998,001 and its Mysterious Recurring Decimals - Numberphile

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Recommends davidlowryduda · 2012-12 · Mathematics Stack Exchange · on “What is special about the numbers 9801, 998001, 99980001 ..?”

Since this was recently brought back to the front page, I wanted to point out a particular numberphile blog video on this exact question [this video] . In this video, Brady Haran interviews Dr. James Grime and they discuss why this happens and how to generalize the result. If I were to briefly summarize the key idea: in the video, they show that this boils down to considering $\dfrac{12345679}{999999999} = \dfrac{1}{81}$, or to evaluating $\dfrac{1}{(1-x)^2}$ at $x = \frac{1}{10}$ and considering a sort of generating functions.

Recommends user170231 · 2015-11 · Mathematics Stack Exchange · on “$1000000 \div 81=12345.67$ - why?”

This Numberphile [this video] video talks about a similar (similar = interesting) pattern that arises from a different number.

Recommends Rhys Hughes · 2018-05 · Mathematics Stack Exchange · on “What does $\frac{1}{9801}$ equal to?”

There is a numberphile video related to this here [this video] . Essentially notice that $9801=99^2$. A similar pattern happens with $\frac{1}{9^2}=\frac{1}{81}$. It's value is $0.\dot{0}1234567\dot{9}$, cycling through all the units except $8$ (To see why, try a long division method). The case with $\frac{1}{99^2}$, it cycles through every two digit number, but excludes $98$ from this list... i.e. it is: $$\frac{1}{99^2}=0.\dot{0}0010203040506\cdots9596979\dot{9}$$ and similar things happen with $\frac{1}{999^2},\frac{1}{9999^2}$, etc.

Cites Lazer · 2012-01 · Mathematics Stack Exchange · on “What is special about the numbers 9801, 998001, 99980001 ..?”

I just saw this post [this video] , and realized that 1/9801 = 0.(000102030405060708091011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495969799)(repeat) That is, after the decimal point, the number $00$ , $01$ , $02$ , up to $97$ appear in order, (followed by $99$ ) and then repeat. Similar properties are also exhibited by numbers 998001 [LINK] , 99980001 [LINK] , .. and so on. Edit: fixed periodic start and end It is not very obvious to me why this happens. Is there some simple explanation to this property?

Cites user399078 · 2017-02 · Mathematics Stack Exchange · on “Proof for sum of this infinite derivative series....”

I was recently watching a Numberphile [LINK] Video [this video] when I saw this 'formula' : $$S = \sum \limits_{n=1} ^ {\infty} n\,x^{n-1} = \dfrac {1} {\big(x-1\big)^2} \,\,\,\forall\,\,\, |x| < 1$$ I was amazed to see this and it immediately struck to my mind that it is really an infinite derivative series: $$S=\sum \limits_{n=1}^{\infty}\dfrac {d}{dx} x^n = \dfrac {d}{dx}\sum \limits_{n=1}^{\infty}x^n$$ Now applying the sum for infinite Geometric Series when $|r|<1$ ($r = $ common ratio) : $$S = \dfrac{d}{dx}\dfrac{1}{1-x} = \dfrac{1}{\big(1-x\big)^2}$$ I am really a beginner at Calculus, so the first question I want to ask is : Is my derivation for the sum $S$ mathematically correct ? Though

Cites orion2112 · 2019-09 · Mathematics Stack Exchange · on “Fractions that have interesting, fun or noteworthy decimal expansions”

I'm looking to discover more fractions that have interesting* decimal expansions. (I'm asking out of curiosity, there is no particular academic reason as far as I'm concerned). Here are a few examples: $\dfrac{1}{99}=0.010101\dots$ $\dfrac{1}{999}=0.001001001\dots$ (and so on...) This post [LINK] talks about: $\dfrac{1}{243}=0.\overline{004115226337448559670781893}$ This Numberphile video [this video] talks about how: $\frac{1}{999^2}=0.00000100200300400500600...$ 'generates' the 3-digit integers (except $998$ ). Similar patterns arise with $1/99$ , $1/9999$ , and so on. *I realize that 'interesting, fun or noteworthy' might make this question a bit open-ended or subjective, hence the 'soft-quest