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John Davis' Home Page ->Free games, eye teasers, quiz, mathemagic,.. Š2003

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These are tricks using mathematics. Fool your friends with these tricks.

Any no. greater than itself

1 = 2 (Trigonometry)

1) To Prove that 1=2

Suppose a=b
Therefore,
a = b
a.a=b.a[multiply 'a' on both sides]
a² = ab
a² - b² = ab - b²[subtract b² on both sides]
(a+b)(a-b) = b(a-b)[identity a² - b² = (a+b)(a-b)]
(a+b) = b -» (1)[cancelling (a+b) on both sides]
Let a=b=1
Then from (1)
a + b = b
1 + 1 = 1
2 = 1
1 = 2

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2) To prove that any number is greater than itself

Assume a>b (a,b are positive)
a > b
a.b > b.b [multiply b on both sides]
ab > b²
ab - a² > b² - a²[subtract a² on both sides]
a(b-a) > (b+a)(b-a)[identity a² - b ²=(a+b)(a-b)]
a > b + a[cancelling (b-a)]
a > b + a
b is positive. So a+b is definitely greater than a
But we have proved that a > a+b.
So a > a >(any no. greater than a)

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3) To prove 1=2 using trigonometry

sin²x + cos²x = 1
cos²x = 1 - sin²x
(cos²x)^3/2 = (1 - sin²x)^3/2[power both sides to ^3/2]
cos³x = (1 - sin²x)^3/2
cos³x + 3 = (1 - sin²x)^3/2+3[add 3]
(cos³x + 3)² = {(1 - sin²x)^3/2+3}² -»(1)

Let x have the value p/2 (90°) (the identity is true for all values)
Then (1) becomes 9=9[cos 90°=0,sin 90°=1], a true result.

Let x have the value p (180°)
Then (1) becomes 2²=4²[cos 180°= -1,sin 180°=0]
2=4
1=2

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