# 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, *n*th even number can be written as E(*n*) = 2*n*. 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 *n*th odd number can be written as U(*n*) = 2*n* – 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 *n*th 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 *n*th square number S(*n*) can be written as S(*n*) = *n*2.

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 *n*2, 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: 2*n* + 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 *n*th cube number C(*n*) can be written as C(*n*) = *n*3. The cube can also be written as Cube = length × width × height.