The following numbers are positive whole numbers. They are also known as counting numbers. 0, 1, 2, 3, 4, 5, 6 …
The following numbers are positive and negative whole numbers … -4, –3, –2, 1, 0, 1, 2, 3, 4 …
Examples of these numbers is where they can be expressed in the form n/m, (n and m are integers). Rational numbers can also be fractions and terminating and recurring decimals, for example, ¾, –0.73, 0.4.
An irrational number cannot be expressed in the form n/m, where n and m are integers. For example, pi and √2.
- A rational number is also written as one integer divided by another
- Terminating and recurring decimals are also written as fractions
- Numbers which cannot be written exactly as fractions or decimals are irrational numbers
- All surds are irrational numbers
Even numbers: numbers which can be exactly divisible by 2.
For example, 24 is an even number. It can be written as
24 = 2 × 12.
Even numbers will have these digits at the end 0, 2, 4, 6 or 8.
The following example are even numbers which are arranged as follows:
Using the example above, nth even number can be written as E(n) = 2n. Note also that E(1) would indicate the first even number, E(2) the second even number etc.
Odd numbers always leaves a remainder of 1 if they are divided by 2.
For example, 15 is an odd number. It can be written as
15 = 2 × 7 + 1
Odd numbers will have these digits at the end 1, 3, 5, 7 or 9.
The following example are odd numbers which are arranged as follows:
Using the example above, the nth odd number can be written as U(n) = 2n – 1. Also note that odd numbers can also be negative.
Triangular numbers can be written as the sum of consecutive whole numbers and they starting with 1.
An example of a triangular number is 21 which can be written as:
21 = 1 + 2 + 3 + 4 + 5 + 6
Triangular numbers can be arranged as follows:
How can we find the value of T(50)? Which is also the 50th triangular number.
Adding all of the numbers from 1 to 50 could be carried out, but this would be a long task! Alternatively, the rule for the nth term, T(n) could be formulated using the following steps:
1. Double the number of counters for each of the triangular arrangements:
2. Then, any rectangular arrangement of counters are then written as the product of two whole numbers:
3. Therefore, from these arrangements we can see that 2T(n) = n(n + 1)
So, for any triangular number T(n)
- Two times a triangular number is equal to the product of the term number and the term number plus 1.
- Each triangular number is equal to half of the product of the term number and the term number plus 1.
The 50th triangular number can now be calculated as follows:
Square numbers is a whole number is multiplied by itself.
For example, 16 is a square number. It can be written as
16 = 4 × 4.
Square numbers can be illustrated as follows:
Making square numbers
The nth square number S(n) can be written as S(n) = n2.
Square number sequences can be generated in the following ways:
- By multiplying a whole number by itself.
For example, 25 = 5 × 5 or 25 = 52.
- By adding consecutive odd numbers starting from 1.
For example, 25 = 1 + 3 + 5 + 7 + 9.
- By adding together two consecutive triangular numbers.
For example, 25 = 10 + 15
- By finding the product of two consecutive even or odd
numbers and add 1.
For example, 25 = 4 × 6 + 1.
Difference between consecutive squares
The square number can be written in the following form if we use the general form for a square number n2, where n is a whole number, the square number will be (n + 1)2.
The difference between two consecutive square numbers can be written as:
(n + 1)2 – n2 = (n + 1)(n + 1) – n2
=n2 + n + n +1 – n2
= 2n + 1
Note: 2n + 1 is always an odd number for any whole number n.
Cube numbers is when a whole number is multiplied by itself and then by itself again.
For example, 64 is a cube number and can be written as
64 = 4 × 4 × 4 or 64 = 43.
Cube numbers can be illustrated using the following image:
The nth cube number C(n) can be written as C(n) = n3. The cube can also be written as Cube = length × width × height.