Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

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27 views

Difference between fraction, decimal and percentage.

Converting between decimals ,percentages and fractions are treated to very trivial. However what I do not understand is the meaning for each of the operations, a slightly detailed reply to each of ...
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1answer
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Taylor's series theorem expansion examples [closed]

Obtain the Taylor's series expansion for the following 1. $\log(z+1)$ $1/z^2$ I've worked out some samples but couldn't get through with the combination
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More or less rigorous proof of the digital root formula

Could someone give me a more or less rigorous proof of the digital root formula? I saw this question. It asks only about intuition and so, the answers there are not helpful for me to understand the ...
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Decimal rounding of division

I have integers 3<=a<100 and 3<=b<10 billion. (The maximums are arbitrary upper bounds) I also have some positive number c expressed as a decimal with no more than 40 digits after the ...
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Show that 1/p has period p-1 iff 10 is a primitive root mod p

I have this excersice, and i want verify my proof: Let $p$ be a prime, then $1/p$ has period $p-1$ iff 10 is a primitive root $\mod p$. My attempt: $\rightarrow)$ Let $\frac{1}{p}=0,\overline{a_1\...
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Sum of the digits of powers of a number

Let $t$ be a positive integers. Find all $t$ such that there exists distinct positive integers $k,n < 12$ such that the sum of the digits $t^k$ is the same as $t^n.$ I don't have any idea how to ...
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94 views

How many prime numbers are there whose squares' decimal representation consist solely of integer squares?

I am on the hunt for prime squares which decimal digits build up squares. The smallest I have found is: $$7^2 = 49 \cases{4=2^2\\ 9=3^2}$$ Another one is $$191^2 = 36481 \cases{36=6^2\\4=2^2\\81=9^2}$$...
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3answers
83 views

Finding the smallest $n$ such that $n^{2}$ ends with $00001$ [duplicate]

The problem is to find the smallest natural number $n$ so that its square's last $5$ digits are $00001$. $n$ and $n^{2}$ cannot begin with $0$. I know that we have a lower bound on $n$, namely $\sqrt{...
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1answer
28 views

What to call “value” of a digit place [closed]

For example, for the tens digit, what do you call the 10? Is there a name for it. I'm looking for the word that fills in the blank in The number $1234$ has a $2$ in the $100$s place; the ____ of the $...
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1answer
29 views

Prove that the difference between numbers with the same sum of digits is a multiple of $9$. [closed]

Prove that the difference between numbers with the same sum of digits is a multiple of $9$.
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1answer
93 views

Proof for how .9999999…=1 [duplicate]

I was just curious for the proof/theorem for a very close decimal (so when you keep adding a decimal) to equal the next integer such as .999999999999....=1. Thank you.
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2answers
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Find all the numbers that are equal to one quarter of the sum of their own digits

The number $1.5$ is special because it is equal to one quarter of the sum of its digits, as $1+5=6$ and $\frac{6}{4}=1.5$ .Find all the numbers that are equal to one quarter of the sum of their own ...
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330 views

What is the sum of number of digits of the numbers $2^{2001}$ and $5^{2001}$

What is the sum of number of digits of the numbers $2^{2001}$ and $5^{2001}$? (Singapore 1970) I attempted to solve this question by working out what each digit must be, and maybe find some pattern, ...
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2answers
98 views

Let $\alpha\in[0,9].$ Is the set of real numbers such that the average of the digits of each of these numbers $=\alpha,$ countable or uncountable?

Let $\alpha \in [0,9]\subset\mathbb{R}.$ Is the following set countable or uncountable: The set of all (nonnegative) real numbers which, when written in decimal expansion form $$k_1\ldots k_m\cdot k_{...
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1answer
41 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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1answer
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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1answer
62 views

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
90 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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1answer
57 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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504 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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108 views

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
94 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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2answers
95 views

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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603 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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3answers
77 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
90 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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1answer
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
74 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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88 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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64 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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1answer
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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1answer
46 views

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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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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1answer
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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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3answers
62 views

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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1answer
319 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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3answers
148 views

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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1answer
92 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
64 views

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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1answer
33 views

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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3answers
105 views

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
90 views

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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43 views

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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