This little guy craves the light of knowledge and wants to know why 0.999… = 1. He wants rigour, but he does accept proofs starting with any sort of premise.
Enlighten him.
and wants to know why 0.999… = 1
\begin{align} 0.999… &= 9\cdot(0.1+0.01+0.001+… ) \ &= 9\cdot( 0.1^1 + 0.1^2 + 0.1^3 + … ) \ &= 9\cdot(\sum\limits_{k=1}^\infty ( \frac{1}{10^k} ) ) \ &= 9\cdot(\sum\limits_{k=0}^\infty ( \frac{1}{10^{(k+1)}}))\ &= 9\cdot(\sum\limits_{k=0}^\infty \frac{1}{10}*(\frac{1}{10^k})) \ &= \frac{9}{10}\cdot (\sum\limits_{k=0}^\infty (\frac{1}{10^k})) \ &= \frac{9}{10}\cdot \frac{1}{(1-(\frac{1}{10}))}\ &= \frac{9}{10}\cdot \frac{10}{9} = 1 \end{align}
The crux rests on a handy result on from calculus: the sum of an infinite geometric series looks likes s = 1/(1-r), when s = \sum\limits_k=0^inf r^k, and |r| < 1.
Sorry for the latex. When will hexbear render latex? This is a bit more readable:
(aesthetic edit for our big beautiful complex analysts)
No. 0.99 is 0.9+0.09. The proof I gave shows that 0.99999999999999999999999999999999999(…) is equal to 1.
He’s hungry
Feed him
0.999…=1 because it’s funny that people get so mad about it
I agree, but the little guy does need rigour, or he will starve under crapitalism.
if 0.333… = 1/3 then 0.999… = 3/3 = 1
I suppose I will post one myself, as I do not expect anybody else to have that one in mind.
The decimals ‘0.999…’ and ‘1’ refer to the real numbers that are equivalence classes of Cauchy sequences of rational numbers (0.9, 0.99, 0.999,…) and (1, 1, 1,…) with respect to the relation R: (aRb) <=> (lim(a_n-b_n) as n->inf, where a_n and b_n are the nth elements of sequences a and b, respectively).
For a = (1, 1, 1,…) and b = (0.9, 0.99, 0.999,…) we have lim(a_n-b_n) as n->inf = lim(1-sum(9/10^k) for k from 1 to n) as n->inf = lim(1/10^n) as n->inf = 0. That means that (1, 1, 1,…)R(0.9, 0.99, 0.999,…), i.e. that these sequences belong to the same equivalence class of Cauchy sequences of rational numbers with respect to R. In other words, the decimals ‘0.999…’ and ‘1’ refer to the same real number. QED.
Had a little trouble following this as plain text, so I wrote it up in LaTeX (it’ll be a bit small if you try to read it inline–you’ll probably want to tap to enlarge on mobile or open the image in a new tab/click this direct link to the image):
I tried to hew as closely to your notation as I could, but let me know if you spot any errors!
Doesn’t that just mean they’re both elements of a Cauchy sequence rather than equivalent? I suck at maths.
Not quite. The wording “equivalence classes of … with respect to the relation R: aRb <==> lim( a_n - b_n) as n->inf” is key.
https://en.wikipedia.org/wiki/Equivalence_class
loosely, an equivalence relation is a relation between things in a set that behaves the way we want an equal sign to
For an element in a set, a, the equivalence class of a is the set of all things in the larger set that are equivalent to a.
So, under the relevant construction of the space of real numbers, every real number is an equivalence class of Cauchy sequences of rational numbers with respect to the relation R outlined in my comment. In other words, under this definition, a real number is an equivalence class that includes all such sequences that for every pair of them the relation R holds (and R is, indeed, an equivalence relation - it is reflexive, symmetric, and transitive, - that is not hard to prove).
We prove that, for the sequences (1, 1, 1,…) and (0.9, 0.99, 0.999,…), the relation R holds, which means that they are both in the same equivalence class of those sequences.
The decimals ‘1’ and ‘0.999…’, under the relevant definition, refer to numbers that are equivalence classes that include the aforementioned sequences as their elements. However, as we have proven, the sequences both belong to the same equivalence class, meaning that the decimals ‘1’ and ‘0.999…’ refer to the same equivalence class of Cauchy sequences of rational numbers with respect to R, i.e. they refer to the same real number, i.e. 0.999… = 1.
