## Is -1 a positive number?

Posted by Jithin on December 20, 2008

I read this in Futility Closet. This is the blog of this week (#BOTW#). Before giving the proof for -1 being a positive number, let me say something about this blog. I stumbled upon it long back. This is not a to-be-followed-seriously blog, but a blog to give life to tired grey cells. A huge collection of anectodes, puzzles, humour, history, science and math.

A very SFW (Safe for Work) blog that gives a good break to your taxing office routine. The collection is wonderful indeed.

What FC says about Itself

No matter how much you love your job, you can’t work for 10 hours straight.

That’s why there’s Futility Closet — 3,000 anecdotes, epigrams, illusions and wonders; puzzles, prodigies, sublimities and horrors.

Read a few and get back to work. We’ll keep finding more.

– Greg Ross, proprietor

Now for the proof.

Let X = 1+2+4+8+16+32+…. (an infinite series)

since there is no chance that there will be negative numbers in the series, hence X should be positive. I agree to this.

Now multiply both sides by 2. Then we have the following.

2X = 2+4+8+16+32+…(another infinite series)

here the right hand side i.e. 2+4+8+16+32+…. is equal to X-1 from the first relation. OK. Seems to be true. Yes, it is correct.

So, mathematically 2X = X-1 is true and this results in X=-1.

From the first statement it is agreed that X is positive. So since -1 is equal to X, hence -1 is **positive**. Huh? 😮

Still pondering.

## pizer said

Of course, the proof is flawed (like every other joke proof). So, in case you want to find the error yourself STOP reading this comment NOW. 🙂

You’re assuming that the infinite series 1+2+4+8+… has a limit value where in fact it has not. So, X isn’t really a number. Both your series (“X”, “2X”) have no bounds. They approach infinity. What you did is subtracting infinity from infinity which has no meaningful definition.

## Jithin K.Rajeev said

Finally I figured that out. Had an afternoon nap that cleared up the thing. When infinity comes into play then mischief comes along. Yes, X is not just positive it is infinity.

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