# Eli5 Why is a negative times a negative a positive?

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I’m thinking of it like money. If I owe 5.00 dollars and then a multiply my debt by 5 dollars, I should be in debt by 25 dollars right? But -5 time -5 is 25. Please help.

Edit: my example was wrong sorry.

Edit: I’m still so confused but I think I got it. If I owe someone a dollar (-1). But that person doesn’t exist (-1). Id have my dollar still. Is that right?

In: Mathematics

Think of it on a gragh. 0 in the middle and 5 and -5 on the sides. The negative makes the number switch which side of the graph the result will be on. If you switch it twice it goes back to the positive side.

Multiplying a debt of 5 dollars (-5) by a factor of 5 (+5) is a negative times a positive, yielding a negative result. Essentially what you’re doing when you multiply two negatives is negating a negative. The opposite of a negative is a positive. Therefore a negative times a negative is a positive.

Thats negative 5 times positive 5 payments in your debt example. -5 dollars paid -5 times would mean someone paid you \$25. It’s easier to think of with ones. -1 × -1 = +1. Just think: what’s the opposite of an opposite? The thing you started with. I can’t really explain well why because it’s self-evident.

You multiplied your debt 5 times. Not negative 5 times.

That said, I don’t have a solution. I’m thinking something about loaning 5 friends each 5 dollars. But I don’t think that works either.

So if you owe 5 people 5 dollars you would be at -25 dollars. However the opposite of owing 5 people 5 dollars is 5 people owing you 5 dollars and therefore you are owed 25 dollars as in you will have positive 25 dollars (once you are paid)

If you multiply your debt by 5, you’re getting a debt of 5 * -5, which is a positive times a negative. Multiplying a debt by *negative* would be starting from 0 and “removing” your debt five times over, so your balance would be -(-5) + -(-5) + -(-5) + -(-5) + -(-5) = 25.

http://mathisvisual.com/more-integer-multiplication/

This guys explains it very well. It’s like you take away 5 groups of negative 5s so with zero pairs you will be left with only the positive 5s

Remember that multiplication is just a bunch of addition.

There’s a couple of things wrong with your discription. First of. If you multiple a dollar by a dollar. You get dollars squares. When you do math you have to do it to the units too. And dollars squares makes no sense.

Anywho, on to the main question. What you example would actually be is “removing 5 dollars of debt five times” in this case *removing* and *debt* are both negative so they cancel out. And in the end you actually gain 25 dollars.

Let’s say you’re going backwards. Then you go BACKWARDS backwards.

See? Now you’re going frontwards!

This is the top answer on basically the same question in this thread:

I give you three \$20 notes: +3 × +20 = +60 for you

I give you three \$20 debts: +3 × -20 = -60 for you

I take three \$20 notes from you: -3 × +20 = -60 for you

I take three \$20 debts from you: -3 × -20 = +60 for you

Credit to u/zerotan

Think of numbers as a line. Positive numbers go off to the right, negative numbers go off to the left. Multiplying a positive number by -1 (for example) means that you turn around and go to the left.

Multiplying a negative number by another negative is like taking someone going left and telling them to turn around and go in the other direction. So they go right. Back in the positive direction.

Counting only works with natural numbers. Negative numbers are different in this respect, you cannot count with those. Your idea with “debt” might only give you a short glimpse, but that isn’t a very accurate one.

Part of the reason why it to so long to discover them.