New Math

Reductionism, the philosophical position that a complex system is nothing other than the sum of its parts, has largely fallen out of favor and for good reason.  Quantum physics and phenomena such as chaos theory and emergent network effects have effectively closed the book on the notion that truth can be found by setting aside the whole and just analyzing fundamental elements.

Given this preamble, I was recently reminded of a pretty cool algebra trick that I first came across about 15 years ago.  Starting with a = b as a given, this series of simple algebraic manipulations proves that 1 = 2; a principle I wish I could apply to my 401k.

a  =  b
ab  =  b2
ab – a2  =  b2 – a2
a(b – a)  =  (b – a)(b + a)
a  =  b + a
a  =  a + a
a  =  2a
1  =  2

Clearly there’s a problem.  For readers who have not seen this and want to solve it, I’ll offer the following assurance without being a spoiler.  There is nothing lame here.  There are no hidden assumptions like this only works if you’re a subatomic particle or if you redefine the symbology of numbers.  The problem is simple and includes everything you need to know.

The frustrating part about this problem is that each individual step from one expression to the next is impeccable in its correctness, but ultimately 1 cannot equal 2.  There is of course an explanation, but it cannot be found solely in the exhaustive analysis of each step.

There is no profound moral to this post; just an observation that connecting dots can be more rewarding than just knowing each dot on a first name basis.  Said another way, if you’re beating your head against a wall because you don’t know where to go, try stepping back and reading the sign.

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2 comments so far

  1. Muhammad Alkarouri on

    “The frustrating part about this problem is that each individual step from one expression to the next is impeccable in its correctness”

    No it’s not.
    Step 5 doesn’t follow from 4. Of course you can cancel out any nonzero factor from both sides, but is (b-a) nonzero?

  2. drjbutler on

    Depends how you look at it. In isolation, this step just divides both sides by a factor, in this case (b-a); nothing wrong with that. But when this is combined with the given that a=b, it means that both sides are being divided by 0. After that, you can claim that a duck equals a garden hose.


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