Example: 2 – (-3) = 5
I learned HOW to solve this type of equation in school…but not the WHY.
In: Mathematics
Your teacher will hate me, but here’s the truth: subtraction does not exist.
What you think of as arithmetic is what mathematicians call a ‘field’. A field obeys six rules:
– Associativity: a + (b + c) = (a + b) + c and a * (b * c) = (a * b) * c
– Commutativity: a + b = b + a and a * b = b * a
– Identities: a + 0 = a and a * 1 = 1, where 0 and 1 are different and unique.
– Additive Inverse: there exists some (-a) for every a where a + (-a) = 0
– Multiplicative Inverse: there exists some (a^-1) for every a except 0 where a * (a^-1) = 1
– Distributivity: a * (b + c) = a * b + a * c
Each of those statements is generalized, where a, b and c can all be any element in your ‘field’ (except where noted).
That’s it. That’s the entirety of the arithmetic you know.
Now, think about your question. There is no subtraction. There is only addition and the additive inverses. So how do you write the equation that’s puzzling you?
Well, first you need to recognize that a, b and c above don’t need to be numbers. They can be vectors – multiple values packed together to represent a single quantity. Indeed, they *have* to be vectors because otherwise you can’t express additive inverses. ‘2’ can be a simple magnitude. But ‘-3’ is actually a magnitude *and* a direction.
So let’s rewrite your equation:
{2,positive} + {1,negative} * {3,negative} = {5, positive}
Lastly, let’s go a bit deeper into what multiplication is. You probably think of it as ‘scaling’ in some sense. If you multiply 5 * 5, you’re scaling up that 5 by a factor of 5. But multiplication is really scaling and rotation. You just don’t see the rotation with counting numbers because they all face the same direction.
So when we multiply {1,negative} * {3,negative}, what we’re really saying is ‘scale up and then turn around the other way’. Which means {1,negative} * {3,negative} = {3,positive}.
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