The Chemistry Maths Book, Second Edition

1.2 Real numbers 3

(ii) Constant and variable.Equation (1.1) contains two types of quantity:

Constant:a quantity whose value is fixed for the present purposes. The quantity

R 1 = 1 8.31447 J K

− 1

mol

− 1

is a constant physical quantity.

2

A constant number is any

particular number; for example,a 1 = 1 0.1andπ 1 = 1 3.14159=

Variable:a quantity that can have any value of a given set of allowed values. The

quantities p, T, and nare the variables of the functionf(p, T, n) 1 = 1 nRT 2 p.

Two types of variable can be distinguished. An independent variableis one whose

value does not depend on the value of any other variable. When equation (1.1) is

written in the formV 1 = 1 nRT 2 p, it is implied that the independent variables are p, T,

and n. The quantity Vis then the dependent variablebecause its value depends on the

values of the independent variables. We could have chosen the dependent variable

to be Tand the independent variables as p, V, and n; that is,T= 1 pV 2 nR. In practice,

the choice of independent variables is often one of mathematical convenience, but it

may also be determined by the conditions of an experiment; it is sometimes easier

to measure pressure p, temperature T, and amount of substance n, and to calculate V

from them.

Numbers are discussed in Sections 1.2 to 1.4, and variables in Section 1.5. The

algebra of numbers (arithmetic) is discussed in Section 1.6.

(iii) A physical quantityis always the product of two quantities, a number and a unit;

for exampleT 1 = 1 298.15 KorR 1 = 1 8.31447 J K

− 1

mol

− 1

. In applications of mathematics

in the sciences, numbers by themselves have no meaning unless the units of the

physical quantities are specified. It is important to know what these units are, but the

mathematics does not depend on them. Units are discussed in Section 1.8.

1.2 Real numbers

The concept of number, and of counting, is learnt very early in life, and nearly every

measurement in the physical world involves numbers and counting in some way. The

simplest numbers are the natural numbers, the ‘whole numbers’ or signless integers

1, 1 2, 1 3,1= It is easily verified that the addition or multiplication of two natural

numbers always gives a natural number, whereas subtraction and division may not.

For example 51 − 131 = 12 , but 51 − 16 is not a natural number. A set of numbers for which

the operation of subtractionis always valid is the set of integers, consisting of all

positive and negative whole numbers, and zero:

• − 3 − 2 − 10 + 1 + 2 + 3 -

The operations of addition and subtraction of both positive and negative integers are

made possible by the rules

m 1 + 1 (−n) 1 = 1 m 1 − 1 n

m 1 − 1 (−n) 1 = 1 m 1 + 1 n

(1.5)

2

The values of the fundamental physical constants are under continual review. For the latest recommended

values, see the NIST (National Institute of Standards and Technology) website at http://www.physics.nist.gov