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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