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What is the sum of all real numbers?

Note by Adharsh M 1 year, 8 months ago

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2 \times 3

2^{34}

a_{i-1}

\frac{2}{3}

\sqrt{2}

\sum_{i=1}^3

\sin \theta

\boxed{123}

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is it 0?

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No, it is not.

It's \(\infty\).

Can you justify why?

@Calvin Lin – Because there are infinitely many real numbers.

@Munem Shahriar – And so?

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

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

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

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?

why doesn't it work?

Read the wiki, esp the last section.

which wiki, which section, please explain

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

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`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

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TopNewestis it 0?

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No, it is not.

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It's \(\infty\).

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Can you justify why?

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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

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

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

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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?

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why doesn't it work?

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Read the wiki, esp the last section.

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which wiki, which section, please explain

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

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