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Let us assume that the square root of 2 was a rational number. It could therefore be expressed as a fraction of x/y where both x and y are integers and share no common factors (i.e. the fraction as expressed cannot be reduced any farther).
Therefore sqroot(2) = x / y
Hence, 2 = x^2 / y^2
Multiplying both sides by the denominator gives us: 2y^2 = x^2
From this equation, we can deduce that x^2 is even, hence x is also even.
Since x is even, we can express x = 2n where n is an integer. Therefore, returning to an earlier equation:
2y^2 = x^2 = (2n)^2, or simply 2y^2 = 4n^2
Divide both sides by 2 gives us:
y^2 = 2n^2, which implies that y is also an even number.
However, that would mean that both x and y are even numbers. This creates a contradiction because we were operating under the definition that x / y cannot have a common factor. If they were both even, they would have a common factor of 2. Since there is a contradiction, the original assumption that the square root of 2 is rational must be false, hence 2 is an irrational number.
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hey tk, can u prove derivatives? just wonderingthere's more to life than meets the eye, but don't let that get you down.
"If someone throws a stone, throw bread back."
-anonymous
"Always do right. This will gratify some people and astonish the rest."
-Samuel Langhorne Clemens
"The true measure of a man is how he treats someone who can do him absolutely no good."
-Samuel Johnson
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Originally posted by MaGi kOz
i bet more than half of the answers of the riddles in here were people that cheated and got it from websites.Jesus Christ on a pogo stick
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To you...who is your father's mother's uncle's sister's aunt's daughter's brother's father's cousin?Trench Wars Broadcast Revolution (TWBR) - Visit: www.youtube.com/fieryfire3d
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How do you hit a blonde and she will never know it....with a thought!
How can you tell a smart blonde from a dumb blonde....the smart blondes have dark roots.
Why don't blondes eat pickles...because they get their heads stuck in the jar.
Why did the blonde get fired from the M & M factory...she threw out all of the W's.
How do you make a blonde laugh on Monday...tell her a joke on Friday.
What do you call a zit on a blonde's butt...brain tumor.
Why don't blondes make kool-aid...can't fit 8 cups of water in the little packages.
What do you call a blonde with half a brain...gifted.
Why do blondes have T.G.I.F. printed on their shoes...stands for Toes G o In First.
How many blondes does it take to change a tire .... 5--2 to get sodas, 2 to cry and 1 to call daddy.
How do you give a blonde a brain transplant .... blow in her ear.
What do blondes and beer bottles have in common .... they're both empty from the neck up.Trench Wars Broadcast Revolution (TWBR) - Visit: www.youtube.com/fieryfire3d
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