What is wrong with this proof?

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i didn't get the mistake in this proof: 2+2=5.Not my proof. Found it on Facebook.

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Note by Priyankar Kumar
5 years, 11 months ago

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There's a subtle mistake in there.

Here's a cryptic hint that might help:

Is x2\sqrt{x^2} always equal to xx?

Mursalin Habib - 5 years, 11 months ago

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exactly..but the usage of (a-b)^2 somehow disguises this.

Priyankar Kumar - 5 years, 11 months ago

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You can't write (4-9/2) as square root of (4-9/2)^2. That is the mistake.

Ram Prakash Patel Patel - 5 years, 11 months ago

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Why not?

Priyankar Kumar - 5 years, 11 months ago

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Ram Prakash is right. 4924-\frac{9}{2} is not equal to (492)2\sqrt{(4-\frac{9}{2})^2}. This is what I hinted at in my initial comment.

Mursalin Habib - 5 years, 11 months ago

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Since 49/24-9/2 is negative, while (49/2)2\sqrt{(4-9/2)^2} is positive.

Michael Tang - 5 years, 11 months ago

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@Michael Tang ok..thanks..i get the point. Thanks to Mursalin too.

Priyankar Kumar - 5 years, 11 months ago

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@Priyankar Kumar You're welcome!

Mursalin Habib - 5 years, 11 months ago

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@Mursalin Habib Look at the last. Assuming the mistake that it is (592)2-\sqrt{(5-\frac{9}{2})^{2}} we will get 5+92+92=4-5+\frac{9}{2}+\frac{9}{2}=4So, it is proved that 2+2=42+2=4 and not 55

Fahad Shihab - 5 years, 10 months ago

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Where is your proof ??

Toan Pham Quang - 5 years, 11 months ago

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it's an external link. Click on the heading of this discussion.

Priyankar Kumar - 5 years, 11 months ago

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It is wrong because x2\sqrt{x^2} is not always equal to xx. It has two values, that are xx and x-x

Akshat Jain - 5 years, 11 months ago

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You should be careful about what you write. x2\sqrt{x^2} [the principal square-root of x2x^2] does not have two values.

Take this for example: what is (13)2\sqrt{(1-\sqrt{3})^2}. Is it (13)(1-\sqrt{3})? Or is it (13)-(1-\sqrt{3})? Or is it both?

According to you, both of these should be correct. Are they?

Also see the solutions for this problem where almost all of them make this same mistake.

Mursalin Habib - 5 years, 11 months ago

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Mmm, that got me.

Akshat Jain - 5 years, 11 months ago

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exactly..but in the subsequent steps, (a-b)^2 disguises this.

Priyankar Kumar - 5 years, 11 months ago

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sqrt( x^2 ) is always | x |

Vinayak k - 5 years, 11 months ago

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I think so that the square roots can't be split...

Fahad Shihab - 5 years, 11 months ago

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(492)=12(4-\frac{9}{2})=-\frac{1}{2}

but,

(592)=12(5-\frac{9}{2})=\frac{1}{2}

in this proof we wrote--

2+2=492+92...........step(i)2+2=4-\frac{9}{2}+\frac{9}{2}...........step (i)

=(592)2+92..........step(viii)=\sqrt{(5-\frac{9}{2})^2}+\frac{9}{2}..........step(viii)

=592+92..........step(ix)=5-\frac{9}{2}+\frac{9}{2}..........step(ix)

so,what we actually doing is

12+92=12+92-\frac{1}{2}+\frac{9}{2}=\frac{1}{2}+\frac{9}{2}..............consider step (i) and (ix)

this is where we made mistake.

So in this case at the time of removing square root of (592)2 \sqrt{(5-\frac{9}{2})^2}

we need to consider (592)2=(592)\sqrt{(5-\frac{9}{2})^2}=-(5-\frac{9}{2})

Gypsy Singer - 5 years, 11 months ago

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ok..i understood thanks.

Priyankar Kumar - 5 years, 11 months ago

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Careful! Gypsy's last statement is incorrect.

(592)2\sqrt{(5-\frac{9}{2})^2} is not equal to (592)-(5-\frac{9}{2}).

However, (492)2=(492)\sqrt{(4-\frac{9}{2})^2}=-(4-\frac{9}{2}).

Mursalin Habib - 5 years, 11 months ago

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(592)2 \sqrt {(5-\frac{9}{2})^{2}} is not equal to (592) -(5-\frac{9}{2})

Tan Li Xuan - 5 years, 11 months ago

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You cant make a negative number positive by just squaring it and taking root. I'm talking about 2nd step, (4-(9/2))

Chirag Pachori - 5 years, 11 months ago

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thanks

Priyankar Kumar - 5 years, 11 months ago

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x2=x \sqrt x^2 = |x| not x x.

Lokesh Sharma - 5 years, 11 months ago

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(492)2 \sqrt{(4 - \frac{9}{2}) ^{2}} is not equal to 492 4 - \frac{9}{2}

Tan Li Xuan - 5 years, 11 months ago

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thanks

Priyankar Kumar - 5 years, 11 months ago

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Did you see the term: 2 x 4 x 9/2? if we cancel 2 then the answer will be 39 but when we multiply the numerators, it is equal to 29. So maybe it is the mistake

James Vincent Llandelar - 5 years, 11 months ago

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No,the calculation is2×4×92 2 \times 4 \times \frac{9}{2}.If we cancel the 2's,then it becomes 4×9=36 4 \times 9 =36 and if we multiply the numerators it is 722=36 \frac{72}{2} = 36. So this is not the mistake.

Tan Li Xuan - 5 years, 11 months ago

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No sir, i don't think so.

Priyankar Kumar - 5 years, 11 months ago

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