Waste less time on Facebook — follow Brilliant.
×

What is the sum of all real numbers?

Note by Adharsh M
4 months ago

No vote yet
1 vote

  Easy Math Editor

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 \( 2 \times 3 \)
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

Sort by:

Top Newest

which wiki, which section, please explain

Adharsh M - 3 months, 3 weeks ago

Log in to reply

The absolutely convergent wiki that I linked to. Read through the entire wiki, esp since it sounds like you are unfamiliar with this concept.

Calvin Lin Staff - 3 months, 3 weeks ago

Log in to reply

why doesn't it work?

Adharsh M - 3 months, 3 weeks ago

Log in to reply

Read the wiki, esp the last section.

Calvin Lin Staff - 3 months, 3 weeks ago

Log in to reply

Mr.Calvin Lin, could you please explain, what actually you want to convey.

Adharsh M - 3 months, 3 weeks ago

Log in to reply

The sum of all real numbers is undefined. This is because the summation is not absolutely convergent, and hence we cannot assign a value to it.

Why does the naive approach of "pairing up \(x\) with \(-x\) to make the sum of infinitely many \( x -x = 0\)" not work?

Calvin Lin Staff - 3 months, 3 weeks ago

Log in to reply

if I say that all the real numbers will have their negative pair ,example 5 and -5,so similarly for infinity there will be -infinity which will sum to 0

Adharsh M - 3 months, 3 weeks ago

Log in to reply

That's a good thought. Under what scenarios can we do? (Maybe the question to answer first is: what are we doing here?)

Calvin Lin Staff - 3 months, 3 weeks ago

Log in to reply

is it 0?

Adharsh M - 4 months ago

Log in to reply

It's \(\infty\).

Munem Sahariar - 4 months ago

Log in to reply

Can you justify why?

Calvin Lin Staff - 3 months, 4 weeks ago

Log in to reply

@Calvin Lin Because there are infinitely many real numbers.

Munem Sahariar - 3 months, 4 weeks ago

Log in to reply

@Munem Sahariar And so?

Calvin Lin Staff - 3 months, 4 weeks ago

Log in to reply

@Calvin Lin Since there are infinitely many real numbers or uncountably many real numbers so we can't determine there sum.

Munem Sahariar - 3 months, 4 weeks ago

Log in to reply

@Munem Sahariar So, you are changing your answer from "\( \infty\)" to "undefined"?

Calvin Lin Staff - 3 months, 4 weeks ago

Log in to reply

@Calvin Lin No. \(\infty\) and undefined both are my answers.

Munem Sahariar - 3 months, 4 weeks ago

Log in to reply

@Munem Sahariar It is very rare for an equation to have 2 answers.

At the very most, it would be "Under different interpretations". For example, \( 1 + 2 + 3 + \ldots \) would be infinity in the usual calculus treatment of infinite sums, but could be \( - \frac{1}{12} \) under the analytic continuation of \( \sum n^s \). As such, can you clarify under which interpretation we get the answer of "\(\infty\)", and under which interpretation we get the answer of "undefined"?

Calvin Lin Staff - 3 months, 3 weeks ago

Log in to reply

No, it is not.

Calvin Lin Staff - 4 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...