Number Patterns In Whole Numbers

Numbers have fascinated humans since ages, be it the mathematicians or statisticians. There is so much that one can do with them and there’s so much that is yet to be discovered. For example, we know that the whole numbers represent the set of all positive numbers, including zero, without any decimal or fractional parts. But did we know that we can derive relationships between the whole numbers by finding some kind of patterns between them? This is why numbers are so interesting or fascinating.

Introduction

Let’s start representing each whole number with a set of dots and arranging these dots in some elementary shape to find number patterns. For arranging these dots, we take strictly four shapes into account. Numbers can be arranged into:

  1. A line
  2. A rectangle
  3. A square
  4. A triangle

Line

Every number can be arranged in a line. Examples:

The number 2 can be represented by Number Patterns

The number 3 can be represented by  Number Patterns

All other numbers can be represented in a similar pattern.

Rectangle

Some numbers can be arranged as a rectangle. Examples:

The number 6 can be arranged as a rectangle with 2 rows and 3 columns as  Number Patterns

Similarly, 12 can be arranged as a rectangle with 3 rows and 4 columns as    Number Patterns

Or as a rectangle with 2 rows and 6 columns as  Number Patterns

Similar it can be formed by 8, 10, 14, 15, etc.

Square

Some numbers can be arranged as squares. Examples:

The number 4 can be represented as Number Patterns and 9 as  Number Patterns

Similar it can be formed by 16, 25, 36, 49 and so on.

Triangle

Some numbers can be arranged as triangles. Examples:

The number 3 can be represented as  Number Patterns  and 6 as   Number Patterns

Similar it can be formed by 10, 15, 21, 28, etc. It is to be noted that the triangle should have its 2 sides equal. Hence, the number of dots in the rows starting from the bottom row should be like 4,3,2,1. The top row should always have one dot.

Number Patterns Observation

Observation of number patterns can guide to simple processes and make the calculations easier.

Consider the following examples which helps the addition and subtraction with numbers like 9, 99, 999, etc. simpler.

  • 145 + 9 = 145 + 10 – 1 = 155 – 1 = 154
  • 145 – 9 = 145 – 10 + 1 = 135 + 1 =  136
  • 145 + 99 = 145 + 100 – 1 = 245 – 1 = 244
  • 145 – 99 = 145 – 100 + 1 = 45 + 1 = 46

Consider another pattern which simplifies multiplication with 9, 99, 999, and so on:

  • 62 x 9 = 62 x (10 – 1) = 62 x 9 = 558
  • 62 x 99 = 62 x (100 – 1) = 62 x 99 = 6138
  • 62 x 999 = 62 x (1000-1) = 62 x 999 = 61938

Consider the following pattern which simplifies multiplication with numbers like 5, 25, 15, etc.:

  • 48 x 5 = 48 x 10/2 = 480/2 = 240
  • 48 x 25 = 48 x 100/4 = 4800/4 = 1200
  • 48 x 125 = 48 x 1000/8 = 48000/8 = 6000

A number of patterns of similar fashion can be observed in the whole numbers which simplifies calculations to quite an extent.

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Practise This Question

The sum of first n consecutive odd numbers is equal to n2.