A math puzzle

[Tiberius]

I'm familiar with the erroneous proof that 1=2, but then I found this on Futility Closet and I can't find where the error is...

[Roger Wilco]

I literally haven't thought about such things since high school, but how exactly do you get from step 2 to step 3?

[sojourner]

Admitting that I suck at math.....

With the top being level 1, the transition from level 2 to level 3 is incorrect. that transition does not take into account 24 on the left and 40 on the right.


ETA: Beaten to the answer while trying to come up with the right words.

[Rhubarbodendron]

the error is in line#1:
9-24 = -15
25-40 = 15

While techically (meaning the way the equation is being treated with completing the square, radicating [correct word??] etc.) the procedure is correct, you start the "proof" under the false premise that -15 = +15
This triggers a cascade of errors with the rest of the calculation, much like a domino-effect:
In Line #2 you get 1 = 31
In Lines #3 and #4 the equation reads as -1 = +1


The important thing - both in mathematics and in life - is to never take anything for granted but always to check the supposed facts yourself, without blindly relying on others.

[Zion Ravescene]

Rhubarbodendron wrote:

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the error is in line#1:
9-24 = -15
25-40 = 15

You start the "proof" under the false premise that -15 = +15 which triggers a cascade of errors with the rest of the calculation, much like a domino-effect.
The important thing - both in mathematics and in life - is to never take anything for granted but always to check the supposed facts yourself, without blindly relying on others.

Actually, the first line is correct.

Given the leap of adding 16 to each side in statement 2, the transition from level 2 to level 3 is also correct.

I think the main mistake is in the 4th statement - both sides should be divided by the same common multiplier to keep things equal, but instead the left is divided by (3-4) and the right by (5-4).

Taking the rest of the sequence beyond that error, you still end up with a difference of 2 instead of equality, whether you physically subtract 4 from each side or just let them cancel each other out.

Do correct me if I am wrong - I've just woken up.

[Rhubarbodendron]

oops, how embarassing! You are right, of course, Zion! Thanks for pointing it out. My only excuse is that I got only 4 hours sleep last night, due to a somewhat loud party in the neighbourhood.
At any rate you proved my point of never taking anything for granted and always checking the facts

The trick with such equations is usually that you discreetly multiply with zero and then cancel down. In this case, however, it works differently:

Line #1 reads as -15 = -15 which is correct
Line #2 says 1 = 1 still correct
Line #3 states -1² = +1² which is basically correct, as both is 1 but
then at line #4 you radicate again which you shouldn't, because -1 is not +1.
So line #4 is definitely wrong, and then the last line automatically gets wrong as well.

[Roger Wilco]

Zion Ravescene wrote:

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Given the leap of adding 16 to each side in statement 2, the transition from level 2 to level 3 is also correct.

Can you explain? What did he do to get from #2 to #3? I'm stumped.

[Shaytan]

#2 to #3 is what we call in French identité remarquable, must have a name in English too, it's supposed to be known by heart to use it to solve simple equations:

(a+b)²= a²+2ab+b²
(a-b)²=a²-2ab+b²
(a-b)(a+b)=a²-b²

[Roger Wilco]

Ah, thank you, now I get it. Wow, that was a long time ago for me that I dealt with that.

[stj]

It's kind of interesting to think about the best way to explain the error.

In line 3, when the expanded terms are reduced to a binomial, neither 3-5 nor 5-4 are binomials in the usual sense of unknown variables that cannot be summed because they are unknown. The usual procedure is to consolidate the constants, which would have given us -1 squared and 1 squared in line 4. Which are indeed equal.

But then, it would have been much more obvious that square roots are commonly limited to the positive roots even though they are both positive and negative. For example, the square roots of 4 are 2 and -2, but only 2 is commonly written down. But here it is arbitrarily written, in effect, that one side the root is only -1 while on the other it is only +1. It should have been +1 and -1 on both sides.

PS Another way of putting it: It is obvious that the square root of a^2 is both plus and minus a. But if we write (a)^2, the parentheses appear to exclude the negative root. The moral here is that parentheses are conveniences, not genuine mathematical operations.

[Lumi]

I love math. I'm terrible at it, but it's always fun trying to figure things out. More topics like this!

[Ood Sigma]

The real problem is in the transition from lines 3 to 4, in which it appears that they are taking the square root of both sides. You cannot do this, as there are two solutions to any square root (for example, the square root of 9 is both 3 and -3).

The third line is correct:
(3-4)^2 = (-1)^2 = 1
(5-4)^2 = (1)^2 = 1

But it is dependent on the fact that 1 has square roots of both 1 and -1.

[Asbo Zaprudder]

Yeah, a square root has two values. The error arises by choosing the wrong sign leading to the difference of 2 on each side. Write line 3 as (3-4)(3-4) = (5-4)(5-4) and it becomes obvious that you can properly divide through by either (3-4) or (5-4), for example:

(3-4) = (5-4)(5-4)/(3-4) = 1x1/-1 = -1

[Rhubarbodendron]

yep, that's what I pointed out in my last post, only less elaborate. It's quite difficult to explain mathematics in a foreign language. There are so many special terms you never learned at school.

Totally off-topic, but I just have to ask: what kind of spider is the one in your avatar, Asbo Zaprudder? It's pretty!

[stj]

A language note: The math term for taking a root is exponentiation. The exponent for taking a square root is 1/2.