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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#HelpMe! #Proofs #Math

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Comments

There's a subtle mistake in there.

Here's a cryptic hint that might help:

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

Mursalin Habib - 7 years, 8 months ago

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

Priyankar Kumar - 7 years, 8 months ago

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

Ram Prakash Patel Patel - 7 years, 8 months ago

Why not?

Priyankar Kumar - 7 years, 8 months ago

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

Mursalin Habib - 7 years, 8 months ago

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

Michael Tang - 7 years, 8 months ago

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

Priyankar Kumar - 7 years, 8 months ago

@Priyankar Kumar – You're welcome!

Mursalin Habib - 7 years, 8 months ago

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

Fahad Shihab - 7 years, 8 months ago

Where is your proof ??

Toan Pham Quang - 7 years, 8 months ago

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

Priyankar Kumar - 7 years, 8 months ago

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

Akshat Jain - 7 years, 8 months ago

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

Take this for example: what is $\sqrt{(1-\sqrt{3})^2}$ . Is it $(1-\sqrt{3})$ ? Or is it $-(1-\sqrt{3})$ ? 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 - 7 years, 8 months ago

Mmm, that got me.

Akshat Jain - 7 years, 8 months ago

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

Priyankar Kumar - 7 years, 8 months ago

sqrt( x^2 ) is always | x |

Vinayak k - 7 years, 8 months ago

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

Fahad Shihab - 7 years, 8 months ago

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

but,

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

in this proof we wrote--

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

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

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

so,what we actually doing is

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

this is where we made mistake.

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

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

Gypsy Singer - 7 years, 8 months ago

ok..i understood thanks.

Priyankar Kumar - 7 years, 8 months ago

Careful! Gypsy's last statement is incorrect.

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

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

Mursalin Habib - 7 years, 8 months ago

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

Tan Li Xuan - 7 years, 8 months ago

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 - 7 years, 8 months ago

thanks

Priyankar Kumar - 7 years, 8 months ago

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

Lokesh Sharma - 7 years, 8 months ago

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

Tan Li Xuan - 7 years, 8 months ago

thanks

Priyankar Kumar - 7 years, 8 months ago

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 - 7 years, 8 months ago

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

Tan Li Xuan - 7 years, 8 months ago

No sir, i don't think so.

Priyankar Kumar - 7 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$