Linked Questions
10 questions linked to/from Why does an argument similiar to 0.999...=1 show 999...=-1?
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Why does an argument similiar to $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...=1$ show that $2+4+8+...=-2$ [duplicate]
See how to prove $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...=1$
$x=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$
$2x=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$
Then:
$x=1$
Now I use the same argument to ...
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why does ...999 apparently equal -1? is this notation valid? [duplicate]
I debated 9... != 1 claims for years now, but the discussion surfaced once again, this time I asked myself: what if I "change the direction" of the recurring digit, i.e. add 9s BEFORE the ...
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How would you convince the child-me this is false: $0.\bar{9} = 1 \implies \bar{9} = - 1$? [duplicate]
So I remember as a child when I was taught: $ . \bar9 =1 $
The proof was taught as:
$$x = 0.\bar{9} \\
10x = 9.\bar{9} \\
10x - x = 9.\bar{9} - 0.\bar{9} \\
9x = 9 \\
x = 1 \\
\therefore 0.\bar{9} = ...
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answers
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$1 + 2 + 4 + 8 + 16 \ldots = -1$ paradox
I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1:
Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 \...
- 841
51
votes
5
answers
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Calculation with infinitely many operands
Recently I have come across the equation
$$x^{x^{x^{x^{...}}}} = 2$$
which I found out to be solvable using substitution:
$$u = x^{x^{x^{x^{...}}}} = 2$$
$$x^u = 2$$
$$x^2 = 2$$
$$x = \sqrt{2}$$
...
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19
votes
4
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Divergent series and $p$-adics
If we naïvely apply the formula $$\sum_0^\infty a^i = {1\over 1-a}$$ when $a=2$, we get the silly-seeming claim that $1+2+4+\ldots = -1$. But in the 2-adic integers, this formula is correct.
Surely ...
- 65.8k
4
votes
3
answers
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Wrong calculation by a calculator
Today I was working on some chemistry problems . Many of you might know it involves some big exponents. So today I just entered in one of my calculators (which might not be strong) as it was inbuilt ...
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Why is it considered that $ | \mathbb{Q} | = | \mathbb{N} | $ is true, but $ | \mathbb{R} | = | \mathbb{N} | $ is wrong?
Please clarify why $|\mathbb{Q}| = |\mathbb{N}|$, but $|\mathbb{R}| > |\mathbb{N}|$? Why do following arguments about $|\mathbb{R}| = |\mathbb{N}|$ are wrong?
Two sets are equinumerous if there is ...
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Why doesn't $\sum_{n=0}^\infty2^n=-1$?
Now of course I'm not stranger to the fact that adding finite (and in many cases - infinite) amount of positive numbers always yeilds a positive number, but in many cases, often the finite limit isn't ...
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Representing negative numbers with an infinite number?
Motivation
We all know that:
$$ .\bar{9} =.999 \dots= 1$$
I was wondering if the following (obviously not rigorous) statement could be defined on the same footing?
Question
$$ x = \bar{9} $$
$$ \...
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