Types of Number (Edexcel GCSE Maths: Foundation)

You will come across vocabulary such as

• Integers 
• 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?

• Integers are whole numbers;
  • They can be positive, negative and zero
  • For example, -3, -2, -1, 0, 1, 2, 3  are all integers

What are rational and irrational numbers?

• A rational number is a number that can be written as a fraction in its simplest form
  • A rational number can be written in the form
    • where and  are both integers, and  is not zero
  • All terminating decimals are rational numbers
    • e.g. 
  • All recurring decimals are rational numbers
    • e.g.  
    • Recurring digits are indicated by dots
      e.g.   
      e.g. 
• An irrational number cannot be written as a fraction
  • All non-terminating, non-recurring decimals are irrational numbers
  • (pi) is an irrational number
    • It has an endless amount of decimal places and there is no pattern or recurring digits
    • Any (simplified) fraction involving , such as, will also be irrational (as in  is not an integer)

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 
• Multiples can be considered as the numbers in a times table
• However multiples go beyond times tables and continue forever
  • For example, the multiples of 3 are 3, 6, 9, 12, 15, ..., 300, ..., 3000, ..., 34 567 896, ... 
• Every non-zero number has an infinite number of multiples
• 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  would be

How do I find the multiples of a number?

• Starting with a particular value, multiples can be listed by counting up in steps of that particular value
  • e.g. the multiples of 7 start with 7, then counting up in 7's will give 14, 21, 28, 35 and so on 
• Multiples form a sequence
  • e.g.  7, 14, 21, 28, 35, ...
• Questions may ask you to state the multiples of a value between certain numbers
  
  • e.g.  the multiples of 7 between 10 and 40 are 14, 21, 28 and 35

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

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, 18

How do I find factors without a calculator?

• Use a divisibility test
  • Some tests 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

How do I test for divisibility by 2?

• A number is divisible by 2 if the last digit is even (a multiple of 2)
  • 126
    6 is even so 126 is divisible by 2
  • 135
    5 is odd so 135 is not divisible by 5

How do I test for divisibility by 3?

• 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

How do I test for divisibility by 4 or 8?

• A number is divisible by 4 if halving the number twice results in an integer
  
  • 128
    128 ÷ 2 = 64;  64 ÷ 2 = 32;  32 is an integer so 128 is divisible by 4
  • 134
    134 ÷ 2 = 67;  67 ÷ 2 = 33.5;  33.5 is not an integer so 134 is not divisible by 4
• A number is divisible by 8 if it can be halved 3 times and the result is an integer

How do I test for divisibility by 5 or 10?

• A number is divisible by 5 if the last digit is a 0 or 5
  • 165
    The last digit is 5;  165 is divisible by 5
  • 230
    The last digit is 0;  230 is divisible by 5
  • 162
    The last digit is 2;  162 is not divisible by 5
• A number is divisible by 10 if the last digit is a 0

What are some other divisibility tests?

• The following are harder divisibility tests
• You don't need to remember them, but they can speed up your working
• You could instead use a formal written method to carry out a division instead
• A useful fact is that if a value is divisible by two numbers, it is also divisible by the product of those two numbers
  • e.g. A number is divisible by 6 if it is divisible by both 2 and 3
  • A number is divisible by 12 if it is divisible by both 4 and 3
• A number is divisible by 7 if you get a multiple of 7 when you double the last digit, and subtract it from the remaining part of the number
  • 245
    • Double 5 is 10
    • 24 - 10 = 14 which is a multiple of 7, so 245 is a multiple of 7
  • 906
    • Double 6 is 12
    • 90-12=78 which is not a multiple of 7, so 906 is not a multiple of 7
• A number is divisible by 9 if the sum of the digits is divisible by 9 (similar to the rule for 3)
• A number is divisible by 11 if you get an answer of 0 or a multiple of 11 when you alternately add and subtract the digits
  • 1364
    • +1-3+6-4=0   so 1364 is a multiple of 11
  • 428
    • +4-2+8=10   so 428 is not a multiple of 11

What are prime numbers?

• A prime number is a number which has exactly two (distinct) factors; itself and 1
  • The first 10 prime numbers are
    • 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
    • You should remember at least the first ten prime numbers
• 1 is not a prime number, there are a few reasons for this such as
  • 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
  • • For example, 27 is often mistaken for a prime number
      • but it is divisible by 3 and 9 (3 and 9 are a factor pair of 27)
      • so 27 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 , the second is and so on
• The first 12 square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144
  
  • Aim to remember at least the first twelve square numbers
• In algebra, square numbers can be written using a power of 2
  • 

What are cube numbers?

• A cube number is the result of multiplying a number by itself, twice
  • The first cube number is , the second is  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
• In algebra, cube numbers can be written using a power of 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 
    
    • e.g.  The square root of 49 would be written as
  • 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.   but
    • Sometimes both positive and negative square roots are of interest and would be indicated by
• The square root of a non-square integer is also called a surd
  • e.g.  is a surd, as 3 is not a square number
  • surds are irrational numbers
    • where possible modern calculators will display irrational numbers as surds
  • is rational, as it is equal to 8
    • 64 is a square number
  • However, is irrational
    • 2 is not a square number

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
    
    • e.g.  The cube root of 64 would be written as

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 
  • The reciprocal of  is 3
  • 
• The reciprocal of  is 
  • The reciprocal of  is 
  • 
• Algebraically the reciprocal of is 
  • The reciprocal of   is a
  • This can also be written using a power of -1
    •