Special types of numbers
In a nutshell
All the numbers that you use can be categorised into different groups.
Integers
Integers are whole numbers - both positive and negative. Zero is also considered an integer.
Odd and even numbers
Odd numbers are integers that end in 1,3,5,7,9. They all leave a remainder of 1 when divided by 2.
Even numbers are integers that end in 0,2,4,6,8. They are all divisible by 2.
Rational and irrational numbers
Rational numbers are numbers that can be written as fractions where the numerator and denominator are both integers. Examples of rational numbers include:
−61,49500,12(=112)
Irrational numbers are numbers that can't be written as fractions where the numerator and denominator are both integers. Examples of irrational numbers include π and any surd.
Square and cube numbers
Square numbers are integers that can be written as the square of another integer.
The first ten square numbers are:
| x
| 1
| 2
| 3
| 4
| 5
| 6
| 7
| 8
| 9
| 10
|
| x2=x×x
| 1
| 4
| 9
| 16
| 25
| 36
| 49
| 64
| 81
| 100
|
Cube numbers are integers that can be written as the cube of another integer.
The first five cube numbers are:
| x
| 1
| 2
| 3
| 4
| 5
|
| x3=x×x×x
| 1
| 8
| 27
| 64
| 125
|
Prime numbers
A prime number is a positive integer that is only divisible by 1 and itself.
Key facts about prime numbers
- 1 is not a prime number - a prime number has to have two different factors (numbers it is divisible by).
- 2 is the only even prime - all the other even numbers are divisible by 2 and hence not prime.
- The single digit prime numbers are 2,3,5,7.
- All prime numbers greater than 10 end in 1,3,7,9.
- Just because a number ends in 1,3,7,9, doesn't necessarily mean it is prime. For example, 33 isn't prime as it is divisible by 3.
- There are an infinite number of primes.
Finding two digit prime numbers
It is very difficult to find large prime numbers, as larger numbers have a higher chance of having additional factors. However, for two digit numbers, there is a procedure.
procedure
| 1.
| Check if the number ends in 1,3,7,9. If it doesn't end with those numbers, it's not prime.
|
| 2.
| Check if the number is divisible by 3 and 7. If it is divisible by either of those numbers, it's not prime.
|
| 3.
| If the number ends in 1,3,7,9 and it is not divisible by 3 and 7, then it's a prime.
|
Note: This procedure only works for two digit numbers.
Example
Verify that 71 is prime.
It ends in 1. So, check if it is divisible by 3 and 7:
71÷3=23 remainder 2.
71÷7=10 remainder 1.
Therefore, 71 is prime because it ends in 1 and is NOT divisible by 3 and 7.
