Here again is something that most people are just told by their math teacher to accept and move on. UNACCEPTABLE. You gotta understand this shit, fam. How we going to colonize Mars with a bunch of people that don’t understand how multiplication works. Nah. Understand math, understand the universe.
We have three basic cases to talk about before we can achieve enlightenment:
- A positive times a positive is a positive
- A positive times a negative is a negative
- A negative times a negative is a positive
A positive times a positive is a positive
This one’s easy to visualize. Say I eat a tray of 3 tacos. Then say I eat 2 more trays of 3. Don’t judge me. 2 groups of 3 is 6. 2 times 3 is 6. Positive times a positive is a positive.
A positive times a negative is a negative
This is a little less intuitive, but basically the same shit. We just need a way to visualize a negative number. I usually use debt to visualize it, since debt is essentially negative money. So let’s say I rack up \$25,000 in GODDAMN CRIMINAL STUDENT DEBT per year. So I essentially gain -\$25,000 per year. If I’m in school for 4 years, then I have -\$100,000 not to mention the anti-anxiety meds I need to fucking get out of bed in the morning. So there you go: a negative (-\$25,000) times a positive (4 years) is a big ass negative (-\$100,000). Please help.
A negative times a negative is a positive
This is the one that fucks with people the most. It’s easy to think about groups of positive and negative numbers, but what the fuck does a negative number of groups look like? I don’t know. This one is going to require a bit of formal logic.
Let’s break it down. Assume $a$ and $b$ are positive real numbers.
So if we’re saying $(-a)(-b)=ab$, we’re basically saying that $(-1)(-1)=1$. Is this true? Well, yeah. Why is it true? It’s true because if it weren’t true it would break shit. To prove it let’s do a quick proof by contradiction.
Let’s assume the opposite of what we’re trying to prove.
Contradiction. How you going to say that $-1=1$. That’s some bullshit. So since we got to a contradiction, that means our original assumption was false. So therefore $(-1)(-1)=1$. Now that we know that, we can confidently say that if $a$ and $b$ are positive real numbers, then $(-a)(-b)=(-1)(-1)(ab)=(ab)$. In other words, a negative times a negative is a positive.
So to sum it all up:
- Why is a positive times a positive a positive? Because it’s obvious.
- Why is a positive times a negative a negative? See above.
- Why is a negative times a negative a positive? Because it fucking has to be.