Is subtraction commutative and consistent with other mathematical operations?

[lLovePhysics]

Are there any communative properties of subtraction because there are many formulas like the slope and distance formulas where you can switch the two terms around right? For example:

Slope Formula: $$m=\frac{y1-y2}{x1-x2}$$

You can switch the terms around so that it would be y2-y1, x2-x1 right?

Also for the distance formula:

$$\sqrt{(x1-x2)^{2}+(y1iy2)^{2}$$


Btw, the numbers are suppose to be subscripts.

 

[cristo]

No, subtraction does not commute, but you could say something like $|x-y|=|y-x|$.

The reason you can swap the terms in the first equation you give is, since y1<y2 and x1<x2, swapping both the values of x on the top and y on the bottom will introduce a minus sign in both the numerator and denominator, which will cancel.

In the distance formula, you are squaring the difference between x1 and x2, and y1 and y2, which will make sure the answer is always positive.

 

[neutrino]

Subtraction is not commutative. In your example of the slope formula, you're just multiplying the numerator and denominator by -1. In the case of the dist. formula, you're using the property the square of any non-zero real number is positive.

P.S. For subsripts, use underscore, as in x_1. $$x_1$$

 

[VietDao29]

Are there any communative properties of subtraction ...

Well, as others have pointed out, the answer is no, there isn't. You can pick up a simple example and see:

3 - 2 = 1
whereas: 2 - 3 = -1.

Well, 1 and -1 are, of course, different. So, no, subtraction is not commutative. :)

 

[CompuChip]

But you can also see that
$$3 - 2 = - (2 - 3)$$
which you can read as shorthand for
$$-1 \times (2 - 3).$$
Now this does always hold and explains why the formulas in your first post work out:

• What happens if you multiply numerator and denominator by the same number in a fraction?
• What happens if you square the opposite of a number (e.g. $x^2 = x \times x$ versus $(-x)^2 = (-x) \times (-x)$.