From 1 + 1 = 2 to Amazon.com

On Quora, someone asked how you could explain to a 5-year-old why 1 + 1 = 2 and not 3, 4, or some other number.  It’s an interesting question, and I’ll repeat my answer, plus a little more insight into why the question is important.

Roughly speaking, this is actually a proof question:  Prove 1 + 1 = 2.  I’ve talked a little bit elsewhere about why mathematicians do proofs, and why proof is important, but a quick justification for why you’d want to prove 1 + 1 = 2 follows.

The reason 1 + 1 = 2 is that mathematically, “+1” means “the thing after.”  So 1 + 1 is “the thing after 1,” which is to say 2.  Likewise, 4 + 1 is “the thing after 4,” namely 5.  The reason this is useful is that once you’ve extended the idea of arithmetic this way, you can develop an entirely new type of arithmetic and ultimately arrive at the basis for internet security.

It’s a long path, but here are the highlights…

Once we recognize “+1” as “the thing after,” we can do arithmetic with anything that has a definite ordering.  Thus Sunday + 1 is “the thing after Sunday,” which we might say is Monday, and write Sunday + 1 = Monday.

We can extend our ideas to +2:  Sunday + 2 = Tuesday, Sunday + 3 = Wednesday, and so on.  However, an interesting thing happens when you go too high:  What’s Sunday + 7? It should be the seventh day after Sunday…which is Sunday again.  So we have Sunday + 7 = Sunday.

Now here’s a peculiar thing:  Sunday + 7 = Sunday.  It seems that “+7” has changed nothing.  In particular, it’s just like adding 0.  In higher math, we have a name for this:  7 is the modulus of the arithmetic system, and the type of arithmetic we produce is called modulo arithmetic.

We can work with other moduli.  The only thing we have to remember is that the modulus is just like 0, so if we add the modulus to a number, we get the same number back.  For example, consider a wacky number system with modulus 12, so we’d write things like 3 + 12 = 3.  Or (since 24 = 12 + 12), we could even write 7 + 24 = 7 and so on.

Now who on Earth would use such a bizarre number system?  The answer is:  you do, every time you check your schedule.  Thus 9 + 4 is the fourth hour after 9:  it’s 1, so we’d add 9 + 4 = 1.  (There are about a million reasons why everyone should learn to tell time on an analog clock…this isn’t one of them, though being able to tell time on an analog clock will make this sort of arithmetic a lot easier)

Let’s change gears a bit, and consider modern internet security.  When you send your credit card number to Amazon.com, you’d really prefer it if no one else could read it.  Thus you encrypt it.  Modern encryption techniques are based around a central idea:  The number you want to keep secret is the solution to a mathematical equation that is very hard to solve.

Unfortunately, in ordinary arithmetic, most equations can be solved easily through the use of what mathematicians call the intermediate value theorem, though the rest of us might call it the Goldilocks Principle.  Consider this:  My secret number solves the equation x^{3} = 729.

Now even if you don’t know how to solve this equation, you can still use the Goldilocks Principle as follows.  Guess any number:  for example, 3.  We observe that 3^{3} = 27, which is too small.  Now guess another number:  for example, 10.  We observe 10^{3} = 1000, which is too big.  Since 3 is too small and 10 is too big, some number between is just right.

So we guess again:  5.  We have 5^{3} = 125, too small.  So now we know 5 is too small and 10 is too big, and again something in between is just right.  If we keep doing this, we’ll be able to find the number that’s just right, and determine the secret number.  Not what you want if the secret number is your credit card number!

So how does modulo arithmetic help?  Modulo arithmetic helps because the intermediate value theorem fails.  Suppose we’re working with a modulus of 12.  Again, suppose my secret number satisfies x^3 = 5.  We try again:  we guess 1, and find 1^{3} = 1, so 1 is too small.  Our next guess is 2, and find 2^{3} = 8, so 2 is too big.

But there’s no number between 1 and 2 that will be just right!

Yet there is a solution:  5^{3} = 5 when we’re working with a modulus of 12.  (That’s because 5^{3} = 125 = 12 \times 10 + 5, and if 12 is like 0, this means 125 = 0 \times 10 + 5 = 5)

The moral of this story is this:  Don’t break into other people’s houses and eat their porridge.  Whups, wrong story.

The moral of this story is this:  Modulo arithmetic allows us to set up equations that are very hard to solve.

Of course, if you’re thinking ahead, you’ll realize that Amazon has to be able to solve the equation.  But that’s a post for another day.

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