×

Problem with numbers

I read somewhere that

"Q is countable but R is not" (reference-R means real numbers and Q means rational numbers)

On thinking i found that this statement doesn't make any sense.Or does it??? Can you explain this statement???

(I WOULD BE PLEASED IF EXPLANATION IS LUCID WITH SOME CRUNCH OF ADVANCED MAtHEMATICS)

Note by Somesh Rout
4 years, 9 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

$$Q$$ and $$R$$ are both infinite sets. But $$Q$$, the set of rational numbers, is countably infinite. On the other hand, $$R$$, the set of real numbers is an example of an uncountable infinity. So far, this shouldn't make much sense, so bear with me.

Even though rational numbers are found everywhere on the number line, they leave a lot of gaps. Think about this: if you had an infinitely thin pin and started to stab a piece of paper with it, you'd make a lot of holes but you'd never be able to destroy the paper. The rational numbers are like those pin pricks on the number line. Using a countable infinity like the set of rational numbers, you can’t construct any kind of continuous set.

However, with real numbers, you can make a continuous set (the number line). This is called an uncountable infinity.

This post here goes into much more detail. Check it out and you should have all your questions answered.

The point to be noted here is that the word 'countable' is not being used in the usual sense. In the usual sense, the elements of an infinite set can't be counted (unless you're given infinite time). Hope this helps!

- 4 years, 9 months ago

I knew the countably and uncountable infinite stuff,but your explanation with the pin is amazing.just loved it.

- 4 years, 9 months ago

You're welcome. But the credit goes to the post I mentioned earlier.

- 4 years, 9 months ago

thanks for the great explanation

- 4 years, 9 months ago

You're welcome.

- 4 years, 9 months ago