### 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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