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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 zero – this
is called “overflow” in computing and “modular arithmetic” in 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 you’re doing regular arithmetic,
so you might think that negative three is just negative three.
But if you’re 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.
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.