### Brain Teasers

# 1 = 2

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 left-hand 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?

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 left-hand 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?

### Hint

Give a and b a value and look at the numbers at each step.### 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.Hide Hint Show Hint Hide Answer Show Answer

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## Comments

BLAH BLHA BLAH BORING

i liked it... maybe im just a nerd

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!

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.

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.

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.

this was soooooooooooooooo boring i hated it i rated it as BORING and o didnt vote on diffuculty

ok i dont get it.im in the third grade here.

I've always wanted to prove this one wrong!!

Eh, didn't get it. `

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

http://www.braingle.com/2013.html

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.

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.

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.

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.

My teacher had to ruin a possibly great problem by introducing and immediately solving this one, when i was first learning algebra.

Nice problem. Didn't find it particularly hard, but still good to work the old grey matter.

More like this, please!

More like this, please!

I'm CONFUZZLED

This wuz sure stupid one.nerdy.

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^2-2ab=ab+a^2-2ab

2a^2-2ab=a^2-ab

2(a^2-ab)=a^2-ab

2=1

Of course, it doesn't work because when you divide by a^2-ab at the end, you're dividing by 0.

a=b

a^2=ab

a^2+a^2=ab+a^2

a^2+a^2-2ab=ab+a^2-2ab

2a^2-2ab=a^2-ab

2(a^2-ab)=a^2-ab

2=1

Of course, it doesn't work because when you divide by a^2-ab at the end, you're dividing by 0.

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