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
0
0
2
4
0
0
4
3
9
0
0
9
4
6
6
5
25
2
5
6
36
3
6
7
49
4
9
8
64
6
2
4
9
81
8
2
0
00
0
2
0
1
21
2
2
2
44
4
2
4
3
69
6
3
9
4
96
9
3
6
5
225
22
3
5
6
256
25
3
6
7
289
28
3
9
8
324
32
4
4
9
361
36
4
20
400
40
4
0
21
441
44
4
22
484
48
4
4
23
529
52
4
9
24
576
57
5
6
25
625
62
5
5
26
676
67
5
6
27
729
72
5
9
28
784
78
6
4
29
841
84
6
30
900
90
6
0
31
961
96
6
32
024
02
6
4
33
089
08
6
9
34
156
15
7
6
35
225
22
7
5
36
296
29
7
6
37
369
36
7
9
38
...
This is a preview of the whole essay
28
784
78
6
4
29
841
84
6
30
900
90
6
0
31
961
96
6
32
024
02
6
4
33
089
08
6
9
34
156
15
7
6
35
225
22
7
5
36
296
29
7
6
37
369
36
7
9
38
444
44
8
4
39
521
52
8
40
600
60
8
0
41
681
68
8
42
764
76
8
4
43
849
84
8
9
44
936
93
9
6
45
2025
202
9
5
46
2116
211
9
6
47
2209
220
9
9
48
2304
230
0
4
49
2401
240
0
50
2500
250
0
0
51
2601
260
0
The first column of number, contains is the numbers that are to be squared. It goes from 0-40. The second column is the numbers squared, from 0-40. The units of the squared numbers make up the fifth column. The number of units follows a sequence of repetition as they go down. 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).
The fifth set of numbers has six numbers with the same difference (the fifth set includes squared numbers from 3234-529). The sixth set has four numbers with the equivalent difference (the sixth set includes squared numbers from 576-729). The remaining numbers go in sets of 6 and then 4 all the way. (All the sets are separated with a line under the number where a specific pattern ends).
The fourth column is concerning difference. Difference is the smaller number subtracted from the larger number.
e.g. 44
4
48
4
52
The sets of numbers in the tens column, pre-mentioned earlier, along with the difference, have a pattern. As the number if sets increase by one, the difference also increases by one.
NUMBER X
SQUARE X^2
NO. OF TENS
DIFFERENCE
NO. OF UNITS
3721
372
3844
384
2
4
3969
396
2
9
4096
409
3
6
4225
422
3
5
4356
435
3
6
4489
448
3
9
4624
462
4
4
I completed the unit column using the sequence the units follow. The number of tens column was completed simply by removing the units from the squared numbers, and then I worked out the difference.
GRAPHICAL REPRESENTATION OF SQUARE NUMBERS:
I drew a graph to represent the squared numbers from numbers 1-40. I came across few patterns as I completed the graph. The squared numbers go through a course of going up and them down and then coming up again, all the way. And right in between (the lines of up and down) there is plane symmetry for all of them. 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.
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
0 A
4/26/07