Pronumerals: Letters for Numbers

In algebra a letter stands for a number, and that letter is called a pronumeral. The name tells you the job it does: pro means in place of, and a numeral is a number symbol, so a pronumeral sits in place of a number. Think of the letter n as a closed box with counters inside. Right now you do not know how many, so you cannot write a digit — but you can still write n and work with it. Later the box might turn out to hold 3, or 7, or 400. If a bag holds n lollies and you eat 2, what is left is n minus 2, and that is a finished answer, not a half-finished one.

Algebra has a shorthand, and the very first rule is that the multiplication sign disappears. Instead of 3 times n you write 3n. The reason is simple: a handwritten times sign looks almost exactly like the letter x, and mixing those two up would be a disaster. A few habits come with the rule. Put the number in front of the letter, so n times 3 is tidied up to 3n. When two letters are multiplied, write them side by side, usually in alphabetical order, so b times a becomes ab. And 1 times n is just n, because writing the 1 adds nothing. If several numbers appear, multiply them first: 5 times k times 2 is 10k.

Here is a picture worth keeping. Three boxes, each holding n, add up to n plus n plus n, and the short way to write that is 3n. So 3n really means three lots of n. Open every box and put 4 in each: you get 4 plus 4 plus 4, which is 12, and 3 times 4 is also 12. The two readings always agree, because multiplying is just repeated adding. Notice how different 2n and n plus 2 are. Two n means the box is doubled, so at n equals 7 it gives 14. But n plus 2 means two extra counters are dropped in, so at n equals 7 it gives only 9.

Two more pieces of shorthand finish the toolkit. Dividing is written as a fraction, so n divided by 5 becomes n over 5, or n slash 5 if you are writing on one line. The division sign almost never appears in algebra. And when a letter is multiplied by itself, you use a power: n times n is written n squared. It is called squared because it is exactly the area of a square whose side is n. Now put n equal to 6 into four expressions and see how far apart they are. n plus 6 is 12, 2n is also 12, n over 2 is 3, and n squared is 36. Two n and n squared are completely different.

Time for the words your teacher will use. A group of numbers and letters joined by operations is an expression, and 3n plus 5 is one. The chunks separated by the plus and minus signs are called terms, so this expression has two terms: 3n and 5. In the term 3n, the number in front is the coefficient, and it tells you how many lots of n you have. The 5 has no letter at all, so it is a constant: it stays 5 whatever n does. One hidden habit to remember is that n on its own really means 1n, so its coefficient is 1. And expressions do real work: a square with side s has perimeter 4s.