is the set of irrational numbers, so-called because, when Pythagoras’s groupies discovered them, they were horrified. According to legend, Pythagoras drowned the individual (a fellow named Hippasus) who originally worked this out (though this story is likely apocryphal).

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