Home > Maths > Proving 1=2 and 2=3

Proving 1=2 and 2=3

By: Suzanne Elvidge BSc (hons), MSc - Updated: 24 Aug 2012 |
 
Mathematics Zero Divide Subtract Add

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.

You might also like...
Share Your Story, Join the Discussion or Seek Advice..
Why not be the first to leave a comment for discussion, ask for advice or share your story...

If you'd like to ask a question one of our experts (workload permitting) or a helpful reader hopefully can help you... We also love comments and interesting stories

Title:
(never shown)
Firstname:
(never shown)
Surname:
(never shown)
Email:
(never shown)
Nickname:
(shown)
Comment:
Validate:
Enter word:
Topics
Latest Comments
Further Reading...
Our Most Popular...
Add to my Yahoo!
Add to Google
Stumble this
Add to Twitter
Add To Facebook
RSS feed
You should seek independent professional advice before acting upon any information on the ScienceProjectIdeas website. Please read our Disclaimer.