# Algebra explained

The line between algebra and [arithmetic](math-arithmetic.md) is most clearly distinguished by the existence of variables. For highly practical minds, it's easy to get lost in algebra because it requires abstractly thinking with a type of [certainty](understanding-certainty.md) about an [unknown](unknown.md) thing (e.g., let x be a not-yet-established number).

With the presence of variables, the equations have *many* options for factoring:

- 5 + x = 12 can become x = 7 or x - 7 = 0
- x^2^ + 5 = 2 can be x^2^ = -3 or x = √-3 or x^2^ + 3 = 0

Most equations require "solve for □" to give an absolute concept, but the vagueness of algebra comes from its relativity, which is probably the largest failure in mathematics [pedagogy](education.md):

- Everything in maths requires understanding the precise [purpose](purpose.md) you wish to accomplish.
- [Rote memorization](mind-memory.md) literally gives the *opposite* effect by provoking people to calculate by [habit](habits.md) without any awareness of the grander picture of *why* the calculation should be done.
- The result of this is that most students can be easily thrown off by the relative thinking necessary to perform algebra.

## What school was supposed to teach

One of the most significant aspects of algebra is how to work with the distributive property [a (b + c) = ab + ac]. It allows more rearranging of values, which becomes *very* useful once multiple variables are introduced.

Another issue with algebra for the uneducated comes through a few issues with [math symbols](math-symbols.md):

- A set of symbols may have a specific meaning in one equation (e.g., f(x) meaning "function of x") but then feeds later into another symbolic representation (e.g., f(x) becomes y).
- It's not uncommon for a parenthesis to represent either an order of operations or the constraint of an idea, context-depending:
  - f(3x) = x + (2y * 7)
- Some symbols (such as π for circles or σ in statistics) are shorthand for *very* specific meanings and purposes.
- All of this is compounded by mathematicians not spelling out each tiny detail regarding *why* an idea exists or its specific rules.

Irrespective, the cumulative algebra education continues:

1. Indicating how proportional correlations in tables can represent as formulas:
   - Given how a cost of 4 correlates to 8 products, 8 correlates to 16, and 50 correlates to 100...
   - ...the same expression could be written as c = 2p.
   - At the same time, proportion can only represent with multiplication and division, *not* addition and subtraction alone.

## Logarithms

While most people are familiar with exponents, many elements of [computing](computers.md) (such as [crypto mining](computers-blockchain.md)) use logarithms, which are the *opposite* of exponents. While an exponent becomes dramatically higher with each iteration (e.g., 3, 9, 27, 81, etc.) logarithms become decreased with each iteration as it progresses infinitely toward that number (e.g., 0.6, 0.8, 0.9, 0.95, etc.).

## Variables

## Linear Algebra

Linear algebra is, quite simply, using formulas as shorthand for tracking where a line is. It establishes an x-axis and y-axis (and sometimes a z-axis), then sets an arbitrary point as (0,0) or (0,0,0). In a mathematical sense, you can set points forward and backward along any of those axes into infinity, but in a computer sense there's a memory limit to how far you can go.

The math operation F represents a line, and often tracks how much that line rises (positively or negatively) relative to how far it runs (positively or negatively). The function of x becomes F(x), and often represents as:

- F(x) = x + 5

In a straight line, any segment of that line will have the exact same properties:

- A 4-foot run with a 1-foot rise means that a 10x-sized run (i.e., 40 feet) will have a 10-foot rise.
- If you add 8 feet to a 4-foot run with a 1-foot rise, then it would have a proportional rise (i.e. 1/4) of 2 feet.

However, the only "linear" algebra is technically multiplication and division, since exponents/squares create curves and addition/subtraction simply translates the line elsewhere while keeping it the same.