Practical Algebra: A Self-Teaching Guide, Second Edition

The following content is from Wikipedia.

Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values, known as variables. This use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.

The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Most quantitative results in science and mathematics are expressed as algebraic equations.

Algebraic notation describes how algebra is written. It follows certain rules and conventions, and has its own terminology. For example, the expression

has the following components:

1 : Exponent (power), 2 : Coefficient, 3 : term, 4 : operator, 5 : constant, x,y : variables

A coefficient is a numerical value, or letter representing a numerical constant, that multiplies a variable (the operator is omitted). A term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators. Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. a,b,c) are typically used to represent constants, and those toward the end of the alphabet (e.g. x,y and z) are used to represent variables. They are usually written in italics.

Algebraic operations work in the same way as arithmetic operations, such as addition, subtraction, multiplication, division and exponentiation, and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a coefficient is used. For example, 3 times x^{2} is written as 3x^{2}, and 2 times x times y may be written 2xy.

Usually terms with the highest power (exponent), are written on the left, for example, x^{2} is written to the left of x. When a coefficient is one, it is usually omitted (e.g. 1x^{2} is written x^{2}). Likewise when the exponent (power) is one, (e.g. 3x^{1} is written 3x). When the exponent is zero, the result is always 1 (e.g. x^{0} is always rewritten to 1). However 0^{0}, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.

Practical Algebra: A Self-Teaching Guide, Second Edition

**Alternative notation.**

Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. For example, exponents are usually formatted using superscripts, e.g. x^{2}. In plain text, and in the TeX mark-up language, the caret symbol “^” represents exponents, so x^{2} is written as “x^2”. In programming languages such as Ada, Fortran, Perl, Python and Ruby, a double asterisk is used, so x^{2} is written as “x**2”. Many programming languages and calculators use a single asterisk to represent the multiplication symbol, and it must be explicitly used, for example, 3x is written “3*x”.

**Variables.**

Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.

- Variables may represent numbers whose values are not yet known. For example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as C=P+20.
- Variables allow one to describe general problems, without specifying the values of the quantities that are involved. For example, it can be stated specifically that 5 minutes is equivalent to 60 times 5 = 300 seconds. A more general (algebraic) description may state the number of seconds, s = 60 times m, where m is the number of minutes.
- Variables allow one to describe mathematical relationships between quantities that may vary. For example, the relationship between the circumference, c, and diameter, d, of a circle is described by pi = c / d.
- Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as (a + b) = (b + a).

**Evaluating expressions.**

Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation).

- Added terms are simplified using coefficients. For example, x + x + x can be simplified as 3x (where 3 is a numerical coefficient).
- Multiplied terms are simplified using exponents. For example, x times x times x is represented as x
^{3} - Like terms are added together, for example, 2x
^{2}+ 3ab – x^{2}+ ab is written as x^{2}+ 4ab, because the terms containing x^{2}are added together, and, the terms containing ab are added together. - Brackets can be “multiplied out”, using the distributive property. For example, x(2x + 3) can be written as x times 2x + (x times 3) which can be written as 2x
^{2}+ 3x - Expressions can be factored. For example, 6x
^{5}+ 3x^{2}, by dividing both terms by 3x^{2}can be written as 3x^{2}(2x^{3}+ 1)

**Equations.**

An equation states that two expressions are equal using the symbol for equality, = (the equals sign). One of the most well-known equations describes Pythagoras’ law relating the length of the sides of a right angle triangle:

c^{2} = a^{2} + b^{2}

This equation states that c^{2}, representing the square of the length of the side that is the hypotenuse (the side opposite the right angle), is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by a and b.

An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as a + b = b + a); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. x^{2} – 1 = 8 is true only for x = 3 and x = -3. The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving.

Another type of equation is an inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: a > b where > represents ‘greater than’, and a < b where < represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped.

**Properties of equality.**

By definition, equality is an equivalence relation, meaning it has the properties (a) reflexive (i.e. b = b), (b) symmetric (i.e. if a=b then b=a) (c) transitive (i.e. if a=b and b=c then a=c). It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the statement will remain true. This implies the following properties:

- if a = b and c = d then a + c = b + d and ac = bd;
- if a=b then a + c = b + c;
- more generally, for any function f, if a=b then f(a) = f(b).

**Properties of inequality.**

The relations less than have the property of transitivity:

- If a<b and b < c then a < c;
- If a<b and c < d then a + c < b + d;
- If a<b and c>0 then ac < bc;
- If a<b and c < 0 then bc < ac.

By reversing the inequation, < and > can be swapped, for example:

- a<b is equivalent to b>a