Proving 1=2 and 2=3

Mathematicians use mathematical proofs to prove that what they say about numbers is true. Some proofs look true but aren’t – these are known as fallacies. Here are a few fallacies:

Proving 1=2

• Start by imagining that a = b
• Multiplying both sides by a, then a x a = a x b
• Adding (a x a) to both sides, then (a x a) + (a x a) = (a x a) + (a x b)
• This can be rewritten as 2(a x a) = (a x a) + (a x b)
• Subtracting 2(a x b) from both sides gives 2(a x a) - 2(a x b) = (a x a) + (a x b) - 2(a x b)
• Simplifying this gives 2(a x a) – 2(a x b) = (a x a) – (a x b)
• Factoring out the 2 on the left hand side gives 2((a x a) – (a x b)) = 1((a x a) – (a x b)
• Dividing both sides by ((a x a) – (a x b)) means that 2 = 1

What’s wrong? The final step cannot work because it is not possible to divide a number by zero, and if a = b, then (a x a) – (a x b) = 0.

All Numbers are Equal

• When multiplying any number by zero, the answer is zero, so 1 x 0 = 0
• Rearranging this gives 1 = 0 / 0
• 2 x 0 = 0
• Rearranging this gives 2 = 0 / 0
• 3 x 0 = 0
• Rearranging this gives 3 = 0 / 0
• Therefore – all numbers are equal

What’s wrong? This cannot work because, as before, it is not possible to divide a number by zero.

Making Four Equal 12

• Start by imagining that a = b
• Multiply both sides by four, so 4a = 4b
• Multiply both sides by 12, so 12a = 12b
• Multiply both sides of the first equation by a, so 4(a x a) = 4(a x b)
• Multiply both sides of the second equation by b, so 12(a x b) = 12(b x b)
• Subtract the second equation from the first equation to make one equation, so 4(a x a) – 12(a x b) = 4(a x b) – 12(b x b)
• Subtract 4(a x b) from both sides and add 12 (a x b) to both sides, so 4(a x a) – 4(a x b) = 12 (a x b) – 12 (b x b)
• Add a x b and subtract (b x b) from both sides, so 4(a x a) – 4(a x b) + (a x b) – (b x b) = 13 (a x b) – 13 (b x b)
• Taking (factoring) out common factors (dividing by (a-b)), so 4a(a – b) + b(a – b) = 13b(a – b)
• Taking (factoring) out common factors, so 4a + b = 13b
• Subtracting b from both sides, so 4a = 12b
• a = b, so substituting a for b, 4b = 12b, so therefore 4=12

What’s wrong? The step of dividing by (a - b) cannot work because it is not possible to divide a number by zero, and if a=b, then (a - b) = 0.