# Sum

What is the sum of all real numbers?

1 year, 8 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:

is it 0?

- 1 year, 8 months ago

No, it is not.

Staff - 1 year, 8 months ago

It's $$\infty$$.

- 1 year, 8 months ago

Can you justify why?

Staff - 1 year, 8 months ago

Because there are infinitely many real numbers.

- 1 year, 8 months ago

And so?

Staff - 1 year, 8 months ago

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

- 1 year, 8 months ago

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

Staff - 1 year, 8 months ago

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

- 1 year, 7 months ago

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?

Staff - 1 year, 7 months ago

why doesn't it work?

- 1 year, 7 months ago

Read the wiki, esp the last section.

Staff - 1 year, 7 months ago

which wiki, which section, please explain

- 1 year, 7 months ago

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

Staff - 1 year, 7 months ago