Math Is Fun Forum

Khushboo

Member

Registered: 2007-10-16

Posts: 47

How are fractions different from the numbers?

Hi

Other day i was trying to explain fractions and the three distinct component. I started by saying fractions as an entity just like the numbers but added that they have three distinct component whereas the numbers donot have. I then found that you cannot compare number with fractions, the reason being the number can be infinitely large and fractions can also be infinitely large so where is the comparision.

This is so very simple but really caught my fancy for sometime.

Regards

Khushboo

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: How are fractions different from the numbers?

I'm not sure that I understand your question completely, but it sounds like you're looking for a way to match each natural number to a fraction, and vice versa.

If that is indeed what you are asking for, you're talking about the "countability" of different types of numbers.  As you pointed out, there are infinitely many natural numbers and infinitely many fractions.  However, we say that the set of fractions is "countably infinite" because we can create a function that maps every fraction to a natural number which is also invertible, meaning that we can create a bijection from the natural numbers to the fractions.  This Wikipedia article explains it further:

http://en.wikipedia.org/wiki/Countably_infinite

Sorry if this isn't what you were asking.

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Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: How are fractions different from the numbers?

I then found that you cannot compare number with fractions, the reason being the number can be infinitely large and fractions can also be infinitely large so where is the comparision.

Sounds like that's a great comparison, not a reason why you can't compare them.

There are spaces between integers (whole numbers).  For example, there aren't any whole numbers between 1 and 2, just a big empty space.  Fractions are numbers which take up these spaces.  And in fact, whole numbers are perfectly valid fractions as well.  5 = 5/1

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."