The Uninteresting Number!

Every number is interesting. Take for example :

  • 11 is neither a prime nor composite
  • 22 is the first & the only even prime
  • 33 is the first odd prime
  • 44 is the first composite number

& so on.

But is there an uninteresting number? Let us find out.

Let UU be the set of all the uninteresting numbers.
Obviously, this set has a smallest element.

But such a number would be interesting because it is the first uninteresting number!

Hence, EVERY number is interesting!

Note by Ameya Salankar
5 years, 6 months ago

No vote yet
1 vote

  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.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. 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 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"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

Obviously, this set has a smallest element.

This is true if the word 'number' means a non-negative integer in this context. The well ordering principle works for non-negative integers only.

This seemingly pradoxical result arises because 'interesting' -ness is not a well-defined property. Would a number be considered interesting if it's uninteresting? Can something have a property AA by not having the property AA?

Mursalin Habib - 5 years, 4 months ago

Log in to reply

This is exactly how Agnishom Chattopadhyay won against me in a debate- " Are all people interesting" He was in the proposition and came up with something similar and said that being a bit uninteresting is in itself an interesting property...thus :P

Krishna Ar - 5 years, 4 months ago

Log in to reply

That was pretty interesting :)

Agnishom Chattopadhyay Staff - 5 years, 4 months ago

Log in to reply

Nice proof!

A Brilliant Member - 5 years, 4 months ago

Log in to reply

what about the 2nd unintresting number??????????? :P

Lokesh Naani - 5 years, 4 months ago

Log in to reply

Its interesting for being the 2nd uninteresting number :D

Shreya R - 5 years, 4 months ago

Log in to reply

@Shreya R quite true!

Ameya Salankar - 5 years, 4 months ago

Log in to reply

Using your method of contradiction, you have thrown away a seemingly extraneous solution.

It is more logically sound to say that all numbers are uninteresting, because in this case, a number isn't interesting, they are all uninteresting.

You've also stated

Obviously, this set has a smallest element

which is an assumption.

So really, no numbers are interesting.

Finn Hulse - 5 years, 4 months ago

Log in to reply

@Finn Hulse,

Obviously, this set has a smallest element

is not an assumption! It's common sense!

How come a set doesn't have a smallest element?

Ameya Salankar - 5 years, 4 months ago

Log in to reply

Careful! A set doesn't have to have a smallest element. There is no smallest element in the set S={x:x is a country}S=\left\{ x : x \text{ is a country} \right\}.

Even if you're talking about sets of numbers, the statement is not always true.

For example: A={x:xR,2<x3}A=\left\{ x : x \in \mathbb{R}, 2<x\leq 3\right\} has no smallest element.

The well ordering principle works on non-negative integers only.

Mursalin Habib - 5 years, 4 months ago

Log in to reply

@Mursalin Habib OK! Looks like we have found a fallacy to the proof!

Ameya Salankar - 5 years, 4 months ago

Log in to reply

@Ameya Salankar That is not the fallacy actually. It is understood that the word 'number' means non-negative integer in this context. The real issue here is that 'interesting' is not well defined.

Here's a more interesting [pun intended!] variant of this.

Is it possible to describe every non-negative integer with 1515 words or less?

Here's an example. 42949672974294967297 is the first Fermat number that is not a prime. This description uses less than 1515 words. Is it possible to do that for every number?

Let's assume the contrary. Then set of numbers that can not be described with 1515 words or less is non-empty.

Now take the smallest element in this set. This is the smallest number that can not be described with 1515 words or less. Wait! That is a way to describe that number! Let's see how many words we used. I'll be damned! We used exactly 1515 words! This number does not belong in this set now. In a similar manner it can be shown that no number can be in this set.

The problem with this whole argument is that 'description' is not well defined. What counts as a 'description'? Is it okay for a description to be inconsistent?

Mursalin Habib - 5 years, 4 months ago

Log in to reply

@Mursalin Habib @Mursalin Habib, I knew this one. But there is something wrong with the proof. (The uninteresting number one)

Ameya Salankar - 5 years, 4 months ago

Log in to reply

@Mursalin Habib Yes,that's absolutely right.

Krishna Ar - 5 years, 4 months ago

Log in to reply

Yup...otherwise you're defying well ordering principle!!!

Krishna Ar - 5 years, 4 months ago

Log in to reply

What if it is an empty set? Then... :O

Finn Hulse - 5 years, 4 months ago

Log in to reply

@Finn Hulse @Finn Hulse, I forgot that one ... :D

Ameya Salankar - 5 years, 4 months ago

Log in to reply

Hi-fi! We have the same number of followers! :D ^_^

Krishna Ar - 5 years, 4 months ago

Log in to reply

@Krishna Ar @Krishna Ar, you have caught up! Initially, you had less number of followers. Great job! Keep it up!

Ameya Salankar - 5 years, 4 months ago

Log in to reply

@Ameya Salankar Thank you ^_^. Initially I had way too less number of followers, yes now I have reached a decent level :D

Krishna Ar - 5 years, 4 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...