Yes, Virginia, 2 DOES equal 1!

Are you ready for this?  A proof that 2 = 1!  Now, don’t freak out, all you need is to be able to follow elementary Algebra.  I’ll help you along the way.  Here goes:

1) Let a = b      (A legitimate assumption one can make in Mathematics)

2) a^2 = ab     (Multiply both sides by a)

3) a^2 – b^2 = ab – b^2   (Subtract b^2  from both sides)

4) (a – b)(a + b) = b(a – b)     (Factoring)

5) a + b = b    (Divide both sides by (a – b)

6) b + b = b    (Substitution, remember step 1?)

7) 2b = b     (Simplifying step 6)

8) 2 = 1 !!! (Divide both sides by b)

        And there you have it, sort of shakes you up a bit, no?  If one can prove that 2 = 1, imagine what other fundamental truths can be undone?  This defies logic, order and everything mathematical and yet, there it is.  Can you discover the fallacy?  I’m not going to give it away too soon but …. if you need a hint, reread my previous blog.

Coming up….. a change of pace from Math to ….. Something else!

10 Comments

  1. I’ve been surprised by how many people intuit that one; that both sets are equally large. Their intuition is based on how both are infinite series, rather than the 1:1: mapping between the sets, but still. Taking them to the next level with real numbers… that’s when they start scratching their heads! 😉

    Do you know about The Hilbert Hotel?

    • Yes, and I plan to take a trip to it soon; possibly traveling through the Phantom Tollbooth to get there. I will travel on my bus with an infinite number of other travelers and I’m sure that we will all find a room! .

      • Sounds like a fun trip to me! I’ll be looking forward to it!

  2. And we thought Godel was smart! (if A=B than A-B=0 and in step five you are dividing by 0, which is a classic no-no)

    • What he said! 🙂

      Sorry, Virginia, 2 ≠ 1. But did you know 0.999… does equal 1?

      It does (really), and it’s easy to prove. (I won’t explain how in case you’re planning a blog post on it.)

      • That’s a really tricky one…..my former students were always shocked when I showed them that…perhaps it will show up in one of my future blogs. Thanks for your interest!

      • No doubt! It’s like Cantor’s proof there is more than one kind of infinity. Some people just can’t wrap their heads around it.

      • Oh wow. Now my head hurts.

    • It’s no surprise to me that you saw this immediately. You’ve got a good head for math (and other interesting things!)

      • More than one kind of infinity — yup! And which set has more numbers: the whole numbers or the even numbers? Students love that one!

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