# Beyond Pythagoras.

## Introduction

For this piece of work I am investigating Pythagoras. Pythagoras was a Greek mathematician. Pythagoras lived on the island of Samos and was born around 569BC. He did not write anything but he is regarded as one of the world's most important characters in maths. His most famous theorem is named after him and is called the Pythagoras Theorem. It is basically a?+b?=c?. This is what the coursework is based on.

I am going to look at the patterns, which surround this theorem and look at the different sequences that can be formed.

### The coursework

The numbers 3, 4 and 5 satisfy the condition 3?+4?=5? because

3?=3x3=9

4?=4x4=16

5?=5x5=25

And so

3?+4?=9+16=25=5?

I will now check that the following sets of numbers also satisfy the similar condition of (smallest number) ?+(middle number) ?=(largest number) ?

a) 5, 12, 13

5?=5x5=25

12?=12x12=144

25+144=169

√169 = 13

This satisfies the condition as

5?+12?=25+144=169=13?

b) 7, 24, 25

7?=7x7=49

24?=24x24=576

49+576=625

√625=25

This satisfies the condition as

7?+24?=49+576=625=25?

The numbers 3,4 and 5 can be the lengths - in appropriate units - of the side of a right-angled triangle.

5

3

The perimeter and area of this triangle are:

Perimeter = 3+4+5=12 units

Area = ?x3x4=6 units ?

The numbers 5,12,13 can also be the lengths - in appropriate units - of a right-angled triangle.

### Perimeter = 5+12+13=30

Area=?x5x12=30

This is also true for the numbers 7,24,25

### Area=?x7x24=84

I have put these results into a table to see if I can work out any patterns.

I have noticed that there seems to be a recurring pattern between the length of the middle side and the length of the longest side. They are always consecutive numbers. I shall now investigate this.

I will assume that the hypotenuse has length b + 1 where b is the length of the middle side. (c=b+1)

b+1

a

b

Using Pythagoras:

a? + b? = (b+1) ?

Expanding this:

a? + b? = b? + 2b+ 1

therefore

a? = 2b +1

This means that a? must be odd because it equals 2b + 1.

Since a? is odd this means that a must be odd also.

(even x even = even)

(odd x odd = odd)

From the previous equation we can work out that

b = (a?-1)/2

The three sides are a, b and b +1 where a is an odd number. Now I will produce a table with the results.

To work out some more formulae I will draw this table again but using an n column.

I have worked out some formulae to find out any number in this sequence using n.

First of all I will work out a formula for the shortest side:

The simplest way to work this out would be to find a common difference between the numbers.

However 3 - 1 = 2

5 - 2 = 3

7 - 3 = 4

There is no common difference in the first difference so I will try the 2nd difference

3-2 = 1

4-3 = 1

This shows that there is a pattern between the numbers. So all I have to do now is find the formula. There might be a pattern if I multiply each number by the same number as in n?.

However:

1?=1

2?=4

3?=9

This cannot be right.

Maybe it would work if I tried multiplying the numbers by 2 as in 2n.

However:

1x2=2

2x2=4

3x2=6

This is wrong. However I have noticed a pattern. Each of the numbers in this pattern is one less than the one I was aiming to get. This must mean that

2n+1

is the correct formula

To check this I will pick a random number between 1 and 25 and check it on the table above the n table. The number will be 25.

2x25+1=51.

This is the same as the number on the table so it must be right.

Next I will work out a formula to work out the middle side:

First I will see if there are any common differences.

There does seem to be a pattern because all the middle side numbers are multiples of 4.

12-4=8

24-12=12

12-8=4.

This again reinforces the fact that there is a pattern as the 2nd difference is 4. I will try squaring n to see if it gets to the number.

1?=1

2?=4

3?=9

These numbers are nowhere near the numbers I was trying ...