Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

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28 views

Differences of # and the reverse of # problem

Millie wrote a five-digit whole number on a blackboard and she also wrote it in reversed order. She considered the difference of her two numbers, and then told Lucy the last three digits of this ...
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24 views

Formula for decimal expansion of a power

Let $a, b > 0$ have decimal representations $$a = a_A \ldots a_1 a_0 \cdot a_{-1} a_{-2} \ldots = \sum_{i=-\infty}^A a_i 10^i,\\ b = b_B \ldots b_1 b_0 \cdot b_{-1} b_{-2} \ldots = \sum_{j=-\infty}^...
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26 views

Where digit sum matches digit sum of the square - formal name?

Just came across this. The UK equivalent is 999, for which the square is 998,001 - which interestingly shares the same digit sum. I just wonder if this has a name and/or any literature? I note ...
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12.40 was rounded to 2 decimal places. What is the lower bound?

12.40 was rounded to 2 decimal places. What is the lower bound? I have tried 12.455 which is not the correct answer
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3answers
54 views

how come square root of 2 times itself equals 2

since square root of $2$ is a irrational number, which we know or assume its decimal part is not a finite number, or doesn't terminate, how come we say that this infinite number (not in terms of being ...
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3answers
85 views

Is it true that the decimal expansion of pi contains any string of digits of any length, infinitely often? [duplicate]

More than 60 years ago I read in a book by Emile Borel that the suite of pi decimals contains any suite of numbers as long as wanted, and this an infinite number of times. The book is lost for a long ...
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1answer
50 views

how many base $10$ decimal expansions can a real number have?

A somewhat unintuitive result of real analysis is that decimal expansions are not unique. For example, $$0.99999...=1.$$ So it can be gathered that every real number has at least one base-$10$ decimal ...
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491 views

Numbers of the kind $0.aaa\ldots =\frac{1}{aaa\ldots a}$

What are the integers $a$ such that $$ 0.\underbrace{aaa\ldots}_{\infty\text{ times}} =\frac{1}{\underbrace{aaa\ldots}_{k\text{ times}}} $$ eg. $$ 0.333\ldots=1/3\\ $$ while $$ 0.1616\ldots\ne1/16\ne1/...
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Problem regarding a MST proof of 0.99999…=1

I came across an answer of this (which is the highest voted, and also awarded bounties worth 50 reputations). To quote, this is what the answer said:- "Suppose this was not the case, i.e. $0.9999....
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1answer
91 views

Repeating decimals in converted fractions, why these increments?

In regards to repeating decimals and cyclic numbers: I understand that many of them, while multiplying by certain integers will produce a number with the same variant of digits, however my question is ...
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How many digits does $2^{25964951}-1$ have?

Mersenne prime numbers are prime numbers of the form $2^n - 1 \,(\mathbb{N}\ni n>1)$such as $2^2 -1$ or $2^{25964951}-1$: How many digits does the latter have? I found this here: $$\log_{10}(2^{...
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589 views

Is $\lim_{x\to\infty} (0.99999…)^x=1$?

Just a brief "simple" question. Is $\lim_{x\to\infty} (0.99999...)^x=1$? According to this question, $0.99999...=1$, so this seems to be true. Is that enough to prove it? It seems slightly ...
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64 views

Infinite binary having infinite decimal [closed]

Can a finite decimal has an infinite binary representation? I have come to a conclusion that it may not be possible based on what I have read from the following: What cannot happen is that the decimal ...
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2answers
88 views

How to multiply 2 Binary Numbers?

Suppose I have two regular numbers $a$ and $b$ in base $10$ like this: (where $N$ is even) $$a=a_{\frac N2}a_{\frac N2-1}\ldots a_1,\qquad b=b_{\frac N2}b_{\frac N2-1}\ldots b_1$$ So the result of ...
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30 views

Modular Numbers: Not Accounting for Decimal Portion in Decimal Expansion

Let n = 10, s = 4. [s] = {s' $\in \mathbb{z}$ | s' $\sim$ s} = {s' $\in \mathbb{z}$ | 10 | s' - 4} = {10k+4 | k $\in \mathbb{z}$ } = {s' $\in \mathbb{z}$ | decimal expansion of s' ends in a 4 if s' is ...
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1answer
69 views

Is it possible to know the sum of the digits of a number (in base 10), without knowing the digits?

Let's say that you have a really big power of 2, that's so big that you can't print it out on a computer. Would it still be possible to find the sum of its digits? There is a similar result that is ...
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1answer
87 views

What is the last non-zero digit of $((\dots(((1!)!+2!)!+3!)!+\dots)!+1992!)!$?

What is the last non-zero digit of $((\dots(((1!)!+2!)!+3!)!+\dots)!+1992!)!$? Clarification of the given expression: Let $A_1=(1!)!$ To get $A_2$, we add $2!$ to $A_1$ then we take the factorial of ...
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3answers
63 views

When are $1/n!$ repeating decimal with single digit repetend

Find all $n$ where $1/n!$ is repeating decimal with single digit repetend(for example $0.4111111...$ but not $0.412121212...$) but cannot be expressed as a terminating decimal (for example $0.9999999$ ...
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63 views

Prior rounding scheme for efficient finite decimal multiplication

This concerns fixed precision finite decimal multiplication. It is about prior rounding which may be computationally beneficial as the result is only needed to be accurate at a specific Order of ...
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Write Multiplication Using Sigma?

Let's suppose I have the following two numbers: $$ a= a_{N/2} a_{N/2-1} \cdots a_1 $$ and $$ b = b_N b_{N-1} b_{N-2} \cdots b_1 $$ Where a_1 is the first digit of a, a_2 the second digit and so on. ...
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Expansion of conditional probability and expected value

The question-Suppose X and Y are discrete random variables. Match up the follow- ing. $$(a) E(X^2Y |Y = 2)$$ $$Answer- 2E(X^2|Y = 2)$$ My approach was to expand which I got $$ 2E(XY |Y=2))$$ I don’t ...
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How to make my algorithm work for case: irrational number $\sqrt{2}$ so that $\left | \frac{p}{q}- \sqrt{2} \right |< 0.001$

On one day, I read a magazine then I'm so interested in the algorithms, one of my favorites is RATCONVERT, i.e., if you have $\dfrac{142}{727}\cong 0.195323246..$ then how do you find $0.195323246.. \...
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Given a natural number $n< 10^{9},$ find the maximum number of the multiple of $3,$ which differs by exactly one digit from the given one

I have an algorithm to solve this problem * given a natural number $n< 10^{9},$ find the maximum number of a multiple of $3,$ which differs by exactly one digit from the given one." My ...
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1answer
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Proof that the set of Pochhammer numbers satisfies Benford's law

Consider the set $S_x$ of the following Pochhammer numbers: $$(x)_n := \frac{\Gamma(x+n)}{\Gamma(x)}\,, \tag{1}$$ with the gamma function: $$\Gamma(n) := (n-1)!\,. \tag{2}$$ From "experiment"...
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Formula for transforming periodical decimal expansion into fraction [duplicate]

How to prove $0.a_1a_2...a_n\dot{b_1}b_2...\dot{b_k}$ is equal to $\frac{a_1a_2...a_nb_1b_2...b_k - a_1a_2...a_n}{99...9 \cdot 10^n}$ ($\text{exatcly}$ $k$ $\text{nines}$) for any $a_i, b_j \in \...
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S(n) properties

Recently, while reading a number theory textbook for Olympiads, i came across the following property; $S(n_1+n_2) \le S(n_1) + S(n_2)$ Where S(A) is the sum of digits of A in base 10. In my textbook, ...
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Prove that $\sum_{n=1}^{\infty} \frac{\mu(n)}{10^n}$ is irrational

First of all, I'm aware that this question has been previously asked, (see: show that $\sum \frac {\mu(n)}{10^n}$ is irrational) however I did not find the solutions there particularly useful. In ...
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Curious short pattern in least common multiple of binomial coefficients

$$f(n) = \text{lcm}\Bigg(\binom n 1, \binom n 2, \dots,\binom n n\Bigg)$$ If we list $f(n) =\; $$\text{A002944}$$(n)$ it starts of kind of boring, but at $n = 14$ we see a curious pattern in base $10$...
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Can fractional/decimal radicals/roots exist?

For questions like "What is the 1/2th root of x would the answer be $x^2$? My logic is that since $$ \sqrt[\cfrac{1}{2}]{x}=x^{1/{(\cfrac{1}{2}})} $$ Which simplifies to $x^2$. So as a general ...
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If 9.999… = 10, then is there a general proof for any number that has infinite trailing 9s?

I've read about $9.999...=10$, and I would say that I understand it. However, I am looking to apply that proof to all real numbers with trailing 9s. For example: $72.999...=73$ I have the following ...
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315 views

Why is $\sum\limits_{n=1}^{\infty}e^{-(n/10)^2}$ almost equal to $5\sqrt\pi-\frac12$ (agreeing up to $427$ digits)?

The following I saw as an exercise in a text on modular forms. I am lacking the understanding for why the following is true, but it is nevertheless astonishing: Why is the numerical value of $$x:=\...
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Last digit in $\sum_{k=1}^{999}k^m$ (olympiad question)

I'm trying to prepare myself for mathematics olympiad. I faced a problem which is kind of interesting, here is the question: Oleg chose a positive integer like $m$ and Andrew found the following ...
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90 views

Why are repeating decimals often self-inversions?

Let $b$ be any base, and let $x$ be some odd integer. Empirically, it seems that for any $b$, the majority of odd $x$ will have the following property. Given a reptend $r$ with $2k$ digits as in $$\...
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1answer
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Calculating Square root of decimal number manually. [duplicate]

https://youtu.be/tRHLEWSUjrQ In general, it will be difficult to compute the square root of a decimal number manually? Examples : 50.73 71.21 156.45
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What does “d-” in decimal number mean

I'm trying to implement some functions over Amazon's Ion Value, while reading its document, I found an example of decimal number is 6.62607015d-34 what does ...
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Why is pi non-repeating?

Ok, I have just learnt the Pigeonhole Principle(PHP) and its application with decimal expansion. To convey my question clearly, I need to convey my understanding of PHP with regards to decimal ...
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1answer
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Do the digits of $\sum_{k=0}^n20^k$ repeat?

Consider the summation $\sum_{k=0}^n20^k$. Do the last digits of this always repeat? For example, with $n=54$ the summation is $$\sum_{k=0}^{54} 20^k \\= 18\,962\,524\,746\,823\,141\,052\,631\,578\,...
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Define an injection $(0,1)^2\rightarrow(0,1)$

Define an injection $(0,1)^2\rightarrow(0,1)$. Is your function surjective? Explain. Hint: use decimal expansions. I am so confused. What does $(0,1)^2$ mean? It's not cartesian product, right? Is (0,...
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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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is the number of digits in the decimal expansion of $2^x$ periodic?

I graphed the number of digits in the base $10$ expansion of the series $2^x$: At first, it looks like a repeating pattern in the plot but when I overlay and shift a sequence on top of that graph, it ...
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A property of digital sum

I was working on a computer program and came up with an intuitive idea that reduces the program module by a considerable length . The idea is intuitive but I never came up with a proof. Claim : For a ...
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How many digits does $x\in \mathbb{Z}$ have if it satisfies $|x|<10$ in base ten?

So I recently got into an argument with someone about this. If $x\in\mathbb{Z}$ satisfies $|x|<10$ in base ten, how many digits does it have? My position is that we can't say for sure, it can have ...
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Find the least positive integer $n$ such that the two digits on the left of $n^{12}$ are equal

Find the least positive integer $n$ such that the two digits on the left of $n^{12}$ are equal. What I tried to find $n^{12}$ for $n=1,2,3,\dots,8$, but non of them was valid and it is tedious to ...
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Show that for an integer $n \ge 2$, the period of the decimal expression for the rational number $\frac{1}{n}$ is at most $n-1$.

I can't solve the following problem. Show that for an integer $n \ge 2$, the period of the decimal expression for the rational number $\frac{1}{n}$ is at most $n-1$. On StackExchange I already found ...
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1answer
79 views

Proof for decimal expansions [closed]

I'm not really getting anywhere with the proof for the following: (a) Let $a$ be a positive infinite decimal expansion. We consider a sequence of a finite decimal expansion $$a^{(n)} = a_0,a_1a_2\...
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1answer
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Why does the the binary expansion of $x$ shift left two decimals for $1-x$?

I am working with ternary expansions for the first time and I do not understand why the expansion shifts left by two positions when one takes $1-x$ with $x \in (0,1)$. We write $x'$ for the ternary ...
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Infinitely many consecutive powers with no consecutive equal digits?

Let $P_k$ be the set of all $n\in\mathbb N$ such that $n^1,n^2,\dots,n^k$ have no consecutive equal digits. In other words, their decimal expansion does not contain '$00$', '$11$' ,'$22$', $\dots$, '$...
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29 views

Given a number between 0 & 1, knowing that the decimal expansion terminates, how could you find out the number of decimal places?

Pretend someone hands you a real number between 0 and 1 (not including 0 or 1). All you know is that its decimal expansion terminates. What could you do to determine the number of decimal places in ...
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2answers
131 views

Is this a valid proof that $0.\overline{9} = 1$?

There are tens of posts already on this site about whether $0.\overline{9} = 1$. This is something that intrigues me, and I have a question about this, including a "proof" which I have found ...
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67 views

Show cardinality between two sets $x=(0,x_1x_2x_3…)_{10}=\sum _{k=1}^{\infty }x_k 10^{-k}$

For every real number $x \in [0,1]$ can be written in decimal form: $$x=(0,x_1x_2x_3...)_{10}=\sum _{k=1}^{\infty }x_k 10^{-k}$$ where $x_i \in \{0,1,2,3...,9\}$ for every $i$. Because of uniqueness ...

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