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  • Suppose you want to know how long it was between the Norman invasion of England and Columbus's

  • arrival in the Caribbean; that is, what’s 1492 minus 1066 – but you hate long subtraction

  • with its borrowing and such.

  • Well, we can subtract by adding: first we just need to replace each digit of the smaller

  • number with 9 minus that digit, except the final digit gets replaced with 10 minus that

  • digit, so 1066 becomes nine minus 1, nine minus 0, nine minus six, ten minus 6.

  • Adding that to 1492 gives 10426, and if we ignore the first digit, we get the answer:

  • 1492 minus 1066 is 426.

  • You can check just to be sure.

  • Coincidentally, it’s roughly 426 times farther from Portugal to the West Indies and back

  • than from France to England.

  • But anyway, the subtraction-by-adding trick works for any positive numbers!

  • 8 minus 6, or 2, is the same as 8 plus 4, ignoring the first digit.

  • 100 minus 1? is 100 plus 999, which is 99, ignoring the first digit.

  • And 424,242 minus 333,333 is 424,242 plus 666,667, or, 90,909 ignoring the first digit.

  • I’ll let you check that one.

  • This trick might seem useless, but suppose you built a machine that can add numbers together,

  • and you wanted to make it subtract?

  • Well, in that case it might be easier to make it subtract by adding: and matter of fact,

  • my friend Hank Green took apart an adding machine and that’s exactly how it subtracts

  • - by adding!

  • Basically, the machine adds numbers by rotating numbered wheels, but there aren’t infinitely

  • many wheels so if you add up past the maximum possible number, you get back to zerothis

  • is calledoverflowin computing andmodular arithmeticin mathematics.

  • But most importantly, getting back to zero by adding means it’s possible to have positive

  • negative numbers, since, for example, negative three is just what you add to three to get

  • zero, and on Hank’s adding machine if you add 3 plus 9,997.00, you get zero, so 9,997

  • is literally negative three!

  • Unfortunately, there are an infinite number of numbers when youre doing regular arithmetic,

  • so you might think that negative three is just negative three.

  • But if youre willing to fudge a little, you can just take the adding machine’s version

  • of negative three, add a bunch of 9s out in front, and when you add that to stuff it’s

  • BASICALLY the same as subtracting three, as long as you don’t look too far.

  • Because, you know, nine gadgillion nine hundred and ninety bajillion nine hundred and ninety

  • seven plus three, is zero.

  • Almost.

  • By the way, this method of subtracting by adding is how computers subtract as well,

  • they just do it in binary which makes it a lot easier.

  • It's called "subtracting using the two's complement," if

  • you want to

  • look it up.

Suppose you want to know how long it was between the Norman invasion of England and Columbus's

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