Political arithmetic

Maths allows economists a greater degree of precision in the description and handling of economic relationships than would otherwise be the case. It also allows them to state their theoretical assumptions neatly and succinctly, and to show clearly the steps in their reasoning so that, for example, any faulty connections can be more easily spotted.

The section on introductory maths for economists of the WinEcon economics program that deals with the basics (chapter 22) explains the contrast between variables (which can be different values) and constants (which have fixed values), and distinguishes between the dependent, or determined, variable on the one hand, and the independent, or autonomous, variable on the other (the one doing the determining). If a change in the one variable always produces the same, unique change in the other, a functional relationship is said to exist between them. The user's grasp of all this is checked and reinforced by means of some simple quizzes.

The basic rules for fractional and algebraic operations, as well as those involving powers, are described, and we are taken step by step through simple illustrative examples.

For fractions, addition and subtraction are easy: multiply the fractions' denominators to find the common denominator, scale up each numerator proportionately, and then add or subtract the numerators as normal; for multiplication of fractions, multiply the numerator and divide the result by the product of the denominators; for division, the second fraction is turned on its head and we proceed as for multiplication.

On the rules for basic algebra, the importance of the order in which subtractions and division are performed is stressed, as is the necessity of performing the same mathematical operations on each side of the "equals" sign (transposition).

A power (otherwise known as an exponent or an index) shows the number of times a variable is to be multiplied by itself and, in algebra, is represented by a letter. For example, if a "b", representing any number, can be multiplied by itself an unspecified number of times (n times), then the number, b, is said to be raised to its power, n, with the power superscripted, so that a positive power is represented in its most general form as bⁿ.

The definitions of powers are as follows:

• for a positive power: bⁿ
• for a fractional power: any number raised to the power of a fraction is equal to the root of that fraction, with the denominator of the fraction indicating the degree of the root (ie whether it is the square or cubed root, or whether it the 4th, 5th or 6th root): b^1/n = ⁿ√b
• for a negative power: any number raised to a negative power is the same as the number raised to that power positively, divided into one: b^-n = 1/bⁿ (eg 16^-3/2 = 1/√16³)
• for a zero power: any number raised to the power of zero equals one: b⁰ = 1

For performing mathematical operations, the rules for powers are as follows:

• to multiply, add the powers: b^m * bⁿ = b^m+n
• to divide, subtract the powers: b^m/bⁿ = b^m-n
• to take exponents of numbers that already have exponents: (a^m)ⁿ = a^mn; eg (2²)³ = 2²*³ = 2⁶ = 64
• the product of any two numbers raised to a power is equal to the same two numbers when the power is a factor: numbers raised to the same power separately and then multiplied together: (a*b)ⁿ = aⁿ * bⁿ; (2*3)² = 2²*3² = 4*9 = 36
• powers cannot be added

The (very) basic principles for plotting co-ordinates on a graph—to show visually how the variable on the vertical axis (y) changes as the horizontal axis (x) does—are given.

Memorising these relationships and rules gives the student a few handy tips when the equations get more complex a little further down the line—as all of this is a way of providing the student with the tools necessary for handling functions and equations. Specifically, we look first at linear, quadratic and simultaneous equations, and then at the special properties of exponents and logarithms, which are useful for modeling more complex economic phenomena. Problems analysed using these equations are usually solved either by graph, or by algebra or by formula.

Linear equations of the type y = ax + b describe the simplest functions, where the relationship between variables is straight and unchanging; "a" gives the slope of the line and "b" the intercept on the axis of the independent variable (the y axis). The slope is a ratio of the degree of change in the (dependent) y variable—how much it goes up—to the (independent) x variable (how much it goes across). Slopes can be positive or negative.

Quadratic equations of the type y = ax² + bx + c are used for modeling non-linear relationships and are the simplest of the polynomials. Graphically, they have characteristic "u" or "hill" shapes, and often have many-to-one relationships: that is, there can be two values of the independent x variable that correspond to a y value of zero. The programme takes us through the manipulations necessary to arrive at the formula needed to calculate these two x values, which is:

± x = −b ± √ b²−4ac/ 2a