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algebra

Branch of mathematics in which the general properties of numbers are studied by using symbols, usually letters, to represent variables and unknown quantities. For example, the algebraic statement:

(x + y)² = x² + 2xy + y²

is true for all values of x and y. For instance, the substitution x = 7 and y = 3 gives:

(7 + 3)² = 7² + 2(7 × 3) + 3² = 100

An algebraic expression that has one or more variables (denoted by letters) is a polynomial equation. A polynomial equation has the form:

f(x) = a_nxⁿ + a_n−1xⁿ⁻¹ + … + a₂x² + a₁x + a₀

where a_n, a_n−1, …, a₀ are all constants, n is a positive integer, and a_n ≠ 0. Examples of polynomials are:

f(x) = 3x⁴ + 2x² + 1

or

f(x) = x⁵ − 18x + 71

or

f(x) = 2x + 3

Algebra is used in many areas of mathematics - for example, arithmetic progressions, or number sequences, and Boolean algebra (the latter is used in working out the logic for computers).

In ordinary algebra the same operations are carried on as in arithmetic, but, as the symbols are capable of a more generalized and extended meaning than the figures used in arithmetic, it facilitates calculation where the numerical values are not known, or are inconveniently large or small, or where it is desirable to keep them in an analysed form.

For example, the following table shows the cost of gas for heating:

There is a connecting rule between the cost and the number of therms used. Gradient = change in cost/change in therms:

= 40 − 20/50 − 10 = 20/40 = £0.5 per therm

Cost intercept = £15 (the intercept is the standing charge).

Since this is a straight line graph, a linear equation connecting the cost and therms used can be created:

cost = 0.5 therms + 15 or c = 0.5 t + 15

A straight line graph can be represented by the general formula:

y = mx + c

where c is the y-intercept, m is the gradient, and (x,y) are the points on the line.

Order of calculation

The simplification of an algebraic equation or expression must be completed in a set order. The procedure follows the rules of BODMAS - any elements in brackets should always be calculated first, followed by power of (or index), division, multiplication, addition, and subtraction.

For example, to solve the equation:

3(2x − x − 1) = 2(x + 3 + 4)

collect the like terms and work out the brackets:

3(x − 1) = 2(x + 7)

multiply out the brackets:

3x − 3 = 2x + 14

collect the xs on the left-hand side of the equation:

3x − 3 − 2x = 14

then solve for x:

x − 3 = 14 x = 14 + 3 x = 17

Inequations or inequalities may be solved using similar rules. When multiplying or dividing by a negative value, however, the direction of the inequality must be reversed, for example: −x > 5 is equivalent to x < −5.

Quadratic equation

A quadratic equation is a polynomial equation of second degree (that is, an equation containing as its highest power the square of a variable, such as x²). The general formula of such equations is:

ax² + bx + c = 0

in which the coefficients a, b, and c are real numbers, and only the coefficient a cannot equal 0.

Some quadratic equations can be solved by factorization (see factor (algebra)), or the values of x can be found by using the formula for the general solution.

x = [−b + √(b² − 4ac)]/2a or

x = [−b − √(b² − 4ac)]/2a

Depending on the value of the discriminant b² − 4ac, a quadratic equation has two real, two equal, or two complex roots (solutions).

When b² − 4ac > 0, there are two distinct real roots.

When b² − 4ac = 0, there are two equal real roots.

When b² − 4ac < 0, there are two distinct complex roots.

Simultaneous equations

If there are two or more algebraic equations that contain two or more unknown quantities that may have a unique solution, they can be solved simultaneously as simultaneous equations. For example, in the case of two linear equations with two unknown variables, such as:

(i) 3y + x = 6 and

(ii) 3y − 2x = 6

the solution will be those unique values of x and y that are valid for both equations. Linear simultaneous equations can be solved by using algebraic manipulation to eliminate one of the variables. For example, subtracting equation (ii) from equation (i) gives:

3y − 3y + x + 2x = 6 − 6

So x = 0, and substituting this value into (ii) gives:

3y = 6

So y = 2.

Another method is to rearrange (i) to give:

x = 6 − 3y

Substituting this into (ii) gives:

3y − 2(6 − 3y) = 6

Multiplying out the brackets gives:

3y − 12 + 6y = 6

So 9y = 18, and y = 2.

‘Algebra’ was originally the name given to the study of equations. In the 9th century, the Arab mathematician Muhammad ibn-Mūsā al-Khwārizmī used the term al-jabr for the process of adding equal quantities to both sides of an equation. When his treatise was later translated into Latin, al-jabr became ‘algebra’ and the word was adopted as the name for the whole subject.

The basics of algebra were familiar in ancient Babylonia (c. 18th century BC). Numerous tablets giving sets of problems and their answers, evidently classroom exercises, survive from that period. The subject was also considered by mathematicians in ancient Egypt, China, and India. A comprehensive treatise on the subject, entitled Arithmetica, was written in the 3rd century AD by Diophantus of Alexandria. In the 9th century, al-Khwārizmī drew on Diophantus' work and on Hindu sources to produce his influential work Hisab al-jabr wa'l-muqabalah/Calculation by Restoration and Reduction.

The development of symbolism From ancient times until the Middle Ages, equation-solving depended on expressing everything in words or in geometric terms. It was not until the 16th century that the modern symbolism began to be developed (notably by François Viète) in response to the growing complexity of mathematical statements that were impossibly cumbersome when expressed in words. Further research in algebra was aided not only because the symbolism was a convenient ‘shorthand’ but also because it revealed the similarities between different problems and pointed the way to the discovery of generally applicable methods and principles.

Quarternions and the idempotent law In the mid-19th century, algebra was raised to a completely new level of abstraction. In 1843 Sir William Rowan Hamilton discovered a three-dimensional extension of the number system, which he called ‘quaternions’, in which the commutative law of multiplication is not generally true; that is, ab ≠ ba for most quaternions a and b. In 1854 George Boole applied the symbolism of algebra to logic and found it fitted perfectly except that he had to introduce a ‘special law’ that a2 = a for all a (called the idempotent law).

Algebraic structures Discoveries like this led to the realization that there are many possible ‘algebraic structures’, which can be described as one or more operations acting on specified objects and satisfying certain laws. (Thus the number system has the operations of addition and multiplication acting on numbers and obeying the commutative, associative, and distributive laws.)

In modern terminology, an algebraic structure consists of a set, A, and one or more binary operations (that is, functions mapping A × A into A) which satisfy prescribed ‘axioms’. A typical example is a structure that had been studied from the 18th century onwards and is known as a group. This structure had turned up in the study of the solvability of polynomial equations, but it also appears in numerous other problems (for example, in geometry), and even has applications in modern physics.

Modern algebra The objective of modern algebra is to study each possible structure in turn, in order to establish general rules for each structure that can be applied in any situation in which the structure occurs. Numerous structures have been studied, and since 1930 a greater level of generality has been achieved by the study of ‘universal algebra’, which concentrates on properties that are common to all types of algebraic structure.

Artin, Emil BODMAS Bombelli, Raffaele

Boolean algebra mathematical logic Nagell, Trygve

Ore, Oystein Weber, Heinrich Wedderburn, Joseph Henry Maclagan

Whitehead, Alfred North XOR