Ad Infinitum And Common Sense

Very often people rely on their "common sense" to answer questions or to make judgements. However good or bad this methodology is will ultimately depend on a person`s knowledge and experience.

Problem:-

Do the 3`s in the evalution of the fraction 1/3 continue forever?

In other words; Are the 3`s in the evaluation of 1/3 infinite?

Calculate 1/3 = 0.3333333333333333333333333333333333333333333...

On a good calculator you could try multiplying by 1000, subtracting 333 and see if you are left with recurring 3`s. The answer is yes.

So, the 3`s are infinite. They do go on forever. True.

Then why doesn`t 1/3 exceed itself?

After all, isn`t it "common sense" that if you add something to a thing, the thing will get bigger. And, if you add an infinite number of things, shouldn`t it become, in the end, infinite?

Yet 1/3 is certainly not infinite!

It is definitely finite. Measureable. For example: 1/3 of 6 = 2.

The solution:-

If I first appeal to your common sense. Imagine that each extra `3` that we place into the recurring pattern is smaller than the one before; infact 10x smaller. Then you can `see` that although we`re adding extra bits on, those bits are getting a lot smaller.

As we add `infinite` bits on, they are themselves becoming infinitely smaller. This balances out the increase with the decrease, causing convergence of the series to a point = 1/3.

Now try typing in on your calculator:-

1/4+1/16+1/64+1/256+1/1024...= 0.333... Series "S"

To generate these fractions we start by imagining a 1 x 1 square.

Split the square into 4 smaller squares [horizontally & vertically].

Shade 1 small square, leaving 3 unshaded. Of the 3, the corner one has dimensions of 1/2 x 1/2; as do they all at this point. It`s area is therefore 1/4.

1/4 is the first number in the above series "S".

Repeat the above proceedure for the remaining, shaded square, and you will form a smaller square of size 1/4 x 1/4 with an area of 1/16.

Half the size of the next square, and you get a samller square of size 1/8 x 1/8 = 1/64.

As you sum these reducing fractions, the calculator display will gradually fill with recurrent 3`s.

Also, as the diagram will display, each shaded square is 1 and the remaining unshaded squares are 3. There exists an inherent ratio of 1:3.

Common Sense?