# A Square Investigation

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Introduction

A SQUARE INVESTIGATION Aim: The aim of this investigation is to find any rules, patterns of square numbers. Patterns/ rules will be discussed step by step below. Firstly a square number is a number multiplied by it. The name becomes apparent if one looks at a diagram of a square. e.g. 2^2 = 2*2= 4sq units; we say 4 is the square of 2. Hence a square number gives the area of a square. Method: the first step was to draw the table below. (Explain what each column means, draw a graph). X NUMBER X^2 SQUARE NO. OF TENS DIFFERENCE NO. OF UNITS 0 0 0 0 1 1 0 0 1 2 4 0 0 4 3 9 0 0 9 4 16 1 1 6 5 25 2 1 5 6 36 3 1 6 7 49 4 1 9 8 64 6 2 4 9 81 8 2 1 10 100 10 2 0 11 121 12 2 1 12 144 14 2 4 13 69 ...read more.

Middle

The sequence is : 0,1,4,9,6,5,6,9,4&1 The unit for 40^2 (1600) is 0, so the unit for 41^2 should be 1, because the rule states that after 0 comes1. The third column is based on the number of tens the squared numbers have. So for example, digits 0-9 have no tens inside them. But digits 10-19, all of these numbers have one ten inside them. e.g. T U 1 9 I realised a pattern within the column of tens. The first and second set of numbers have the same number of difference, (the "column of difference" will be explained later. The first set includes squared numbers from 0-9, and this has the same number of difference as set 2, which includes squared numbers from 16-49). The third and the fourth set of numbers also have the same number of difference (the third set includes squared numbers from 64-124 and the fourth set includes squared numbers from 169-289). ...read more.

Conclusion

As the squared numbers are moving up, the peak they reach is always unit number 9. Another pattern I realised was that on the top line (the points form) has four points. The second line (the points form) also has four points, and the third line (formed by the points) has two points. And the lines with the same number of points repeat again, i.e. 4,4,2-4,4,2 The last pattern I discovered was that as the squared umbers were going down, the lowest they fell to was as follows: 2,10,22 & 40. 1 0 2 2 2 3 10 8 6 4 22 12 4 5 40 18 6 So I worked out a formula to work out the lowest reached squared numbers: -2x + 2x^2-2 = 2x^2 - 2x -2 So for example, if I wanted to find out the lowest reached square number for the third number, which works out to be 10 as you can see on the top , then I would do as follows: 2*3^2-2*3-2 2*9-6-2 18 - 6 - 2 = 10 CONCLUSION: ?? ?? ?? ?? MOUDUD HUSSAIN 10 A 4/26/07 ...read more.

This student written piece of work is one of many that can be found in our GCSE Consecutive Numbers section.

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