Why there are letters in mathematics
A letter is a sentence about every number at once.
The moment letters appear, a lot of people decide mathematics has stopped making sense. It is worth saying plainly what the letters are for, because it is not obvious and it is rarely explained.
A letter is a statement about all numbers
You can check that 3 + 5 equals 5 + 3. And 7 + 2 equals 2 + 7. You could keep going forever and never finish.
a + b = b + a says it for every pair of numbers at once - including ones nobody has written down. The letters are not hiding a number you have to find. They are how you say something about all numbers in one line.
That is the same leverage as proof by induction: finite writing, infinite content.
The three jobs letters do
- Unknown - a specific number you do not know yet. 'x + 3 = 10' has one answer.
- Variable - something that genuinely varies. In 'y = 2x', x is not hiding a value; it ranges over all of them.
- Parameter - fixed for now, adjustable later. In 'y = mx + c', m and c stay put while x varies, and changing them gives a different line.
Much of the confusion in early algebra comes from these three being written identically and meaning quite different things. It is worth asking, of any letter, which of the three it is.
The unhelpful bit of school algebra
'Solve for x' gets the most classroom time and is the least representative of what algebra is for. It gives the impression that the point is finding a hidden number.
The point is describing relationships. Every formula you use is algebra doing its real job:
- speed = distance / time - true for every journey there has ever been
- area = pi r squared - every circle, at once
- a squared + b squared = c squared - every right-angled triangle
Each of those is infinitely many facts written once. Without letters you would need a table, and the table would never be finished.
The three examples suggest the answer. The line with n proves it, and shows why - the n cancels at the last step. Arithmetic can make you confident; algebra tells you what is going on.
Reading a formula out loud
A formula is a sentence and can be read as one. y = mx + c says: start at height c, and for every step right, go up by m. Once you hear it that way, m being the steepness and c being where it crosses is not a fact to memorise but a description of what you just said.
x is not sacred. It is used because a 17th-century French printer had plenty of spare x blocks in his type case when printing Descartes. Use any letter. Physicists use t, v and a because they are naming things - time, velocity, acceleration - and naming your letters after what they mean makes formulas much easier to read than x and y ever will.
Why it is worth pushing through
Algebra is where mathematics stops being a set of procedures for getting answers and becomes a language for stating rules. Every later subject - physics, economics, statistics, anything involving code - is written in it.
And the underlying move is the same one the reasoning puzzles train: going from 'here are some examples' to 'here is the rule that generates them'. In a matrix puzzle you say the rule in English. In algebra you say it in symbols so it can be manipulated. Same step, different notation.