Why does multiplying two negative numbers equal a positive number?
In: 1210
My favourite explanation:
If you love to love = you love.
If you love to hate = you hate.
If you hate to love = you hate.
If you hate to hate = you love.
Hate (-) means reversing the direction, love (+) is reinforcing it. That’s it.
If you remove something negative, that’s positive. If you remove (minus) debt (negative money) you end up with actually gaining wealth. Or put like this: if you remove something weighing you down it no longer weighs you down.
If you visually think of the numbers as in a straight line, ordered from smaller to higher with the negatives to the infinite left and the positives to the infinite right.
Multiplying by a positive numbers is like stretching that number line. So, for example 2×3, is like stretching the number line to triple it’s length. The position of 2 is now in the position of 6, so the result is 6.
Multiplying by negative one (-1) is like mirroring that line with respect to zero. So 3x(-1) is a flip around zero. The position of number 3 is where number -3 used to be, so the result is -3. A negative number is then the flipped version of a number.
So, if you take any number. And you do the flip operation twice you are back where you started.
So that two negative numbers, when multiplied wouldn’t return the same as a positive and negative multiplied.
How else should it be?
If you don’t not have something it means you do have something
Word words words words to meet the minimum word length words words words (why does ELI5 have a minimum world length? Short explanations are better) words words words
Multiplication is repeated addition, so negative multiplication is repeated subtraction. If you repeatedly remove $5 in debt, 6 times you just gained $30 in value.
Negative number are just regular numbers with the minus sign in front. Lets try using a different sign to show a negative number. Lets use @.
1 is positive
@1 is negative
@1+@1=@2
@1-@1=0
@1-@2=1
Multiplication is just addition
3*4= 3+3+3+3 = 12
@3*4= (@3)+(@3)+(@3)+(@3) = @12
@3*-4= (@3)-(@3)-(@3)-(@3) = 12
A trick I was taught to remember what the result should be: Positive numbers have an implied + in front of them. If the symbols ( – or +) in front of the numbers you are multiplying match, the result will be positive. If they don’t, the result will be negative.
The identity property of multiplication: Any number multiplied by 1 is equal to itself. If you have one $20 bill, you have $20. 20 x 1 = 20.
The identity property also applies to negative numbers: -5 x 1 = -5. If you have one negative five, you have one negative five.
Nonzero real numbers are either negative or positive; there’s no other directions you can go on the number line. If a nonzero real number is not positive, it must be negative. Therefore, if a negative times a negative does not equal a positive, it must equal a negative.
But if you let a negative times a negative equal a negative, that would mean -5 x -1 would equal -5 again. You could then substitute that into the original equation we had in the second step and that would give you -5 x 1 = -5 x -1. Divide both sides by -5 and now you have -1 = 1, which cannot be true.
Therefore, by contradiction, a negative times a negative cannot be a negative, and must be a positive.
Math is a tool we use to measure the universe. Negatives are a concept, not a universal truth. If we based our numbers with zero being absolute nothingness, negative would be an impossible number, so we would always have to be positive, so we’d have to define our initial reference point if we ever had to deduct from our initial reference. This becomes a problem if you have no way to define the initial reference point, so we make the reference point 0 and allow anything under it to be a negative. So we have to make a “rule” on how to measure from this initial reference point. That role is that multiplying two negatives is positive, and we formulate equations around this rule.
We also could redefine our order of operations (and we do in many fields), but then we need to formulate our equations to match that order of operations.
So the reason is because it’s a rule we’ve all agreed on to make our measurements of the universe able to be communicated in a standardized way.
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