The undone effect/inverse. Multiply to divide, plus to minus

[Niaboc67]

Forgive me for my lack of mathematical understanding! but the whole idea here with these mathematical concepts is that it must have it's numerical opposite, correct? Such as to add you must be able to subtract, to divide you must be able to multiply. Similarly to use a logarithm you can square it. It seems in mathematics there is a series of inverse qualities. Why is this?

 

[micromass]

Similarly to use a logarithm you can square it.

The inverse of a logarithm is not the square, but rather the exponent. The inverse of a square is the square root (somewhat, see below).

Why is this?

Why questions are pretty difficult to answer. I'm not sure what kind of answer you're looking for, which is the problem with why questions. See also my posts in this thread: https://www.physicsforums.com/showthread.php?t=754154

Now, it is a natural thing that if you have some action, then you look for something that will "undo" the action. This is not only true in mathematics, but everywhere in life. You can put a light on, but that means that there should also be a mechanics to put a light off.

In physics, you can calculate what happens in the future. For example, if I throw a ball with a certain speed upwards, then you can calculate quite precisely when it will hit the ground again and with what speed. Curiously, in physics you can usually go back in the past too! If you know enough information about the ball, then you can usually go back and figure out when the ball is thrown and with what speed.

So it shouldn't be a surprise that we want to do the same in mathematics. Now, we find in mathematics that there are some operations which are reversible. For example, addition is something that can be undone by subtraction. Such operations usually form an abstract structure called a group. There are many examples of groups out there, even in real life and nature. For example, the game Rubik's cube forms a group in a certain way. Indeed, every move or sequence of moves in a Rubik's cube can be undone.

Now, there are also many operations which can not be undone. One example is multiplication. This cannot be undone by division since division by $0$ is undefined. So if we start from $x$ and we multiply by $0$, then we get $0$, but there is no real mechanism to end up back to $x$. If we leave out the number $0$, then multiplication is a reversible operation.

Squares are not reversible. Indeed, we can get from $1$ to $1$ by squaring. But we can also get from $-1$ to $1$ by squaring. So if we want to "undo" the squaring, then we should get from $1$ to both $1$ and $-1$ somehow. This is not a well-defined function. If we restrict attention to the positive numbers, then squares are reversible by square roots.

Cubes are reversible by cube roots. Think about why the problem with squares and square roots does not occur here.

Exponentials are reversible by logarithms (where they are defined).

So we see that any of the above operations is reversible, at least if we restrict our attention to special numbers. For example, for squares we had to restrict to positive numbers.

In mathematics, we don't really speak of "reversible". Rather, we speak of inverse functions and bijectivity. Some functions have an inverse, some only have a partial inverse. Finding inverses of functions is a very huge part of mathematics and shows up truly everywhere. Whenever functions are not invertible, problems arise and things get more complicated (= more interesting).