I think the the question is pretty self explanatory.
With positive numbers you can let’s say put 3 balls in a box and there you have it, a representation of the positive number 3.
Same goes with 0, a box with 0 balls represent the amount 0.
But how do you do that with a negative number?
In: 0
With the concept of debt. If you owe me $5, you have -$5. If you then pay me $3 you still have -$2
Now let’s say you want to represent not just the number of balls in a basket but more specifically the number of balls that you own. If you borrowed 3 balls (so you owe 3 balls to someone else), then the number is -3. That’s actually how it’s done in accounting.
And let’s say you don’t want to just describe lengths but also direction. In one direction numbers are positive and in the opposite direction they are negative… and for movements on a 2-D plane you just add another axis. That’s actually how it’s done in reality.
I just ELI5-ed a concept that is learned in primary school if not earlier 🫤
Anything where you’re measuring from a base level, so a garden table sits 1 meter above ground level and the hole next to it is -0.5 meters above ground level.
You represent positive numbers by putting 3 balls into a box. You represent negative numbers by taking 3 balls out of a box, leaving it empty.
-3 is the additive inverse of 3. When you add -3 to 3, you get the additive identity 0 (0 is the additive identity because eg 4 + 0 = 4, similarly 1 is the multiplicative identity). That’s ultimately the “point” of negative numbers, to the extent they have one.
People usually like to try to represent them as a “thing” rather than an “action”, but why restrict things like that? Numbers or other mathematical objects usually aren’t very interesting unless they’re acting on something or being acted upon.
there’s many ways, debt has been mentioned as has height.
but the problem of representation is real, and it took people a long time to recognise that they existed. it also took a while to recognise the number 0, for the same reason: what does it *mean*?
my best take on it is just the number line. if you count from left to right and mark down 1,2,3 etc, it goes off to infinity. now count from 3,2,1 … and keep going, what actually stops you? you get 0,-1,-2,-3 and it keeps going to the left forever too.
and the spaces between the numbers give you the real numbers too.
i know it’s not a *physical* representation, but to me, really does make the most sense.
and if you think about the representation of height, what’s happening is we’re putting the zero of the number line on the ground, then measuring height as positive, anything below the ground surface would be in the negative numbers on that line.
same with debt, if your account is positive, it’s value is marked on the right of the number line, if you add a positive number, you move right, if you add a negative number, you move left. if you go into debt, your value goes to the left of zero.
easiest is probably height, for visualization I like to use “heigh above water”.
when I stand on the shore my head is ~ 2m above the water.
when I dive, it’s now “-1m above” the water, or “1m below” the water
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