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.
That's exactly what eventually helped me understand it.
To multiply the negative by a negative is like an instruction to "reverse the circumstance that created the negative and then keep 'reversing' forward", so to speak.
You come across a hole in the ground. You see a shovel and 5 piles of dirt. That hole in the ground represents where that dirt used to be.
You can "add" more depth to the hole by digging, i.e. continuing to remove dirt and create more piles.
But you can also reverse what was done by "un-digging", I.e. putting the dirt back into the hole.
So if you "un-dug" the hole with the 5 piles of dirt, you'd have 0 piles, and 0 holes.
But if you "un-dug" the hole 5 times in a row, you've filled the hole and started creating a pile on top of it with dirt from somewhere else.
Itâs 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
Oh this is a really cool way to think about it! Thanks for sharing
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.
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.
Because it works
It's the sign rather than the absolute value of the multiplicator that defines a rotation of 180° for
-or 360°/0° for+.
âIt absolutely, definitely makes senseâ
Turn Around. Every now and then I get a little bit lonely and you're never coming 'round...
Permanently Deleted
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!
If you are curious about the math logic side of this like I was, here's the explanation.
Multiplying is just like addition.
3 * 3 = 3 + 3 + 3 = 9
Simple enough but what if one is negative?
3 * (-3) = (-3) + (-3) + (-3) = -9
Also easy, all we changed was the signs of the 3's being added together. But what do you change if you make both of them negative? The only thing left to change is the operation sign. Thus multiplying two negatives is like subtracting negatives.
(-3) * (-3) = - (-3) - (-3) - (-3) = 3 + 3 + 3 = 9
Notice that I placed a negative sign in front of the first (-3). That first one has to be subtracted as well so you can imagine a zero in front of the operation.
Edited: Some formatting.
Copy pasted from my other comment:
This doesn't work if you have to deal with multiplication of numbers that are not integers. You can adjust your idea to work with rational numbers (i.e. ratios of integers) but you will have trouble once you start wanting to multiply irrational numbers like e and pi where you cannot treat multiplication easily as repeated addition.
The actual answer here is that the set of real numbers form a structure called an ordered field and that the nice properties we are familiar with from algebra (for ex that a product of two negatives is positive) can be proved from properties of ordered fields.
Before i ask my question, know that my math is all the way in the back of my head and i didnt get too far in math at school.
Wdym irrational numbers dont work? -3 * -pi would be the same as 3*pi, no?
I always assumed if all factors of the multiplication are negative, it results in the same as the positive variant, no matter the numbers ( real, fractal, irrational, .. )Fun fact... a formal definition of irrational numbers didn't exist until the 1880s (150+ years after Newton died). There were lots of theories before that time (including that they didn't exist) and they were mostly ignored. Iirc, it was Euler's formal definition of complex numbers and e (an irrational number) that led to renewed interest.
The snark in that first reply is glorious.
TIL math logic
I wrote about this after pondering in my blog: http://hypotheticallyy.blogspot.com/2023/08/why-is-negative-times-negative-positive.html
I like how no one here seems to get the concept of the number line You walk 5 steps in the left direction, then you go five steps on the opposite direction and repeat this five times and you end up at the right side with 20 steps,
But wait the magnitude should be 25, well i guess fuck it then
It's not that hard.
If you have -3 -3s and I give you one, you now only have -2 -3s. If you want to get to a total of -6, I have to hand over 4 more -3s to get there, the first 3 of them just being what's needed to get you to 0 and out of deficit. Now you get to hold onto the next two I hand over, and now you have 2 -3s which total -6. But that's 15 worth of -3s I had to hand over to get you there and -6 + 15 = 9, like -3 Ă -3 does too.
Negative numbers aren "real". Like 0, they're just a concept used to represent something, deficit.
umm
Headhurts
I like the greentext explanation better.
This doesn't work if you have to deal with multiplication of numbers that are not integers. You can adjust your idea to work with rational numbers (i.e. ratios of integers) but you will have trouble once you start wanting to multiply irrational numbers like e and pi where you cannot treat multiplication easily as repeated addition.
The actual answer here is that the set of real numbers form a structure called an ordered field and that the nice properties we are familiar with from algebra (for ex that a product of two negatives is positive) can be proved from properties of ordered fields.
Don't confuse the wording "set of real numbers" here, this is just the technical name for the collection of numbers people use from elementary algebra on through to calculus.
