Example: 2 – (-3) = 5

I learned HOW to solve this type of equation in school…but not the WHY.

In: Mathematics

If you consider a number line painted on the ground under you, with positive numbers ahead of you and negative numbers behind you, you’ve got the framework necessary to understand.

Subtracting a number can be thought of as taking a step backwards.

Subtracting a *positive* number can be thought of taking a step backwards *while facing the positive direction.*

So, if you turn around and face the negative direction, and then start walking backwards, you end up moving in the positive direction.

Think of it with money. You have 2 dollars and you get rid of 3 dollars you don’t have. Now you have 5 dollars.

Its just a different perspective of adding 3 dollars. By working and getting a paycheck you are taking away the absence of money.

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}.

Think of subtracting like turning around (to face the opposite direction) and a negative number as walking backwards. If I tell you to walk forward 2 steps, turn around (subtract) then walk *backwards* 3 steps (negative 3), you are now 5 steps *ahead* of where you started.

If I told you to take 2 steps, turn around, then walk forwards 3 steps ,you end up 1 step *behind* where you started (2 – 3 = -1).

If, instead, I told you to take 2 steps, then walk forward 3 steps, you end up 5 steps *ahead* of where you started.

Most simply, when you “subtract a negative”, the end result is the exact same as if you added a positive.

The same reason why adding a positive makes a positive. You are SUBTRACTING the NEGATIVE. Which means you are ADDING the POSITIVE. Hope that helps. It really helped me when I learned this stuff.