Mathematical Conundrums

Conundrums are known as problems that are difficult or confusing. One of the more widely known mathematical conundrum is whether 0.999... is equals to 1.  Below is an idea of it for those of you who do not know what it is about:

(Image from Wikipedia.)

Comparing the first and last sentences from the above, you see that 0.999... = 1. (Amazing!!)

There are many more mathematical conundrums that you can sprawl through on the internet. For now, we will just share with you two of the more easier to understand ones.

The first one
What is 1 + 2 + 4 + 6 + 8 + 16 + ...?

Here, the "..." means continuing to sum forever, adding twice as much each time. You would think that this sum must be infinite. Think again...

Let's multiply the sum above by 2. This will give us 2 x (1 + 2 + 4 + 6 + 8 + 16 + ...) = 2 + 4 + 6 + 8 + 16 + ... which looks a lot like the original sum with the 1 missing at the start.

This brings us to conclude that 2 x (1 + 2 + 4 + 6 + 8 + 16 + ...) is 1 less than (1 + 2 + 4 + 6 + 8 + 16 + ...). That is to say that

2 x (1 + 2 + 4 + 6 + 8 + 16 + ...) - 1 x (1 + 2 + 4 + 6 + 8 + 16 + ...) = -1.

Simplifying the left hand side will give us

1 + 2 + 4 + 6 + 8 + 16 + ... = -1.

How is it that adding bigger and bigger numbers can lead us to negativeland a.k.a negative one? Confused yet? If not, read on...

The second one

What is the value of the infinite sum 1 - 1 + 1 - 1 + 1 - 1 + ...?

If we add in brackets, we can observe that 

(1 - 1) + (1 - 1) + (1 - 1) + ... = 0 + 0 + 0 + ... 

which means that the value of the infinite sum is 0. 

Then again, if we express 1 - 1 + 1 as 1 - (1 - 1), then we can rewrite the infinite sum as 

1 - (1 - 1) - (1 - 1) - (1 - 1) ... = 1 - 0 - 0 ...

which means that the value of the infinite sum is 1.

So is it 0 or 1? Or 0 half the time and 1 the other half the time? You must be confused by now...

We shall end here to let your confused minds ponder over things. 

If you want to know more about the above two conundrums, you can read about the first conundrum here and the second conundrum here. Links open in a new windows.

Reference: Jordan J. Ellenberg, J. (2014). How Not To Be Wrong The Hidden Maths Of Everyday Life. Great Britain: Penguin. (Great book about mathematics by the way! Go read it!)