Why does multiplying two negative numbers equal a positive number?
In: 1210
So -5 x -6 = 30
If we talk about money that could be described as: I remove $5 dollars of debt 6 times. That means I have $30 less debt which is also known as “having $30 more dollars.”
Removing it six times is a -6 and five dollars in debt is a -5
That’s how I’ve always thought of it anyway, “removing” negatives a given number of times.
There’s two ways to show that, a logical way and a graphical way.
The logical one is basically that “negative” means “loss”. If you’re adding a negative number you’re actually subtracting, and multiplying a positive by a negative number turns a “gain” into “loss”.
Well, if you multiply negative twice, you’re “losing loss”, so you’re gaining.
For the graphical way, imagine a number line with zero on it. Left you have negatives, right you have positives. What you’re doing when you multiply by a negative is a very specific operation: you’re reflecting the number from right to left. What was positive is now negative.
Well, if you multiply by negative twice, you’re reflecting around zero twice. What was on the right is in the left, then it’s back on the right. Back to positive.
The way that math is taught, being taught memorization or formula means you think of multiplication as it’s own operation that does a specific thing. But it isn’t.
Multiplication is just super addition. Like 3 x 4 is 3+3+3+3. But if it’s -3 x 4, that’s -3 + -3 + -3 + -3.
This actually happens on the next level up too. 3^4. Exponentiation is just super multiplication. Or super super addition.
3 x 3 x 3 x 3 which is really 3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3
*turn around* (negative number)
*turn around again* (another negative)
You’re back to facing forward
Turn around 180 degrees. Then do it again. Now you’re facing the same way you started. Also, you can almost think of it like this:
Imagine 3×4 as saying “take 3 steps forward. Do this 4 times.”
Then -3×4 could be thought of as saying “turn around. Then take 3 steps forward. Repeat the 3 steps 4 times.” In this case, you’ll walk the same amount, just in the opposite direction.
So now, think of -3 × -4 as “turn around. Take 3 steps backwards. Repeat the 3 steps 4 times.” Since you turned around but then walked backwards, you’ll end up in the same place as the 1st scenario. Thus, -3 x -4 must be the same as 3 x 4.
**Edited to fix some typos
If you turn yourself around so you face backwards and then take five steps backwards, you end up five steps forwards of where you started because the two negatives become a positive.
If you forgive (-) your buddy of four five dollar debts (-), his net worth goes up by $20. Negative number times a negative number results in a positive.
If you found $10 dollars thats +$10.
If you lost $10 dollars thats -$10.
If someone is ordered to pay you a credit of $10, thats +$10.
If you owe someone a debt of $10, thats -$10.
If someone lost that credit to you, that would be -1*$10 = -$10.
If you lost that $10 debt you owe someone, thats -1*-$10 = $10.
You gained a net of $10.
If you have five boxes with $5 cash in each. You are $25 dollars richer. 5 X 5 = 25
If you have five boxes with $5 IOU’s in each You are $25 in debt 5 X -5 =-25
If you burn the five boxes with the IOU’s you are $25 better off -5 X -5 = 25
If you burn the box with cash you are $25 worse off -5 X 5 = -25
Maths has to be consistent and beautiful. Things you don’t know must follow from things you already know to be true.
We know that positive times negative is negative:
2 * -3 = -6
Similarly negative over positive is negative
-6 / 2 = -3
And so is positive over negative
6 / -2 = -3
But for the latter to be true, it’s necessary that:
-2 * -3 = 6
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