Types of Numbers (Cambridge International IGCSE Mathematics)

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 and 18

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

Worked Example Show that 51 is not a prime number.

Answer:

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 and 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 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

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

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 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:

    • ∛(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.

Answer:

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 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
  • $3\times\frac{1}{3}=\frac{1}{3}\times3=1$

• 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: $\frac{3}{7}\times\frac{...}{...}=1$

Answer:

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

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