Classifying numbers
Classifying numbers
Natural numbers
The following numbers are positive whole numbers. They are also known as counting numbers. 0, 1, 2, 3, 4, 5, 6 …
Integers
The following numbers are positive and negative whole numbers … -4, –3, –2, 1, 0, 1, 2, 3, 4 …
Rational numbers
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.
Irrational numbers
An irrational number cannot be expressed in the form n/m, where n and m are integers. For example, pi and √2.
Learning Tip:
- 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
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
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
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)
Learning tip:
- 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
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
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.
