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## Monday, January 14, 2008

### 4 = 5

One of Professor Egghead's undergrad students had tried to prove that 2 is equal to 1, but he saw through that trick very quickly. A few days later the professor decided to turn the tables on his class. He brought in the following proof and wrote it on the chalkboard. He then told the class that the first student who could explain where the proof failed and why would get an automatic 'A' on the next test.

Given:
-20 = -20

1. Rewrite the two given values as differences:

16-36 = 25-45

2. Replace the four values from step 1 as products:

4² - (9x4) = 5² - (9x5)

3. Add 81/4 to each side:

4² - (9x4) + 81/4 = 5² - (9x5) + 81/4

4. Factor both sides of the equation:

(4 - 9/2)² = (5 - 9/2)²

5. Take the square root of each side:

4 - 9/2 = 5 - 9/2

6. Add 9/2 to each side and we end up with:

4 = 5

The students tried and tried to figure out where the problem was but none of them could solve the puzzle. Can you? Anthony Hurst said...

The square root of a negtive number is imaginary; basically impossible.

Actually, I'd like to post a different answer: The problem is not taking the square root of a negative number. In step 5, when we take the square root, we're taking the square root of (4-9/2)^2, in other words, sqrt((-1/2)^2) = sqrt(1/4). So we're not taking the square root of a negative number, the problem is not specifying which root we're using. On the left, we're using the negative root of 1/4, on the right, we're using the positive root of 1/4.

Chris said...

Ok my confusion occurs between steps three and four on the left side of the equation

Step 3) breaks down as 16-15.75=.25 a positive number

Step 4) 4-4.5=-.5 a negative number

so 4^2 in step 4 equals the 16 in step 3

but

-4.5^2 (20.25) in step 4 does not equal the -15.75 in step 3

Erik said...

Chris,

You cannot break down a square like that.

(4 - 9/2)² <> (4)² - (9/2)²

Think of it like this:

(5)² - (4)² = 25 - 16 = 9

which is not the same as:

(5 - 4)² = (1)² = 1

You have to solve all the additions and subtractions inside the parenthesis before you square the value. The square function is not distributive.

Adam correctly pointed out that there really wasn't anything mathematically wrong with any of the steps except that in step 5, when we took a square root, we chose a different valid root on either side of the equation, which cannot be done.

The square of +5 and the square of -5 are both +25. Whenever you take a square root of a positive number the result COULD be either positive OR negative. It's easier to deal with positive numbers so we don't always think about the other valid answer. Anonymous said...

The whole point is just that x^2 = y^2 does not imply x = y.

payal bansal said...

(4 - 9/2)² = (5 - 9/2)²
This implies (4-9/2)=-(5-9/2)
For square root of a square can be either positive or negative.
This means
(4-9/2)=(-5+9/2)
So this theory has a flaw.

:-) Anonymous said...

The whole point is: when you take a SQRT of the left side (step 4), you should end up with i(0.5), which is NOT equal to (0.5 for the right hand side (step 5).

Phani said...

Nice one! Can anyone prove that 1/2 is not 0.5? Anonymous said...

The problem is that you get a negative number squared, which you then square root. Anonymous said...

****Here is the answer****

Erik said it best.

The square root of x^2 is either positive or negative x.

In this case, only one side will yield a positive root and the other, a negative. Try it. It works.

When taking a square root, there are two answers. We normally disregard one and the other we keep depending on what correctly fits the equation. Suman said...

ok, the answer for 4=5 has a mistake in step 5 and Payal, there's nothing wrong with d theory coz
square root of x^2 is neither x nor -x. It is actually mod x. And this is a proved fact.
so square root of (4-9/2)^2 is actually |4-9/2| which is 9/2-4.
therefore, 9/2-4 = 5- 9/2 , which is
9=9.
so, the proof for 4=5 is wrong. Wilfred said...

Nice one!
There are all sorts of tricks which appeal to people who have not seen them before. The real problem is that when people see a trick performed, they do not see the sleight-of-hand which is performed by the trickster.
Even when a problem like this maths poser is presented with a “supposed solution”, people who have not really understood the rules of mathematics are likely to be tricked into believing an erroneous statement.

Step 5 asks us to obtain the square root for each side of the equation:
(4 - 9/2)² = (5 - 9/2)²
And provides us with the answer:
4 - 9/2 = 5 - 9/2

This is a little bit like asking us to obtain the square root for each side of the equation:
x2 = y2
And saying that the result should be x = y. However we need to remember that the square root of x2 could be (plus x), or it could be (minus x).
The suggested answer implies that a square root of a number always gives a positive result, and quite clearly this is not true.
Let us briefly consider the left hand side of the suggested answer, namely:
(4 minus 9/2) which evaluates to (minus ½)
Likewise consider the right hand side of the suggested answer, namely:
(5 minus 9/2) which evaluates to (plus ½)
It is true that the square of (minus ½) is equal to the square of (plus ½), but that does not mean that (minus ½) is equal to the (plus ½), and neither is 4 equal to 5. Anonymous said...

Ok pay attention
the problem is not in the square roots of the problem....people always try to blame what they don't understand. Chris was the closest in that the problem with the proof happens from step 3 to step four. The factoring that was done was improper. Just set the left side of 3 equal to the left side of step four and you will see that the equations changed proving that the process is not valid.

brennan graham said...

ok people, this information is correct because what was said before about the whole (4 minus 9/2) and (5 minus 9/2) it is NOT 0.5 because if you use the math that they teach you in, like 7th grade you will know that the way the problem is set up could mean that because you are adding 9/2 to each side you are eliminating the 9/2 on each side resulting in 4=5 telaykimi03 said...

yeah. I agree to what "anonymous" has said..x squared=y squared is not simply as x=y..for instance,
1^2=(-1)^2..if we are to get the square root of both sides..it would be 1=-1,which is obviously wrong.

Aragorn said...

So much debate about such a simple problem, and some one even got the answer right. Adam is right. Period.

kim said...

step 4 is incorrect. Cannot do a factorial that way

Charles said...

The error is that when you take the square root of a squared number the result is an absolute value. So it should say:
|4-9/2|=|5-9/2| which is true.

Charles said...
This comment has been removed by the author.
Stranger said...

actually there is no error in calculation. all things done good. but the error is only when we revert the calculation there always error occurs which are only logically. you can get this by counting back to your fingers. like count down from your second hand like 10, 9 , 8, 7 , 6 then if you add this 6 fingers into first hand fingers you will get 11 fingers which is not possible.:)) anyways. dont think so much on this problem. becoz there are many of like this...
Regards
Basharat Martin

Narendra said...

the answer is that in step 5 , we can't take sqrt of both sides, b'coz by taking sqrt of both sides we remove power from both sides which base are not equal . base of LHS is -1/2 and RHS is +1/2 .Therefore we can't remove power because bases are different

Narendra Gupta ,Teacher , govt secondary school diwakari , Alwar , Rajasthan , India
Ph: 9460472836

Abdulelah Fallatah said...

There is something wrong with step 4. Calculate what is between the parentheses of each side of the equation and you'll find out that they do not equal each other, even if you go to step 5.

karan manral said...

Wrong

Rahul said...

Check the maths proof mistakes 4=5. And get the best solution for this.
https://youtu.be/vvKbVCvTIk8

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Unknown said...

We will open square root on 2nd last step after taking their LCM ... then the answer will be same 1/4 = 1/4