The nature of algebra

Algebra is a generalization of arithmetic. Each statement of arithmetic deals with¹ particular numbers: the statement (20+4)²=20²+2*20*4+4²=576 explains how the square of the sum of the two numbers, 20 and 4, may be computed².It can be shown³ that the same procedure applies if the numbers 20 and 4 are replaced by any two other numbers. In order to state the general rule, we write symbols, ordinarily letters, instead of⁴ particular numbers. Let the number 20 be replaced⁵ by the symbol a, which may denote any number, and the number 4 by the symbol b. Then the statement is true⁶ that the square of the sum of any two numbers a and b can be computed by the rule (a+b)²=a²+2a*b+b².

This is a general rule which remains true no matter what⁷ particular numbers may replace the symbols a and b. A rule of this kind is often called a formula.

Algebra is the system of rules concerning the operations with numbers. These rules can be most easily stated as formulas in terms of letters, like the rule given above for squaring the sum of two numbers.

The outstanding characteristic of algebra is the use of letters to represent numbers. Since the letters used represent numbers, all the laws of arithmetic hold for⁸ operations with letters.

In the same way, all the signs which have been introduced to denote relations between numbers and the operations with them are likewise used with letters.

For convenience⁹ the operation of multiplication is generally denoted by dot as by placing the letters adjacent to each other. For example, a*b is written simply as ab.

The operations of addition, subtraction, multiplication, division, raising to a power and extracting roots are called algebraic expressions.

Algebraic expressions may be given a simpler form by combining similar terms. Two terms are called similar, if they differ only in their numerical factor (called a coefficient).

Algebraic expressions consisting of more than one term are called multinomial’s. In particular, an expression of two terms is a binomial, an expression of three terms is a trinomial. In finding the product of multinomial’s we make use of the distributive law.

Notes:

¹ to deal with – иметь дело с; рассматривать

² may be computed – может быть вычислена

³ it can be shown – можно показать

⁴ instead of – вместо

⁵ let the number 20 be replaced – давайте заменим число 20

⁶ then the statement is true – тогда справедливо утверждение

⁷ no matter what – не зависимо от того, какой

⁸ to hold for – годится для

⁹ for convenience – для удобства