Arrows

Have you ever wondered, why is it when you multiply any number by itself, the result is a positive number? We all know that "minus times minus is plus", but why?

To answer, let’s talk about arrows. For example, +1 looks like this.

We can reverse, or negate, the arrow and make it go in the opposite direction. Negative numbers are arrows pointing left.

To add two numbers, we place them one after another, like our feet.

+3 is the same as 1 * 3, because * means repeated addition.

Here is another example:

Here we see that +1 added to –1 leads back to where we came from – the sum is an arrow of zero length.

I think it’s very interesting that we had this notion of direction – right or left, and when we got zero we lost it, so that the direction is now unclear.

Subtracting an arrow is really adding a reversed arrow. The above example shows adding +1 to –1, but how is it different from 1-1, i.e. subtracting 1 from 1? It’s not – we just have this word called subtraction that refers to addition of a negated arrow.

Back to the question: why is it when you multiply two negative numbers, you get a positive number? You know the rule, we all learned it in school, but could you show where that rule follows logically from?

When we negate an arrow, we turn it 180 degrees. You can confirm this by looking at the picture above. 1 and –1 are the same arrow, just pointing in the opposite direction. Negation is rotation.

Each of our arrows so far has had a direction and a length.

Above, we have an arrow of length 3 pointing left (180°), corresponding to the number -3, and an arrow of length 2 pointing in the default direction (0°), corresponding to number +2.

I assigned the angle of 0° to positive numbers, but his choice does not affect anything. I could have called positive numbers 90° and negative 270°. Alternatively, instead of assuming that the "full circle" is 360°, we could say that it's really 2, then positive numbers would correspond to 0°, and negative to 1°.

To multiply 3 (180°) by 2 (0°), we use repeated addition for the length (3 * 2), and add the angles.

3 (180°) * 2 (0°), which is

3*2(180°+0°), which is

6 (180°)

That’s an arrow of length 6 pointing in direction of 180°, which makes it a negative number. So multiplication is also rotation, plus repeated addition of lengths.

Similarly, if we multiply 2(180°) by 2(180°), we get 4(180° + 180°), which is the same as 4(360°), which is 4(0°) or +4. Why is 360° the same as 0°? Because you make a full circle and come back to where you were. 360° is the same as 720°, same as 1080°, same as 3600° – they all represent the same direction.

We got a glimpse of multiplication as rotation from arrow reversal – after all, it’s true that reversing something is rotating it by 180°.

Now we know why multiplying by –1 is the same as negating:

a * -1 = -a

That’s because we’ve just added 180° to the number’s angle, which makes it point in the opposite direction without any scaling.

The angles also explain the way the signs behave when we multiply numbers:

positive * positive = positive, because 0° + 0° = 0° = positive

negative * negative = positive, because 180° + 180° = 360° = 0° = positive

positive * negative = negative, because 0° + 180° = 180° = negative

negative * positive = negative, because 0° + 180° = 180° = negative

This covers all the combinations of all signs, and we see the addition of angles agrees with what we learned in school.

You may well wonder, 0° and 180° is good, but what about all the other angles, do they go unused? The answer is of course not, and that’s the topic for the next post.