Beautiful Mathematics

amit1986 said:
Yes, I understand, but we are working sequentially from up to down in solving the equation. The mathematical steps are correct, but the assumption is not.
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there is nothing wrong with a = b and everything is tickety boo (= fine/happy/correct) up to

2(a^2 - ab) = a^2 - ab. (*)

You cant then divide by a^2-ab because a = b and so (a^2-ab) = 0

The correct way to proceed is to rearrange (*) to get

2(a^2-ab) - a^2-ab = 0 and factorising this gives (2-1)(a^2-ab) = 0

and this implies either 2-1 = 0 (which it doesnt) or a^2-ab = 0 which it does as a = b so no contradiction
 
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so if 1+1 isnt equal to 2, then I am expecting my house and all the buildings in the world to collapse at any time. There is quite a lot of important stuff that depends on that sum:)
 
Scotty2Cues said:
there is nothing wrong with a = b and everything is tickety boo (= fine/happy/correct) up to

2(a^2 - ab) = a^2 - ab. (*)

You cant then divide by a^2-ab because a = b and so (a^2-ab) = 0

The correct way to proceed is to rearrange (*) to get

2(a^2-ab) - a^2-ab = 0 and factorising this gives (2-1)(a^2-ab) = 0

and this implies either 2-1 = 0 (which it doesnt) or a^2-ab = 0 which it does as a = b so no contradiction
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Yes, what you say is right. But remember, your argument depends critically on substituting a for b in equation (*). We could have done that much earlier in the proof but chose to proceed with the original variables as they are, throughout the proof.

The division is correct - dividing both sides of an equation with the same number/variable is okay to do in this.

I learned this proof in my first year of undergrad. As a challenge, our prof had asked us to determine what was wrong with the proof, and those who got the answer right would a bonus mark. After all submissions, our professor had revealed to us that the fallacy of the proof lied not in the math, but only in the assumption that a = b.
 
amit1986 said:
Yes, what you say is right. But remember, your argument depends critically on substituting a for b in equation (*). We could have done that much earlier in the proof but chose to proceed with the original variables as they are, throughout the proof.

The division is correct - dividing both sides of an equation with the same number/variable is okay to do in this.

I learned this proof in my first year of undergrad. As a challenge, our prof had asked us to determine what was wrong with the proof, and those who got the answer right would a bonus mark. After all submissions, our professor had revealed to us that the fallacy of the proof lied not in the math, but only in the assumption that a = b.
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The division is wrong! What is the solution to x^2*sinx = sinx ?

I knew there was a reason why I stayed away from t2w
 
new_trader said:
This one always causes controversy

0.999~ = 1
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was just seeing a man about a dog and realised that 0.999.. is the infinite geometric series:

9/10 + 9/100 + 9/1000 .... we multiply by 1/10 to get the next term

The sum for a geometric series to converge is a/(1-r) where a = 9/10 (the first term) and r = 1/10 (the ratio between successive terms) so a/(1-r) = 9/10/(1-1/10) = 1

To the pubs!!
 
Here's another nice one.


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Ambrose Ackroyd said:
It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect but true


Bertrand Russell (1872-1970), Autobiography, George Allen and Unwin Ltd, 1967, v1, p158
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And let's not forget another great mathematician, Leonardo Da Vinci, and his centuries old (ahead of his time) quote on trading and mathematics;

"Maths? It's all fookin mumbo jumbo and voodoo mate, what you need is Level II and DOM..That Lenny Fibonacci of Pisa bloke was off his head btw..I reckon they must have used some of his cracked maths for the leaning tower.."
 
And let's not forget another great mathematician, Leonardo Da Vinci, and his centuries old (ahead of his time) quote on trading and mathematics;
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So true .... he was designing the parachute, aeroplanes of a kind and many other things long long before most people had even conceptualised these things.


Just read this short synopsis .....


Leonardo da Vinci, the most versatile genius of the Renaissance, is best remembered as the painter of the Mona Lisa (c. 1503) and The Last Supper (c. 1495). But he is almost equally famous for his astonishing multiplicity of talents: architecture, sculpture, music, engineering, geology, hydraulics and the military arts, all with success, and in his spare time doodled sketches for working parachutes and flying machines like helicopters that resembled inventions of the 19th and 20th centuries. He made detailed drawings of human anatomy which are still highly regarded today. Was also known for his engineering of canal locks, cathedrals, and engines of war. Leonardo also was quirky enough to write notebook entries in mirror (backwards) script, a trick which kept many of his observations from being widely known until decades after his death.


True and utter genius.
 
In the area of very large phenomena when the time period of each planet's revolution around the sun is compared in round numbers to the one adjacent to it, their fractions are Fibonacci numbers!

Beginning with Neptune and moving inward toward the sun, the ratios are 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34.

These are the same as the spiral arrangement of leaves on plants !

I find that awesome and humbling ...... sounds to me like a part of an overall creation rather than a scientific accident.

God only knows !!!!!!

Literally .........:D
 
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