Algebra

Algebra¹³ is a reasoning process that is one of the fundamental cornerstones of mathematical reasoning. As far as we are concerned, it consists of two things:

• Representing numbers of one sort or another (where we could without loss of generality assume that they are complex numbers, since real numbers are complex, rational and irrational numbers are real, integers are rational, and natural numbers are integer) with symbols. In physics this representation isn't only a matter of knowns and unknowns - we will often use algebraic symbols for numbers we know or for parameters in problems even when their value is actually given as part of the problem. In fact, with only a relatively few exceptions, we will prefer to use symbols as much as we can to permit our algebraic manipulations to eliminate as much eventual arithmetic (computation involving actual numbers) as possible.
• Performing a sequence of algebraic transformations of a set of equations or inequalities to convert it from one form to another (desired) form. These transformations are generally based on the set of arithmetic operations defined (and permitted!) over the field(s) of the number type(s) being manipulated.

That's it.

Note well that it isn't always a matter of solving for some unknown variable. Algebra is just as often used to derive relations and hence gain insight into a system being studied. Algebra is in some sense the language of physics.

The transformations of algebra applied to equalities (the most common case) can be summarized as follows (non-exhaustively). If one is given one or more equations involving a set of variables $a, b, c, ... x, y, z$ one can:

1. Add any scalar number or well defined and consistent symbol to both sides of any equation. Note that in physics problems, symbols carry units and it is necessary to add only symbols that have the same units as we cannot, for example, add seconds to kilograms and end up with a result that makes any sense!
2. Subtract any scalar number or consistent symbol ditto. This isn't really a separate rule, as subtraction is just adding a negative quantity.
3. Multiplying both sides of an equation by any scalar number or consistent symbol. In physics one can multiply symbols with different units, such an equation with (net) units of meters times symbols given in seconds.
4. Dividing both sides of an equation ditto, save that one has to be careful when performing symbolic divisions to avoid points where division is not permitted or defined (e.g. dividing by zero or a variable that might take on the value of zero). Note that dividing one unit by another in physics is also permitted, so that one can sensibly divide length in meters by time in seconds.
5. Taking both sides of an equation to any power. Again some care must be exercised, especially if the equation can take on negative or complex values or has any sort of domain restrictions. For fractional powers, one may well have to specify the branch of the result (which of many possible roots one intends to use) as well.
6. Placing the two sides of any equality into almost any functional or algebraic form, either given or known, as if they are variables of that function. Here there are some serious caveats in both math and physics. In physics, the most important one is that if the functional form has a power-series expansion then the equality one substitutes in must be dimensionless. This is easy to understand. Supposed I know that $x$ is a length in meters. I could try to form the exponential of $x$ : $e^x$ , but if I expand this expression, $e^x = 1 + x + x^2/2! +...$ which is nonsense! How can I add meters to meters-squared? I can only exponentiate $x$ if it is dimensionless. In mathematics one has to worry about the domain and range. Suppose I have the relation $y = 2 + x^2$ where $x$ is a real (dimensionless) expression, and I wish to take the $\cos^{-1}$ of both sides. Well, the range of cosine is only $-1$ to $1$ , and my function $y$ is clearly strictly larger than 2 and cannot have an inverse cosine! This is obviously a powerful, but dangerous tool.