Rational Number

Hey Everyone! Is it correct to speak that the denominator of 0 is not defined, as 0 = 0/1 = 0/2 = 0/3.... Actually my teacher asked me the denominator of and I said that its not defined but she said it 1. Please reply soon and Thank You for answering in advance.

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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

Math	Appears 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}`	$\boxed{123}$

Comments

Excellent question, Anoopam!

You might be thinking that the definition for rational number fails at the case of zero. Rational numbers are usually defined as $\{ n\in \mathbb{Q}\iff n=\frac{p}{q}$ and $(p,q)=1 \}$ . So if $0=\frac{0}{n}, n\ge 1$ then there's no problem as such.

But your definition does have some problems. If you see here you'll see the definition of $(0,n)$ is not universal. So the $(p,q)$ condition might fail.
Also the $(p,q)$ condition is imposed to to get unique representation for each rational number which fails in your case. In fact this 'unique' representation property is very useful to show bijections from rational to different sets.
If we make a more subjective treatment and see how we 'construct' rational numbers from positive integers, we'll see the usual bijection which is followed is $f(1)=0, f(2)=1, f(3)=-1 , f(4)=-2, f(5)=-\frac{1}{2}...$ . Thus $0$ is in fact taken as $\frac{0}{1}$ . [See fig. for the bijection]

But in my opinion, there is no problem if we define zero to be vacuously a rational number and it doesn't 'need' to be represented as $\frac{p}{q}$ .

Abhishek De - 8 years, 2 months ago

Please review your posting. You first mention "denominator of 0 is not defined", but in your example you list it with 0 as the numerator, i.e. " $0 = 0/1 = 0/2 = 0/3 \ldots$ ." Both of your statements are true, but express vastly different ideas.

I am also not certain what "but she said it 1" means.

Calvin Lin Staff - 8 years, 2 months ago

She said it 1

He means that she ( his teacher) had said that the denominator should be 1.

Abhishek De - 8 years, 2 months ago

Math	Appears 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}`	$\boxed{123}$