Number Patterns

Introduction

When we think of mathematics, the first thing that comes to our mind is numbers. We are aware of different kinds of numbers that have been defined such as natural numbers, whole numbers, decimals, fractions and so on. Each set of a number has its own unique characteristic that makes it a set. For instance, the set of even numbers comprises of all numbers that are divisible by 2. Similarly, prime numbers are the numbers that are not completely divisible by any other number other than themselves and the number 1. By seeing these examples can we say that the numbers can be put in a sort of a pattern? Let us find out.

Numbers and Primary Patterns

The number pattern tells the common relationship between the given set of numbers. In other words, a Number Pattern is the pattern or sequence in the given series of numbers. Let us now discuss some patterns that are formed by primary numbers.

Even and Odd Number Patterns

We are aware of natural numbers. Natural numbers are the numbers that begin from 1 and go on up to infinity. They are in the form of 1 , 2 , 3, 4 and so on. Natural numbers form two types of patterns, depending upon the fact whether they are odd or even. Recall that odd numbers are the numbers which when divided by 2 will leave 1 as a remainder. In other words, odd numbers are the numbers that are not divisible by 2. So, how is the pattern of odd numbers defined? The number pattern of odd numbers is defined by the numbers starting from 1 ,3 , 5 , 7 , 9  and so on. Mathematically, this pattern for natural numbers can also be represented as the following on a number line.

Therefore, the pattern of odd natural numbers is 1, 3, 5, 7, 9, 11 and so on. 

Similarly, let us recall what we mean by even numbers. Even numbers are the numbers which when divided by 2 will leave 0 as a remainder. In other words, even numbers are the numbers that are completely divisible by 2. So, how is the pattern of even numbers defined? The arithmetic pattern of even numbers is given by 2, 4, 6, 8, 10 and so on. Mathematically, this pattern for natural numbers can also be represented on a number line as  – 

Therefore, the pattern of even natural numbers is 2, 4, 6, 8, 10, 12 and so on. 

Now, let us learn about patterns of some set of numbers.

Number Patterns of Prime and Composite Numbers

One of the most used number patterns in our primary classes is prime and composite numbers. Let us recall what we mean by prime numbers. Numbers that are greater than 1 and have only two factors, i.e. the number 1 and the number itself are called prime numbers. Therefore, a number having only two factors is called a prime number. On the other hand, the numbers which have more than two factors are called composite numbers. Considering the numbers from 1 to 10, we can say that the numbers 2, 3, 5 and 7 are prime numbers while the numbers 4, 6, 8, 9 and 10 are not prime numbers. The numbers that are not prime numbers are called composite numbers. However, 1 is a number that is neither prime nor composite.

Although a definite pattern is not formed by prime numbers and composite numbers, they still form a pattern according to their respective definitions. 

Following is the pattern of prime and composite numbers from 1 to 100. Note that 1 is neither prime nor composite. The numbers marked in yellow are prime numbers while the numbers marked in blue are composite numbers. We can see that all numbers ending with 5 and 0 form a pattern.

Order of Number Patterns

Ordering is the general concept that is used to define number patterns. There are two ways in which the number patterns can be ordered – 

1. Ascending Order
2. Descending Order

Let us understand these in detail

Ascending Order of Number Patterns

Ascending order of number patterns is the pattern where the numbers are written from the smallest number to the largest number. For example, suppose we have a number pattern that contains all odd natural numbers from 2 to 10. The numbers in this pattern will be 3, 5, 7, 9. If this pattern is to be written from the smallest to the largest number, the pattern will be 3, 5, 7 and 9. This is known as a number pattern in ascending order.

Descending Order of Number Patterns

Descending order of number patterns is the pattern where the numbers are written from the largest number to the smallest number. For example, suppose we have a number pattern that contains all odd natural numbers from 2 to 10. The numbers in this pattern will be 3, 5, 7, 9. If this pattern is to be written from the largest number to the smallest number, the pattern will be 9, 7, 5 and 3. This is known as a number pattern in descending order.

How to Identify Patterns of Numbers

We are aware of the four operations of mathematical operators, namely, addition, subtraction, multiplication and division.  Most of the number patterns are based on these four mathematical operations only. However, there are some patterns that involve a combination of these operations. 

Let us understand this by an example.

Suppose, we have been given the number pattern 1, 3, 5, 7, 9,  …………. 

What arithmetic pattern is followed by the above sequence? Let us find out.

Observe each of the terms carefully. We can see that – 

First Term = 1 = 2 x 0 + 1

Second Term = 3 = 2 x 1 + 1

Third Term = 5 = 2 x 2 + 1

Fourth Term = 7 = 2 x 3 + 1

Fifth Term = 9 = 2 x 4 + 1 and so on.

Therefore, we can identify the number pattern in the given sequence as 2 n + 1, where n≥1. We can clearly see that this sequence involved a combination of two operators, “ x “ and “ + “.

Through this example, we have learnt that a number of different combinations of operators can be used to define the number pattern in a sequence. So, what do we do that helps us in the identification of the number pattern in a sequence? There are some standard patterns of numbers that we should be aware of. Now, that we have understood what are number patterns and how they are ordered; let us move to learn about the common types of number patterns. 

Types of Number Patterns Based on their Order

In the example above, we have seen that addition can be used to generate an arithmetic pattern. Is this the only way to generate a number pattern? Following are the different types of arithmetic patterns that are most commonly in use – 

• Growing Pattern – As the name suggests, the growing pattern is the arithmetic pattern where the numbers are present in an increasing order. 
• Reducing Pattern – Again, as the name signifies, a reducing pattern is the arithmetic pattern in which the numbers are present in the decreasing order.   
• Recurring Pattern – In the recurring pattern of numbers, the same set of numbers keep repeating themselves to form a pattern of numbers.

Types of Number Patterns based on the Type of Numbers

There are different number patterns that are commonly used for various mathematical operations. Let us discuss some of the common number patterns

Arithmetic Number Pattern

As the name suggests, this pattern is formed by using the basic arithmetic operations between the numbers. Recall that there are four basic arithmetic operators, namely, addition, subtraction, multiplication and division. This means that a number pattern can be formed by either adding or subtracting a common digit to the previous term to obtain the next term of the pattern.

Let us understand this by an example.

Suppose we have a number, say 2.

We add 3 to this number. We have, 2 + 3 = 5

Now, we again add 3 to the result that we obtained in the previous step. We will have, 5 + 3 = 8.

Similarly, we again add 3 to the result that we obtained in the previous step. We will have, 8 + 3 = 11

Hence, we can see that a pattern is formed by the numbers as 2, 5, 8, 11 and so on.

Triangular Number Pattern

Triangular numbers are generated from a pattern of dots that form a triangle. In other words, the triangular number sequence is the representation of the numbers in the form of an equilateral triangle. The pattern formed by the triangular numbers is such that the sum of the previous number and the order of the succeeding number results in the sequence of triangular numbers. In other words, the number of the pattern of dots that form a triangle will give a triangular number sequence. This arrangement is represented as below –

Number Pattern of Square Numbers

Square numbers are the numbers obtained when a number is multiplied by itself. For instance 2 x 2 = 4, therefore, 4 is the square of 2. Similarly, 3 x 3 = 9, therefore, 9 is the square of 3. The pattern of square numbers is given by 1, 4, 9, 16, 25, 36 and so on. In other words, a square number pattern is a sequence in which the numbers are forming the pattern in the form of a square.

The number pattern of square numbers is formed as follows – 

1 ² = 1 x 1 = 1

2 ² = 2 x 2 = 4

3 ² = 3 x 3 = 9

4 ² = 4 x 4 = 16

5 ² = 5 x 5 = 25

Number Pattern of Cube Numbers

Cube numbers are the numbers obtained when a number is multiplied twice with itself. For instance 2 x 2 x 2 = 8, therefore, 8 is the cube of 2. Similarly, 3 x 3 x 3 = 27, therefore, 27 is the cube of 3. The number pattern of cube numbers is given by 1, 8, 27, 12, 64, 125 and so on. In other words, the cube number pattern is the sequence of numbers, which form the pattern in the form of the cube. A cube is a three-dimensional figure.

The number pattern of cube numbers is formed as follows – 

1 ³ = 1 x 1 x 1 = 1

2 ³ = 2 x 2 x 2 = 8

3 ³ = 3 x 3 x 3 = 27

4 ³ = 4 x 4 x 4 = 64

5 ³ = 5 x 5 x 5 = 125

Let us now understand the concept of the pattern of numbers through an example.

Example Find the eighth term of the following number pattern –

Solution We have been given a number pattern and we need to find the eighth number of that pattern. Let us first understand how the number pattern has been formed.

We can see that the first number of the given number pattern is 0 ……… ( 1)

Now, the first arrow of the pattern points to the number 5. Therefore, the second number of the pattern is 5.

Can we see a relation between the first term and the second term?

Let us check their difference, we have 5 – 0 = 5 ……………………… ( 2 )

Now, the second arrow of the pattern points to the number 10. Therefore, the second number of the pattern is 10.

Can we see a relation between the second term and the third term?

Let us check their difference, we have 10 – 5 = 5 ……………………… ( 3 )

Again, the third arrow of the pattern points to the number 15. Therefore, the third number of the pattern is 15.

Can we see a relation between the third term and the fourth term?

Let us check their difference, we have 15 – 10 = 5 ……………………… ( 4 )

From ( 1 ), ( 2 ), ( 3 ), and ( 4 ), we can see that the difference in any two terms in the pattern = 5. This means that the given number pattern has been formed by adding 5 to the previous number to obtain the next number. We must therefore, obtain more number of the series as  –

First Number = 0

Second Number = 0 + 5 = 5

Third Number = 5 + 5 = 10

Fourth Number = 10 + 5 =15

Fifth Number = 15 + 5 = 20

Sixth Number = 20 + 5 = 25

Seventh Number = 25 + 5 = 30

Eighth Number = 30 + 5 =35

Hence, the eighth number in the given number pattern will be 35.

Key Facts and Summary

1. Number Pattern is the pattern or sequence in the given series of numbers.
2. Odd numbers are the numbers which when divided by 2 will leave 1 as a remainder. The number pattern of odd numbers is defined by the numbers starting from 1 ,3 , 5 , 7 , 9  and so on.
3. Even numbers are the numbers which when divided by 2 will leave 0 as a remainder. The number pattern of even natural numbers is defined by the numbers starting from 2, 4, 6, 8, 10, 12 and so on. 
4. Numbers that are greater than 1 and have only two factors, i.e. the number 1 and the number itself are called prime numbers.
5. The numbers which have more than two factors are called composite numbers.
6. Ascending order of number patterns is the pattern where the numbers are written from the smallest number to the largest number.
7. Descending order of number patterns is the pattern where the numbers are written from the largest number to the smallest number.
8. Triangular numbers are generated from a pattern of dots that form a triangle.
9. Square numbers are the numbers obtained when a number is multiplied by itself. A square number pattern is a sequence in which the numbers are forming the pattern in the form of a square.
10. Cube numbers are the numbers obtained when a number is multiplied twice with itself. The cube number pattern is the sequence of numbers, which form the pattern in the form of the cube.