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.

Lmao not gonna lie, this would be a very intuitive way of teaching a kid negative values.

“It absolutely, definitely makes sense”

Turn Around. Every now and then I get a little bit lonely and you're never coming 'round...

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

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.

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