Types of Numbers (Cambridge International IGCSE Maths)

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

  • 0, 1, 2, 3, 4, … are the natural numbers

    • Notice that 0 is included

What are multiples?

• A multiple is a number which can be divided by another number, without leaving a remainder

  • For example, 12 is a multiple of 3

    • 12 divided by 3 is exactly 4

• A common multiple is multiple that is shared by more than one number

  • 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 number is a value that divides the given number exactly, with no remainder

  • 6 is a factor of 18

    • because 18 divided by 6 is exactly 3

• Every integer greater than 1 has at least two factors

  • The integer itself, and 1

• A common factor is a factor that is shared by more than one number

  • 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

• 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 some useful divisibility tests?

• A number is divisible by 2 if the last digit is even (a multiple of 2)

• A number is divisible by 3 if the sum of the digits is divisible by 3 (a multiple of 3)

  • 123 1 + 2 + 3 = 6; 6 is a multiple of 3, so 123 is divisible by 3
  • 134 1 + 3 + 4 = 8; 8 is not a multiple of 3, so 134 is not divisible by 3

• A number is divisible by 4 if halving the number twice results in an integer

• A number is divisible by 8 if it can be halved 3 times and the result is an integer

• A number is divisible by 5 if the last digit is a 0 or 5

• A number is divisible by 10 if the last digit is a 0

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

Worked Example Show that 51 is not a prime number.

If we can find a factor of 51 (that is not 1 or 51), this will prove it is not prime

51 is not even so is not divisible by 2 Next use the divisibility test for 3

5 + 1 = 6; 6 is divisible by 3; therefore 51 is divisible by 3 51 ÷ 3 = 17

The factors of 51 are 1, 3, 17 and 51

Final answer: 51 is not prime as it has more than two (distinct) factors

What are square numbers?

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

  • The first square number is $1\times1=1$, the second is $2\times2=4$ 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 $1\times1\times1=1$, the second is $2\times2\times2=8$ 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 opposite of squaring

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

    • e.g. The 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

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

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 opposite 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:

    • ∛(1 comma ∛(8 comma ∛(27 comma ∛(64 comma ∛(125 comma ∛(1000

Worked Example Write down a number which is both a cube number and a square number, and hence express this number in two different ways using index notation.

Listing the first 12 square numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

Listing the first 5 cube numbers

1, 8, 27, 64, 125

64 appears in both lists, it is the 8^thsquare number and 4^th cube number

Final answer: 64 is both a square and cube number 64 = 8²and 64 =4³

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 3 is $\frac{1}{3}$

  • The reciprocal of $\frac{1}{3}$ is 3
  • $3\times\frac{1}{3}=\frac{1}{3}\times3=1$

• 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

    • $\frac{1}{a}=a^{-1}$

Worked Example Write down a fraction that completes this calculation: $\frac{3}{7}\times\frac{...}{...}=1$

Recall that a number multiplied by its reciprocal is equal to 1

$\frac{3}{7}\times\frac{7}{3}=1$