A bit of maths - or how to prove Pythagoras
05-11-2008, 09:14 PM
#1
PassionFord Post Whore!!
Thread Starter
Join Date: Jun 2003
Location: Exeter
Posts: 3,744
Likes: 0
Received 0 Likes
on
0 Posts
A bit of maths - or how to prove Pythagoras
Hi all
I have been reading a book recently on Fermats Last Theorem, and I came across something that intrigued me, and hopefully will others on the forum!
I have been reading a book recently on Fermats Last Theorem, and I came across something that intrigued me, and hopefully will others on the forum! It does require a basic grasp of maths algebra, and equation simplifications, particularly given that I am useless at explaining this stuff!
I love this kind of thing, but if you don't, sorry for wasting your time!
Basically, the purpose of this thread is to explain the idea of Mathematical proofs. What is this? If you have an equation, you need to be able to prove that it will work with all variables. This difficulty in proof depends on the equation. So…..
Pythagoras equation – For every right angled triangle, the square of the length of the hypotenuse is always equal to the sum of the square of the adjacent side, and the square of the opposite side. The simple way of stating this is that a˛ + b˛ = c˛
So do you remember that from school? If not, draw a right angled triangle – the flat line along the bottom 3cm long, and the long side vertically from the right angle 4cm long. When you join the two points together to make a triangle, the length of the long line will be 5cm. (a˛ + b˛ = c˛, or 3˛ + 4˛ = 25, which is 5˛ )
The issue here is that this only proves that the theorem works for the numbers 3, 4 and 5. We know because we have been told this that the theorem works for any right angled triangle, but how do they know? There are an infinite number of sides lengths, and it would be impossible to check them all - it would take forever!
BUT – it is possible to prove it, and it is quite effective, and reasonably simple to follow.
I am going to use a drawing to help me here. Look at the drawing below…
What you can see above is a square in a square. The way that they are arranged creates four little triangles outside the middle square. This is important - all four of those triangles are identical, and also they are all right angle triangles. Now we know that we can label the sides a (shortest side), b (middle sized), and c(longest size).
If I add the letters as above, we end up with the square as labelled below....
The secret to the proof is in the equation.
What I am going to do is work out the area of the big square, which can be done in two different ways :
Way 1
The first way is to work out the length of one of the sides of the square, and multiply it by itself. (a+b) X (a+b). That will immediately give you the area of the square.
Way 2
The second way would be to work out the area of the middle square (cXc – or c˛), and add the area of the four triangles. Each triangle is (aXb)/2, and there are four triangles, so you would multiply this equation by 4.
So area via second way would be 4X((aXb)/2)+c˛
So based on the above, we will know that both of the equations given equal the area of the large square. OR
(a + b) X (a + b) = 4 x ((a x b)/2) + c˛
We now need to simplify the above equations. I will do this one step at a time, starting with the left. You may remember from school that there are rules on how bracketed equations with squares get solved, but generally, it is as below
a˛ + b˛ + 2ab (trust me – this is how it works!)
Now we do the right hand side, but I will leave the above on the left so you can follow the transformation….
a˛ + b˛ + 2ab = 4 x ((a x b)/2) + c˛
to
a˛ + b˛ + 2ab = 2 x ((a x b)) + c˛
to
a˛ + b˛ + 2ab = 2ab + c˛
now simply simplify the above by deducting the 2ab from both sides and look at what you are left with!
a˛ + b˛ = c˛
And hey presto! You have successfully proved Pythagoras. Why? Because you have established with algebra without assigning a specific value to any of the variables. As such this proves that it must work in all occasions. The key is that we didn’t need to put numbers into the squares, so we didn’t need to know how big they were!
Didn’t follow my stupid explanation? Post and I will try to elucidate!
Hope this in some way lifts your day - if not - I did try to warn you at the start!
JJ
I have been reading a book recently on Fermats Last Theorem, and I came across something that intrigued me, and hopefully will others on the forum!
I have been reading a book recently on Fermats Last Theorem, and I came across something that intrigued me, and hopefully will others on the forum! It does require a basic grasp of maths algebra, and equation simplifications, particularly given that I am useless at explaining this stuff!
I love this kind of thing, but if you don't, sorry for wasting your time!
Basically, the purpose of this thread is to explain the idea of Mathematical proofs. What is this? If you have an equation, you need to be able to prove that it will work with all variables. This difficulty in proof depends on the equation. So…..
Pythagoras equation – For every right angled triangle, the square of the length of the hypotenuse is always equal to the sum of the square of the adjacent side, and the square of the opposite side. The simple way of stating this is that a˛ + b˛ = c˛
So do you remember that from school? If not, draw a right angled triangle – the flat line along the bottom 3cm long, and the long side vertically from the right angle 4cm long. When you join the two points together to make a triangle, the length of the long line will be 5cm. (a˛ + b˛ = c˛, or 3˛ + 4˛ = 25, which is 5˛ )
The issue here is that this only proves that the theorem works for the numbers 3, 4 and 5. We know because we have been told this that the theorem works for any right angled triangle, but how do they know? There are an infinite number of sides lengths, and it would be impossible to check them all - it would take forever!
BUT – it is possible to prove it, and it is quite effective, and reasonably simple to follow.
I am going to use a drawing to help me here. Look at the drawing below…
What you can see above is a square in a square. The way that they are arranged creates four little triangles outside the middle square. This is important - all four of those triangles are identical, and also they are all right angle triangles. Now we know that we can label the sides a (shortest side), b (middle sized), and c(longest size).
If I add the letters as above, we end up with the square as labelled below....
The secret to the proof is in the equation.
What I am going to do is work out the area of the big square, which can be done in two different ways :
Way 1
The first way is to work out the length of one of the sides of the square, and multiply it by itself. (a+b) X (a+b). That will immediately give you the area of the square.
Way 2
The second way would be to work out the area of the middle square (cXc – or c˛), and add the area of the four triangles. Each triangle is (aXb)/2, and there are four triangles, so you would multiply this equation by 4.
So area via second way would be 4X((aXb)/2)+c˛
So based on the above, we will know that both of the equations given equal the area of the large square. OR
(a + b) X (a + b) = 4 x ((a x b)/2) + c˛
We now need to simplify the above equations. I will do this one step at a time, starting with the left. You may remember from school that there are rules on how bracketed equations with squares get solved, but generally, it is as below
a˛ + b˛ + 2ab (trust me – this is how it works!)
Now we do the right hand side, but I will leave the above on the left so you can follow the transformation….
a˛ + b˛ + 2ab = 4 x ((a x b)/2) + c˛
to
a˛ + b˛ + 2ab = 2 x ((a x b)) + c˛
to
a˛ + b˛ + 2ab = 2ab + c˛
now simply simplify the above by deducting the 2ab from both sides and look at what you are left with!
a˛ + b˛ = c˛
And hey presto! You have successfully proved Pythagoras. Why? Because you have established with algebra without assigning a specific value to any of the variables. As such this proves that it must work in all occasions. The key is that we didn’t need to put numbers into the squares, so we didn’t need to know how big they were!
Didn’t follow my stupid explanation? Post and I will try to elucidate!
Hope this in some way lifts your day - if not - I did try to warn you at the start!
JJ
Last edited by JjCoDeX75; 05-11-2008 at 09:18 PM.
05-11-2008, 09:22 PM
#7
PassionFord Post Whore!!
Thread Starter
Join Date: Jun 2003
Location: Exeter
Posts: 3,744
Likes: 0
Received 0 Likes
on
0 Posts
And if you are not all well behaved, I will tell you all why the square root of 2 is an irrational number, and as a consequence cannot be definitavely solved!
JJ
JJ
Trending Topics
05-11-2008, 09:30 PM
#10
PassionFord Post Whore!!
Thread Starter
Join Date: Jun 2003
Location: Exeter
Posts: 3,744
Likes: 0
Received 0 Likes
on
0 Posts
05-11-2008, 09:30 PM
#11
15K+ Super Poster!!
I remember this theory was proved when i was studying for my maths GCSE. From what i remember it states that there are no non zero intergers (whole numbers) that an+bn=cn when n is higher than 2 (read as to the power of, ie n is superscript)
Rick
Rick
05-11-2008, 09:31 PM
#12
PassionFord Post Whore!!
Thread Starter
Join Date: Jun 2003
Location: Exeter
Posts: 3,744
Likes: 0
Received 0 Likes
on
0 Posts
05-11-2008, 09:34 PM
#14
PassionFord Post Whore!!
Thread Starter
Join Date: Jun 2003
Location: Exeter
Posts: 3,744
Likes: 0
Received 0 Likes
on
0 Posts
That is pretty much spot on!
It relates to the desire of matheticians to deal in integers, and Fermat claimed that he had a proof for the statement. I believe that the problem itself pre-dates Fermat - he was the first to claim a proof, which he did not write down.
Simple equation - fucking complicated proof!
one day I'll have a brain and you had all better watch out when that day comes!!!!!!!
05-11-2008, 09:36 PM
#18
PassionFord Post Whore!!
Thread Starter
Join Date: Jun 2003
Location: Exeter
Posts: 3,744
Likes: 0
Received 0 Likes
on
0 Posts
05-11-2008, 09:50 PM
#24
PassionFord Post Whore!!
Thread Starter
Join Date: Jun 2003
Location: Exeter
Posts: 3,744
Likes: 0
Received 0 Likes
on
0 Posts
06-11-2008, 07:17 AM
#27
PassionFord Post Whore!!
Join Date: Jun 2003
Location: Saving the planet
Posts: 5,749
Likes: 0
Received 0 Likes
on
0 Posts
06-11-2008, 10:09 AM
#31
06-11-2008, 12:30 PM
#35
PassionFord Regular
Join Date: Sep 2004
Location: Essex
Posts: 379
Likes: 0
Received 0 Likes
on
0 Posts
The statement a2+b2=c2 is very true.
You can prove this by using the cos rule or the sin rule. Depending on what figures you know when you begin will determine if you use the sin or cos rule.
If you make up some angles and lengths, then hide a few, for any figures, using sin or cos rule, u will always get correct answers.
When working out angles remeber to use sin-1 to get the true angle figure.
Just to add some maths in!!!!!
You can prove this by using the cos rule or the sin rule. Depending on what figures you know when you begin will determine if you use the sin or cos rule.
If you make up some angles and lengths, then hide a few, for any figures, using sin or cos rule, u will always get correct answers.
When working out angles remeber to use sin-1 to get the true angle figure.
Just to add some maths in!!!!!
06-11-2008, 12:41 PM
#37
PassionFord Regular
Join Date: Sep 2004
Location: Essex
Posts: 379
Likes: 0
Received 0 Likes
on
0 Posts
Oh sorry, yeah misread that bit!!
I think with stuff like this, trying to find the proof, i remember when i started doin maths ad physics A level, i used to always question everything. My lecture just used to say...just accept it, it is what it. The algebraic equations are the proof!!!!
I think with stuff like this, trying to find the proof, i remember when i started doin maths ad physics A level, i used to always question everything. My lecture just used to say...just accept it, it is what it. The algebraic equations are the proof!!!!
06-11-2008, 12:49 PM
#38
............
Oh sorry, yeah misread that bit!!
I think with stuff like this, trying to find the proof, i remember when i started doin maths ad physics A level, i used to always question everything. My lecture just used to say...just accept it, it is what it. The algebraic equations are the proof!!!!
I think with stuff like this, trying to find the proof, i remember when i started doin maths ad physics A level, i used to always question everything. My lecture just used to say...just accept it, it is what it. The algebraic equations are the proof!!!!
You didnt pay much attention then by the sounds of it
06-11-2008, 12:59 PM
#39
PassionFord Regular
Join Date: Sep 2004
Location: Essex
Posts: 379
Likes: 0
Received 0 Likes
on
0 Posts
You know don’t ya, those college girls were the biggest distraction in the world lol.
If i were to dig out my old text books, there will probably be loads of statements and examples, explaining and proving it, just like the above.
To be honest, i use this stuff every day in my job, the proof for me that it works is, that it does work, we know it works, because it has been proven, otherwise no one would use it.
When Pythagoras stated his theory, and anyone who quested that was wrong or insane....could there be any other theories?? That’s the question!
If i were to dig out my old text books, there will probably be loads of statements and examples, explaining and proving it, just like the above.
To be honest, i use this stuff every day in my job, the proof for me that it works is, that it does work, we know it works, because it has been proven, otherwise no one would use it.
When Pythagoras stated his theory, and anyone who quested that was wrong or insane....could there be any other theories?? That’s the question!
Last edited by S2Dan; 06-11-2008 at 01:13 PM.
