CXC CSEC Mathematics

The number sets

Numbers are grouped into sets, each one containing the one before it.

A rational number can be written as a fraction of two integers, which means its decimal form either stops or repeats. So $0.25 = \dfrac{1}{4}$, $-7 = \dfrac{-7}{1}$ and $0.\dot{3} = \dfrac{1}{3}$ are all rational. An irrational number cannot be written as such a fraction, and its decimal never stops and never repeats. Examples are $\pi$ and $\sqrt{2}$. A square root is irrational unless the number under it is a perfect square, so $\sqrt{49} = 7$ is rational while $\sqrt{50}$ is not.

The sets are nested, and the relationship is written with the subset symbol:

Every natural number is therefore also a whole number, an integer, a rational number and a real number. The irrational numbers sit inside $\mathbb{R}$ but outside $\mathbb{Q}$, so $\mathbb{R}$ is the union of the rational and irrational numbers.

To place a number, choose the smallest set that holds it. For example $-3$ is an integer, $0$ is a whole number, $\dfrac{5}{8}$ is rational, and $\sqrt{3}$ is irrational and real.

Factors, multiples and types of number

A factor of an integer divides it exactly, and a multiple is the result of multiplying it by an integer. The factors of $12$ are $1, 2, 3, 4, 6, 12$ and, counting negative factors as well, $-1, -2, -3, -4, -6, -12$. The first multiples of $12$ are $12, 24, 36, 48, \dots$

Several kinds of number have their own names.

• An even number is a multiple of 2, and an odd number is an integer that is not.
• A square number is the square of an integer: $1, 4, 9, 16, 25, \dots$
• A prime number has exactly two positive factors, 1 and itself: $2, 3, 5, 7, 11, 13, \dots$ The number 1 is not prime because it has only one factor, and 2 is the only even prime.
• A composite number has more than two positive factors, such as $4, 6, 8, 9, 10$.

Divisibility tests save trial and error. A number is divisible by 2 if it ends in an even digit, by 5 if it ends in 0 or 5, by 10 if it ends in 0, by 3 if its digits add to a multiple of 3, by 9 if its digits add to a multiple of 9, and by 4 if its last two digits form a multiple of 4. So $7\,164$ is divisible by 9 because $7 + 1 + 6 + 4 = 18$, and by 4 because 64 is a multiple of 4.

Prime factorisation, HCF and LCM

Every integer greater than 1 is a product of primes in exactly one way. To find it, divide repeatedly by the smallest prime that goes in.

The highest common factor (H.C.F.) of two or more positive integers is the largest integer dividing all of them. The lowest common multiple (L.C.M.) is the smallest positive integer that is a multiple of all of them. With the prime factorisations written using powers, the H.C.F. takes each prime present in every number to its lowest power, and the L.C.M. takes every prime present in any number to its highest power.

For $84 = 2^2 \times 3 \times 7$ and $120 = 2^3 \times 3 \times 5$:

For two numbers the H.C.F. multiplied by the L.C.M. equals the product of the numbers, which checks the work: $12 \times 840 = 10\,080 = 84 \times 120$.

The H.C.F. is used to simplify, for example to cut an $84$ cm by $120$ cm board into the largest possible equal squares, which have sides of $12$ cm. The L.C.M. is used when events repeat, for example two lights flashing every $84$ and $120$ seconds flash together again after $840$ seconds.

Properties of operations

Five properties describe how addition and multiplication behave on a set of numbers.

Multiplication is distributive over addition: $a(b + c) = ab + ac$. Subtraction and division have none of the first two properties, since $5 - 3 \ne 3 - 5$ and $8 \div 2 \ne 2 \div 8$.

A set is closed under an operation when the result of the operation on any two members is always in the set. The natural numbers are closed under addition and multiplication, but not under subtraction, because $3 - 5 = -2$ is not natural. The integers are closed under subtraction but not division, because $1 \div 2$ is not an integer. The rational numbers are closed under all four operations, apart from division by zero.

These properties justify mental shortcuts. To find $25 \times 17 \times 4$, regroup as $(25 \times 4) \times 17 = 100 \times 17 = 1\,700$, and to find $6 \times 98$, write $6 \times (100 - 2) = 600 - 12 = 588$.

Ordering real numbers

To order numbers of different forms, write each as a decimal to enough places for the differences to show. To put $2.2$, $\sqrt{5}$, $\dfrac{9}{4}$ and $2.24$ in ascending order:

The order is $2.2,\ \sqrt{5},\ 2.24,\ \dfrac{9}{4}$. Descending order is the same list reversed.