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Beyond Pythagoras
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Introduction
Maths Pure Coursework 1
Beyond Pythagoras
Maths Pure Coursework 1
By: Ben Ingram
10R
Beyond Pythagoras
Pythagoras Theorem:
Pythagoras states that in any right angled triangle of sides ‘a’, ‘b’ and ‘c’ (a being the shortest side, c the hypotenuse): a2 + b2 = c2
E.g. 1.
32 + 42 = 52
9 + 16 = 25
52 = 25
2. 52 + 122 = 132 3. 72 + 242 = 252
25 + 144 = 169 49 + 576 = 625
132 = 169 252 = 625
All the above examples are using an odd number for ‘a’. It can however, work with an even number.
E.g. 1. 102 + 242 = 262
100 + 576 = 676
262 = 676
N.B. Neither ‘a’ nor ‘b’ can ever be 1. If either where then the difference between the two totals would only be 1. There are no 2 square numbers with a difference of 1.
32 9 42 16 52 25 | 62 36 72 49 82 64 | 92 81 102 100 112 121 |
As shown in the above table, there are no square numbers with a difference of anywhere near 1.
Part 1:
Aim: To investigate the family of Pythagorean Triplets where the shortest side (a) is an odd number and all three sides are positive integers.
Middle
Investigation:
Patterns in ‘a’: The smallest numbers always increase by 2 in this family.
Relations between ‘n’ and ‘a’: The only pattern that I can see in these two sets of numbers is that ‘a’ is always double ‘n’ plus 1. E.g. 1 and 3
1 x 2= 2
2 + 1= 3
This works with all of the above pairs. It can be expressed algebraically as: 2n + 1=a
Patterns in ‘b’: The numbers in the ‘b’ column increase uniformly in a pattern that in itself increases by 4 every time.
8 12 16 20 24 28
4 4 4 4 4
This however cannot be used to form a formula as it could not be worked out for the nth term.
Relations between ‘a’ and ‘b’: The relationship here is that multiplying ‘a’ by ‘n’ then adding ‘n’ always leaves you with ‘b’. E.g. 1 and 5
5 x 1= 5
5 + 1= 6
This works for each one and can be expressed as the formulae: an + a= b
Relations between ‘b’ and ‘c’: The obvious relation here is that ‘c’ is always one more than ‘b’. E.g. 5 is one more than 4
13 is one more than 12
and so on…
Expressed algebraically this is: b + 1= c
However, these formulae only work if you do them in order. I.e.
Conclusion
The fact that the two smaller numbers (when squared) add up to the same as the third number (when squared), proves my theory that multiples of Pythagorean triplets will also adhere to the rules of Pythagoras. My formulae can be changed to suit this:
y (2n + 1 = a)
y (2n2 + 2n = b)
y (2n2 + 2n + 1 = c)
y (4n2 + 6n + 2 = perimeter)
y (2n³ + 3n² + n = area)
In all the above cases, ‘y’ is a substitute for any number, when replaced it can be worked the same as the normal formula, multiplied and still produce a Pythagorean triplet.
E.g. 3 x ((2 x 4) + 1) = 27
3 x ((2 x 4)² + 8) = 120
120 + 3 = 123
Once again the triplets from this family can be put into a table showing not only themselves, but also the fact that they conform with the rules of Pythagoras. In this case it is the multiples of my primary triplet; 3,4,5.
a | b | c | a² + b² | c² |
3 6 9 12 15 18 21 24 27 30 | 4 8 12 16 20 24 28 32 36 40 | 5 10 15 20 25 30 35 40 45 50 | 25 100 225 400 625 900 1225 1600 2025 2500 | 25 100 225 400 625 900 1225 1600 2025 2500 |
As you can see, when squared, the first two columns add up to the third number squared. This proves that my hypothesis is correct and multiples of triplets do conform with the rules of Pythagoras. This is true with both odd and even numbers being inserted into the ‘a’ column.
Ben Ingram Page of 10R
This student written piece of work is one of many that can be found in our GCSE Beyond Pythagoras section.
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