Why does multiplying two negative numbers equal a positive number?
In: 1210
Basically because a negative number is supposed to behave in the opposite way to a positive number when it comes to addition.
So we know that for any number, say -1, we have
-1 × (-1 + 1) = -1 × 0 = 0
Now, we know that for multiplication and addition we can say
0 = -1 × (-1 + 1) = (-1 × -1) + (-1 × 1)
We know that -1 × 1 = -1 therefore the equation is only fulfilled if -1 × -1 = 1.
So we decide the rules that addition and multiplication have to fulfill and then define -1 as the thing that fulfills the rules and is the opposite of 1 for addition.
If you have a video of someone walking backwards (a negative) and you play that video backwards (negative) the person will seem to walk forward (positive) that’s why multiplying a negative by a negative is a positive
Consider 2 x 3 = 6. “Two times three is six.”
2 x -3 = -6 -> “Two, times *the opposite of* three, is *the opposite of six*.”
-2 x 3 = -6 -> “**The opposite of** two, times three, is **the opposite of** six.”
-2 x -3 -> “**The opposite of** two, times *the opposite of* three, is **the opposite of** *the opposite of* six.” -> = -(-6) = 6.
Saw something like this earlier, but this should help. Nothing I read here is super convincing.
Negative numbers are the opposite of numbers so that a number and it’s opposite combined are zero (combined meaning added – if you go to the store from your house, (let’s call this 1) you have to go the same distance to go back home (let’s call this -1). It’s just a description from a single perspective.
2 x 3 is 6 (2+2+2)
2 x -3 is -6: the negative tells us the how to look at this, so let’s rewrite this as -(2 x 3). Or the opposite of 6.
-2 x -3 is 6: rewrite like -(-(2 x 3)). It’s the same thing as the previous step, but with an extra “opposite.” So the opposite of the opposite of 6.
The thing is, negative is a concept and that can get a bit strange when you start using it, just like how me saying “I’m not not going to go to the store and won’t come back without milk.” It’s easier to say “I’m going to go to the store and come back with milk.”
So now that we understand it’s a way to think about things, we don’t have to think about real life just yet. You can think how do I “simplify” or “solve” this to make it clearer THEN figure out what it actually means.
In that last example, we could simplify the 6 to mean making 6 trips. But if we didn’t simplify it, it would look like making the opposite of 2 trips home the opposite of 3 times. It wouldn’t make sense to do a trip an opposite of any time, right?
So, we simplify the math part to get to the meaning. This is why math can be tough. It’s because the way we end up thinking about math doesn’t actually line up perfectly with the concept IN the math. So, let’s make it easier to understand the math first, like “it’s not not Opposite Day. No it is.”
So, simplify first. Make sense kiddo?
Think about it THIS WAY. 3 x -3 = -9, because it is literally three negative threes (-3 + -3 + -3).
to answer your question, -3 x -3 is negative three going a reverse negative three from zero (–3 + –3 + –3). Thus, in reality, 3 x 3 or 9
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