1 = 2
Math brain teasers require computations to solve.
Bob and Sue were having a discussion one day about 1 and 2. Sue reckoned she had seen a way that you could make 1 = 2 but Bob who wasn't called 'logical Bob' for nothing was obviously adamant that this just wasn't possible. Sue said "Ok smarty pants here's the equation to prove you wrong"...
Assumptions: There are two numbers, a and b, and they are equal.
Proof:
a = b
Multiply the equation by b
ab = b^2
Subtract a^2 from the equation
ab  a^2 = b^2  a^2
Factor difference of squares
ab  a^2 = (b  a)(b + a)
Factor a on the lefthand side
a(b  a) = (b  a)(b + a)
Divide by common factor
a = (b + a)
Substitute a for b
a = (a + a)
Simplify
a = 2a
Divide by common factor
1 = 2
Question:
Bob didn't take long and found the line where there is a crucial mistake in the working though was understanding of Sue's issue as it is well hidden... can you see it?
HintGive a and b a value and look at the numbers at each step.
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Answer
Going from a(b  a) = (b  a)(b + a) to a = (b + a) is not allowed, because the equation is divided by (b  a), which is equal to 0, given that a = b.
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Comments
katiebeth123
Mar 10, 2005
 BLAH BLHA BLAH BORING 
cdizzlefoshizzl
Mar 10, 2005
 i liked it... maybe im just a nerd 
sosweet487
Mar 11, 2005
 I did this in my math class a couple years ago. It duped me then and I didn't get it now until I saw the hint. The hint gave it away... but still a good one! 
castanz
Mar 11, 2005
 BTW People here say things are nerdy or geeky. Well, Duh, look at the name of the site. Yea, I is a nerd to and proud of it. 
castanz
Mar 11, 2005
 Yes, I know. The rest of the nerds are saying "to" is supposed to be "too" in my last comment. And the really nerdy ones are saying it should be "I am a nerd". I like math and chess btw. 
(user deleted)
Mar 11, 2005
 You can prove 0.999... = 1. Say y = .999..., then 10y = 9.999..., So, 10y  y = (9.999...)  (0.999...) = 9y. Then, 9y = 9, that implies y = 1. Therefore, 0.999... = 1. 
poopie
Mar 12, 2005
 this was soooooooooooooooo boring i hated it i rated it as BORING and o didnt vote on diffuculty 
runscapekiller
Mar 12, 2005
 ok i dont get it.im in the third grade here. 
melilyn
Mar 12, 2005
 I've always wanted to prove this one wrong!! 
koolkat57
Mar 16, 2005
 Eh, didn't get it. ` 
Sane
Mar 20, 2005
 Thankyou for duplicate searching...I can't believe this slipped by even when one of the editors was the one who submitted the original one.
http://www.braingle.com/2013.html 
munchtipq
Mar 21, 2005
 Whoops, I did try to search, but I (stupidly) was searching on 1=2, not 2=1, so I missed the title of the other one. My bad. 
Sane
Mar 21, 2005
 lol, ya I don't blame ya. It just surprises me how Mogmatt16, an editor (and creator of the original), let it slip.
Just be sure when seaching for a duplicate, don't just enter stuff like 1=2. Because if it even was "1 = 2" the search wouldn't pick it up because of the spaces. I think I searched for "what is wrong" or something along those lines in the answer field. 
sotnakitt
Mar 23, 2005
 This is a fairly old problem i believe, lol i mean i knew the answer before i read the problem.
My teacher had to ruin a possibly great problem by introducing and immediately solving this one, when i was first learning algebra. 
robinmckenzie
Mar 25, 2005
 Nice problem. Didn't find it particularly hard, but still good to work the old grey matter.
More like this, please! 
trumpetduck13
Apr 03, 2005
 I'm CONFUZZLED 
Ms50cent
Apr 17, 2005
 This wuz sure stupid one.nerdy. 
brianz
Apr 23, 2005
 By the way, there's another one that's similar to this.
a=b
a^2=ab
a^2+a^2=ab+a^2
a^2+a^22ab=ab+a^22ab
2a^22ab=a^2ab
2(a^2ab)=a^2ab
2=1
Of course, it doesn't work because when you divide by a^2ab at the end, you're dividing by 0. 
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