That’s surprisingly accurate, as people here are highlighting (it makes geometrical sense when dealing with complex numbers).
My nephew once asked me this question. The way that I explained it was like this:
- the friend of my friend is my friend; (+1)*(+1) = (+1)
- the enemy of my friend is my enemy; (+1)*(-1) = (-1)
- the friend of my enemy is my enemy; (-1)*(+1) = (-1)
- the enemy of my enemy is my friend; (-1)*(-1) = (+1)
It’s a different analogy but it makes intuitive sense, even for kids. And it works nice as mnemonic too.
I teach maths and one of the analogies is use is watching a film of someone walking forwards and backwards. If you play the film forwards (multiplying by positive), you can see the person walking forwards and backwards as normal. If you play the film backwards (multiplying by negative) you see the opposite. So multiplying by negative reverses whatever was happening before. Hard to put into words but the visuals (hopefully) seem to explain it well enough.
It’s a different analogy but it makes intuitive sense, even for kids.
Its good maths but terrible realpolitick.
Realpolitik is terrible no matter what.
My math teacher in middle school explained it with love/hate, but same set up.
If you hate to love you’re a hater If you love to hate you’re a hater
I mean, you can negotiate a peace agreement and free trade deals, you don’t have to be enemies.
This is basically the staple way of explaining the topic in my country. It was a very bizzare concept for 13 year old me so it made understanding it a lot easier.
Sorry for the question, but where are you from? I learned this with my mother, so I don’t know if it’s something common here (Brazil) or something that she picked from her Polish or Italian relatives.
Turn Around. Every now and then I get a little bit lonely and you’re never coming 'round…
Oh man I read the OP earlier but didn’t click to the comments and I couldn’t figure out why this song was stuck in my head.
A pretty general explanation is that a number consists of an length and an angle on the number line. Positive numbers have angle = 0. Negative numbers have angle = pi (or 180° if you want to work with degrees instead of radians).
Multiplication is an operation where you add together the angles to retrieve the resulting angle and multiply together lengths to get the resulting length (yes, kinda recursive, but we’re only working with purely positive numbers here).
So 3 * (-3) means
Length = 3 * 3 = 9
Angle = 0 + pi = pi (or 0 + 180° = 180°)Of course this is very pedantic, but it works in more complex scenarios as well (pun intended).
Imaginary numbers have angle pi/2 (or 90°) or 3pi/2 (or 270°). So if you for instance want to find the square root of i, you can solve it by finding the length:
1 = x * x
And angle:
pi/2 = y + y
(can use modulus 2pi to acquire 2 solutions here)Solving the equations and resolving the real and imaginary part with trigonometry, we get
1/sqrt(2) + 1/sqrt(2)*i
And
-1/sqrt(2) - 1/sqrt(2)*i
It’s all just circles all the way down
I’m sure circles fit into string theory somehow too.
I’ve never thought of numbers having a direction in a number line, that’s great. Thank you for explaining it this way!
Lmao not gonna lie, this would be a very intuitive way of teaching a kid negative values.
How is multiplying akin to rotating?
Fun fact: exponents and multiplication DO work like rotation … in the complex domain (numbers with their imaginary component). It’s not a pure rotation unless it’s scalar, but it’s neat.
I know I explained that the worst ever, but 3blue1brown on YT talks about it and many other advanced math concepts in a lovely intuitive way.
The snark in that first reply is glorious.
“It absolutely, definitely makes sense”
I wrote about this after pondering in my blog: http://hypotheticallyy.blogspot.com/2023/08/why-is-negative-times-negative-positive.html
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