Comment by MikeDelta - Hacker Neue

MikeDelta Dec 1, 2023 parent

Maybe you will find the proof that the infinite series 0.9999... exactly equals 1 interesting:

https://en.wikipedia.org/wiki/0.999...

jcul Dec 1, 2023

Wow, can't believe I've never realised this. How counterintuitive.

The 1/3 * 3 argument, I found the most intuitive.

quchen Dec 1, 2023

It's a flawed psychological argument though, because it hinges on accepting that 0.333...=1/3, for which the proof is the same as for 0.999...=1. People have less of a problem with 1/3 so they gloss over this - for some reason, nobody ever says "but there is always a ...3 missing to 1/3" or something.

Scarblac Dec 1, 2023

The problem is that there are two different ways to write the same number in infinite decimals notation. (0.999... and 1.000...).

Thats what's counter intuitive to people, it's not an issue with 1/3. That has just one way to write it as decimals, 0.333...

nly Dec 1, 2023

Another intuition:

All the decimals that recur are fractions with a denominator of 9.

E.g. 0.1111.... is 1/9

0.7777.... is 7/9

It therefore stands to reason that 0.99999.... is 9/9, which is 1

hoosieree Dec 1, 2023

That's a good one! Might replace my current favorite which is:

Let x = 0.99...

Then 10*x = 9.99...

And if we subtract x from both sides, we get:

10x - x = 9.99... - x

And since we already defined x=0.99... when we subtract it from 9.99..., we get

9x = 9

So we can finally divide both sides by 9:

x = 1

thaumasiotes Dec 1, 2023

I like the argument that observes "if you subtract 0.99(9) from 1, you get a number in which every decimal place is zero".

The geometric series proof is less fun but more straightforward.

As a fun side note, the geometric series proof will also tell you that the sum of every nonnegative power of 2 works out to -1, and this is in fact how we represent -1 in computers.

nly Dec 1, 2023

How can the sum of a bunch of positive powers powers of 2 be a negative number?

Isn't the sum of any infinite series of positive numbers infinity?

twiceaday Dec 1, 2023

https://youtu.be/krtf-v19TJg?si=Tpa3EW88Z__wfOQy&t=75

You can 'represent' the process of summing an infinite number of positive powers of x as a formula. That formula corresponds 1:1 to the process only for -1 < x < 1. However, when you plug 2 into that formula you essentially jump past the discontinuity at x = 1 and land on a finite value of -1. This 'makes sense' and is useful in certain applications.

xigoi Dec 1, 2023

The infinite sum of powers of 2 indeed diverges in the real numbers. However, in the 2-adic numbers, it does actually equal -1.

https://en.wikipedia.org/wiki/P-adic_number

Dylan16807 Dec 1, 2023

Eh, P-adic numbers basically write the digits backwards, so "-1" has very little relation to a normal -1.

2 More Comments →

defrost Dec 1, 2023

\1 is a good question that deserves an answer.

\2 is "not always" ..

Consider SumOf 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ...

an infinite sequence of continuously decreasing numbers, the more you add the smaller the quantity added becomes.

It appears to approach but never reach some finite limit.

Unless, of course, by "Number" you mean "whole integer" | counting number, etc.

It's important to nail down those definitions.

thaumasiotes Dec 1, 2023

> \1 is a good question that deserves an answer.

The same argument I mentioned above, that subtracting 0.99999... from 1 will give you a number that is equal to zero, will also tell you that binary ...11111 or decimal ...999999 is equal to negative one. If you add one to the value, you will get a number that is equal to zero.

You might object that there is an infinite carry bit, but in that case you should also object that there is an infinitesimal residual when you subtract 0.9999... from 1.

It works for everything, not just -1. The infinite bit pattern ...(01)010101 is, according to the geometric series formula, equal to -1/3 [1 + 4 + 16 + 64 + ... = 1 / (1-4)]. What happens if you multiply it by 3?

        ...0101010101
      x            11
    -------------------
        ...0101010101
     + ...01010101010
    -------------------
       ...11111111111

You get -1.

2 More Comments →

mr_mitm Dec 1, 2023

In an introductory course to String Theory they tried to tell me that 1+2+3+4+... = -1/12.

There is some weird appeal to the Zeta function which implies this result and apparently even has some use in String Theory, but I cannot say I was ever convinced. I then dropped the class. (Not the only thing that I couldn't wrap my head around, though.)

thaumasiotes Dec 1, 2023

The result isn't owed to the zeta function. For example, Ramanujan derived it by relating the series to the product of two infinite polynomials, (1 - x + x² - x³ + ...) × (1 - x + x² - x³ + ...). (Ok, it's the square of one infinite polynomial.)

Do that multiplication and you'll find the result is (1 - 2x + 3x² - 4x⁴ + ...). So the sum of the sequence of coefficients {1, -2, 3, -4, ...} is taken to be the square of the sum of the sequence {1, -1, 1, -1, ...} (because the polynomial associated with the first sequence is the square of the polynomial associated with the second sequence), and the sum of the all-positive sequence {1, 2, 3, 4, ...} is calculated by a simpler algebraic relationship to the half-negative sequence {1, -2, 3, -4, ...}.

The zeta function is just a piece of evidence that the derivation of the value is correct in a sense - at the point where the zeta function would be defined by the infinite sum 1 + 2 + 3 + ..., to the extent that it is possible to assign a value to the zeta function at that point, the value must be -1/12.

https://www.youtube.com/watch?v=jcKRGpMiVTw is a youtube video (Mathologer) which goes over this material fairly carefully.

This item has no comments currently.

Preferences

Keyboard Shortcuts

Story Lists

Navigation

Miscellaneous