Negative numbers: subtraction

Mathematicians tell us that there are positive and negative numbers. Only then do they talk about the absolute value of the number (the modulus of the number). I don't trust mathematicians. Are there negative numbers in nature? What do the plus and minus signs mean in math? We'll try to figure this out.

Negative numbers appear in arithmetic when a larger number is subtracted from a smaller one. That's what mathematicians say and that's how they teach us.

If you always subtract a smaller number from a larger number, then negative numbers will never occur. This rule is no better or worse than the first rule. But we are not taught this rule.

Subtraction

The minus signs fell off like dried mud. What remains is a crystal clear addition. As you can see, negative numbers cannot be obtained by subtraction.

I did everything right. That's how they taught me at school. If minus x equals minus one, then x equals one.

-x = -1
x = 1

What is the secret of this trick? Mathematicians solve the problem of subtraction incorrectly. If we know the result of the addition and one of the terms, then the second term can be found by subtracting the known term from the sum. The minus signs indicate that we are confused.

Let's see what my manipulations on the number axis look like.

Subtraction on the number line

On the left side of the figure, the numerical axis is fixed. I change the position of the segments relative to zero. On the right side of the figure, the segments remain motionless. I am changing the position of zero. In any case, the sign of the number changes when passing through zero. In algebra, this is equivalent to moving numbers through the equals sign. In the future, we will return to the numerical axis and consider it carefully. An equals sign and zero is a mathematical boundary where spies become scouts and scouts become spies.

We are done with subtraction. Next, we'll see if negative numbers come up when we borrow something.