Types of Number (WJEC GCSE Maths & Numeracy (Double Award): Higher): Revision Note

Types of number

You will come across vocabulary such as

• Integers and natural numbers

• Rational and irrational numbers

• Multiples

• Factors

• Prime numbers

• Squares, cubes and roots

• Reciprocals

Knowing what each of these terms mean is essential.

What are integers and natural numbers?

• Integers are whole numbers;

  • They can be positive, negative and zero

  • For example, -3, -2, -1, 0, 1, 2, 3  are all integers

• Natural numbers are the positive integers

  • They can be thought of as counting numbers

  • 1, 2, 3, 4, … are the natural numbers

    • Notice that 0 is not included

What are multiples?

• A multiple of a given integer is a number that can be formed by multiplying the integer by another positive integer

  • For example, 12 is a multiple of 3 because 12 = 3 × 4

• A common multiple of two or more integers is a number that is a multiple of the integers

  • For example, 12 is a common multiple of 4 and 6

• Even numbers (2, 4, 6, 8, 10, ...) are multiples of 2

• Odd numbers (1, 3, 5, 7, 9, ...) are not multiples of 2

• Multiples can be algebraic

  • For example, the multiples of $k$ would be $k,2k,3k,4k,5k....$

What are factors?

• A factor of a given integer is a value that divides the given number exactly with no remainder

  • 6 is a factor of 18 because 18 ÷ 6 = 3

• Every integer greater than 1 has at least two factors

  • The integer itself, and 1

• A common factor of two or more integers is a number that is a factor of the integers

  • For example, 3 is a common factor of both 21 and 18

How do I find factors?

• Finding all the factors of a particular value can be done by finding factor pairs

• For example, when finding the factors of 18

  • 1 and 18 will be the first factor pair

  • Divide by 2, 3, 4 and so on to test if they are factors

    • 18 ÷ 2 = 9, so 9 and 2 are factors

    • 18 ÷ 3 = 6, so 6 and 3 are factors

    • 18 ÷ 4 = 4.5, so 4 is not a factor

    • 18 ÷ 5 = 3.6, so 5 is not a factor

    • 18 ÷ 6 would be next, but we have already found that 6 was a factor

    • So we have now found all the factors of 18: 1, 2, 3, 6, 9

How do I find factors without a calculator?

• Use a divisibility test

  • Some tests are easier to remember, and more useful, than others

• Once you know that the number has a particular factor, you can divide by that factor to find the factor pair

• If a number is not divisible by an integer, then the number is also not divisible by any multiple of that integer

  • 119 is not divisible by 3

    • so 119 is not divisible by 6, 9, 12, etc, ...

• If a number is divisible by two integers, and the only common factor of those integers is 1, then the number is also divisible by the product of the integers

  • 594 is divisible by 2 and 3

  • The only common factor of 2 and 3 is 1

    • so 594 is divisible by 2×3=6

• Instead of a divisibility test, you could use a formal written method to divide by a value

  • If the result is an integer; you have found a factor

What are prime numbers?

• A prime number is a number which has exactly two (distinct) factors; itself and 1

  • You should remember at least the first ten prime numbers:

    • 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

• 1 is not a prime number, because:

  • by definition, prime numbers are integers greater than or equal to 2

  • 1 only has one factor

• 2 is the only even prime number

• If a number has any factors other than itself and 1, it is not a prime number

What are square numbers?

• A square number is the result of multiplying a number by itself

  • The first square number is $1\times 1=1$, the second is $2\times 2=4$ and so on

• The first 15 square numbers are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

  • Aim to remember at least the first fifteen square numbers

• In algebra, square numbers can be written using a power of 2

  • $a\times a=a^2$

What are cube numbers?

• A cube number is the result of multiplying a number by itself, twice

  • The first cube number is $1\times 1\times 1=1$, the second is $2\times 2\times 2=8$ and so on

• The first 5 cube numbers are 1, 8, 27, 64 and 125

  • Aim to remember at least the first five cube numbers

  • You should also remember 10³ = 1000

• In algebra, cube numbers can be written using a power of 3

  • $a\times a\times a=a^3$

What are square roots?

• The square root of a value, is the number that when multiplied by itself equals that value

  • For example, 4 is the square root of 16 

  • It is the inverse of squaring

  • Square roots are indicated by the symbol $\sqrt{}$

    • e.g.  The positive square root of 49 would be written as $\sqrt{49}$

  • Square roots can be positive and negative

    • e.g.  The square roots of 25 are 5 and -5

  • If a negative square root is required then a - sign would be used

    • e.g.  $\sqrt{25}=5$ but $-\sqrt{25}=-5$

    • Sometimes both positive and negative square roots are of interest and would be indicated by $\pm \sqrt{25}$

• The square root of a non-square integer is also called a surd

  • e.g. $\sqrt{3}$ is a surd, as 3 is not a square number

  • Surds are irrational numbers

  • $\sqrt{64}$ is rational, as it is equal to 8

  • However, $\sqrt{2}$ is irrational, as 2 is not a square number

• You should aim to remember the square roots of the first 15 square numbers

  • $$\begin{gathered} \sqrt{1},\sqrt{4},\sqrt{9},\sqrt{16},\sqrt{25},\sqrt{36},\sqrt{49},\sqrt{64}, \\ \sqrt{81},\sqrt{100},\sqrt{121},\sqrt{144},\sqrt{169},\sqrt{196},\sqrt{225} \end{gathered}$$

What are cube roots?

• The cube root of a value, is the number that when multiplied by itself twice equals that value

  • For example, 3 is the cube root of 27

  • It is the inverse of cubing

  • Cube roots are indicated by the symbol $\sqrt[3]{}$

    • e.g.  The cube root of 64 would be written as $\sqrt[3]{64}$

  • You should remember the values of the following cube roots:

    • $\sqrt[3]{1},\sqrt[3]{8},\sqrt[3]{27},\sqrt[3]{64},\sqrt[3]{125},\sqrt[3]{1000}$

What is a reciprocal?

• The reciprocal of a number is the result of dividing 1 by that number

  • Any number multiplied by its reciprocal will be equal to 1

    • The reciprocal of an integer is $\frac{1}{\mathrm{integer}}$

    • The reciprocal of the fraction $\frac{a}{b}$ is $\frac{b}{a}$

    • The fraction is flipped upside-down!

• The reciprocal of 3 is $\frac{1}{3}$

  • The reciprocal of $\frac{1}{3}$ is 3

  • $3\times \frac{1}{3}=\frac{1}{3}\times 3=1$

• The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$

  • The reciprocal of $\frac{3}{2}$ is $\frac{2}{3}$

  • $\frac{2}{3}\times \frac{3}{2}=\frac{3}{2}\times \frac{2}{3}=1$

• Algebraically the reciprocal of $a$ is $\frac{1}{a}$

  • The reciprocal of $\frac{1}{a}$  is $a$

  • This can also be written using a power of -1

    • $\frac{1}{a}=a^{-1}$