Questions tagged [decimal-expansion]
For questions about decimal expansion, both practical and theoretical.
1,034
questions
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votes
1answer
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 ...
0
votes
0answers
24 views
Formula for decimal expansion of a power
Let $a, b > 0$ have decimal representations
$$a = a_A \ldots a_1 a_0 \cdot a_{-1} a_{-2} \ldots = \sum_{i=-\infty}^A a_i 10^i,\\
b = b_B \ldots b_1 b_0 \cdot b_{-1} b_{-2} \ldots = \sum_{j=-\infty}^...
0
votes
1answer
26 views
Where digit sum matches digit sum of the square - formal name?
Just came across this.
The UK equivalent is 999, for which the square is 998,001 - which interestingly shares the same digit sum. I just wonder if this has a name and/or any literature? I note ...
-2
votes
1answer
42 views
12.40 was rounded to 2 decimal places. What is the lower bound?
12.40 was rounded to 2 decimal places.
What is the lower bound?
I have tried 12.455 which is not the correct answer
0
votes
3answers
54 views
how come square root of 2 times itself equals 2
since square root of $2$ is a irrational number, which we know or assume its decimal part is not a finite number, or doesn't terminate, how come we say that this infinite number (not in terms of being ...
3
votes
3answers
85 views
Is it true that the decimal expansion of pi contains any string of digits of any length, infinitely often? [duplicate]
More than 60 years ago I read in a book by Emile Borel that the suite of pi decimals contains any suite of numbers as long as wanted, and this an infinite number of times. The book is lost for a long ...
5
votes
1answer
50 views
how many base $10$ decimal expansions can a real number have?
A somewhat unintuitive result of real analysis is that decimal expansions are not unique. For example,
$$0.99999...=1.$$
So it can be gathered that every real number has at least one base-$10$ decimal ...
8
votes
2answers
491 views
Numbers of the kind $0.aaa\ldots =\frac{1}{aaa\ldots a}$
What are the integers $a$ such that
$$
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/...
1
vote
1answer
97 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....
2
votes
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 ...
4
votes
2answers
93 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:
$$\log_{10}(2^{...
12
votes
4answers
589 views
Is $\lim_{x\to\infty} (0.99999…)^x=1$?
Just a brief "simple" question. Is $\lim_{x\to\infty} (0.99999...)^x=1$? According to this question, $0.99999...=1$, so this seems to be true.
Is that enough to prove it? It seems slightly ...
5
votes
3answers
64 views
Infinite binary having infinite decimal [closed]
Can a finite decimal has an infinite binary representation? I have come to a conclusion that it may not be possible based on what I have read from the following:
What cannot happen is that the decimal ...
2
votes
2answers
88 views
How to multiply 2 Binary Numbers?
Suppose I have two regular numbers $a$ and $b$ in base $10$ like this: (where $N$ is even)
$$a=a_{\frac N2}a_{\frac N2-1}\ldots a_1,\qquad b=b_{\frac N2}b_{\frac N2-1}\ldots b_1$$
So the result of ...
1
vote
1answer
30 views
Modular Numbers: Not Accounting for Decimal Portion in Decimal Expansion
Let n = 10, s = 4.
[s] = {s' $\in \mathbb{z}$ | s' $\sim$ s}
= {s' $\in \mathbb{z}$ | 10 | s' - 4}
= {10k+4 | k $\in \mathbb{z}$ }
= {s' $\in \mathbb{z}$ | decimal expansion of s' ends in a 4 if s' is ...
1
vote
1answer
69 views
Is it possible to know the sum of the digits of a number (in base 10), without knowing the digits?
Let's say that you have a really big power of 2, that's so big that you can't print it out on a computer. Would it still be possible to find the sum of its digits?
There is a similar result that is ...
-1
votes
1answer
87 views
What is the last non-zero digit of $((\dots(((1!)!+2!)!+3!)!+\dots)!+1992!)!$?
What is the last non-zero digit of $((\dots(((1!)!+2!)!+3!)!+\dots)!+1992!)!$?
Clarification of the given expression:
Let $A_1=(1!)!$
To get $A_2$, we add $2!$ to $A_1$ then we take the factorial of ...
1
vote
3answers
63 views
When are $1/n!$ repeating decimal with single digit repetend
Find all $n$ where $1/n!$ is repeating decimal with single digit repetend(for example $0.4111111...$ but not $0.412121212...$) but cannot be expressed as a terminating decimal (for example $0.9999999$ ...
-2
votes
1answer
63 views
Prior rounding scheme for efficient finite decimal multiplication
This concerns fixed precision finite decimal multiplication. It is about prior rounding which may be computationally beneficial as the result is only needed to be accurate at a specific Order of ...
0
votes
1answer
45 views
Write Multiplication Using Sigma?
Let's suppose I have the following two numbers:
$$ a= a_{N/2} a_{N/2-1} \cdots a_1 $$
and
$$ b = b_N b_{N-1} b_{N-2} \cdots b_1 $$
Where a_1 is the first digit of a, a_2 the second digit and so on.
...
0
votes
0answers
26 views
Expansion of conditional probability and expected value
The question-Suppose X and Y are discrete random variables. Match up the follow- ing.
$$(a) E(X^2Y |Y = 2)$$
$$Answer- 2E(X^2|Y = 2)$$
My approach was to expand which I got
$$ 2E(XY |Y=2))$$
I don’t ...
2
votes
1answer
122 views
How to make my algorithm work for case: irrational number $\sqrt{2}$ so that $\left | \frac{p}{q}- \sqrt{2} \right |< 0.001$
On one day, I read a magazine then I'm so interested in the algorithms, one of my favorites is RATCONVERT, i.e., if you have $\dfrac{142}{727}\cong 0.195323246..$ then how do you find $0.195323246.. \...
1
vote
1answer
84 views
Given a natural number $n< 10^{9},$ find the maximum number of the multiple of $3,$ which differs by exactly one digit from the given one
I have an algorithm to solve this problem * given a natural number $n< 10^{9},$ find the maximum number of a multiple of $3,$ which differs by exactly one digit from the given one." My ...
1
vote
1answer
41 views
Proof that the set of Pochhammer numbers satisfies Benford's law
Consider the set $S_x$ of the following Pochhammer numbers:
$$(x)_n := \frac{\Gamma(x+n)}{\Gamma(x)}\,, \tag{1}$$
with the gamma function:
$$\Gamma(n) := (n-1)!\,. \tag{2}$$
From "experiment"...
-1
votes
1answer
21 views
Formula for transforming periodical decimal expansion into fraction [duplicate]
How to prove $0.a_1a_2...a_n\dot{b_1}b_2...\dot{b_k}$ is equal to $\frac{a_1a_2...a_nb_1b_2...b_k - a_1a_2...a_n}{99...9 \cdot 10^n}$ ($\text{exatcly}$ $k$ $\text{nines}$) for any $a_i, b_j \in \...
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votes
2answers
42 views
S(n) properties
Recently, while reading a number theory textbook for Olympiads, i came across the following property;
$S(n_1+n_2) \le S(n_1) + S(n_2)$
Where S(A) is the sum of digits of A in base 10. In my textbook, ...
0
votes
1answer
93 views
Prove that $\sum_{n=1}^{\infty} \frac{\mu(n)}{10^n}$ is irrational
First of all, I'm aware that this question has been previously asked, (see: show that $\sum \frac {\mu(n)}{10^n}$ is irrational) however I did not find the solutions there particularly useful.
In ...
2
votes
1answer
26 views
Curious short pattern in least common multiple of binomial coefficients
$$f(n) = \text{lcm}\Bigg(\binom n 1, \binom n 2, \dots,\binom n n\Bigg)$$
If we list $f(n) =\; $$\text{A002944}$$(n)$ it starts of kind of boring, but at $n = 14$ we see a curious pattern in base $10$...
1
vote
1answer
35 views
Can fractional/decimal radicals/roots exist?
For questions like "What is the 1/2th root of x would the answer be $x^2$?
My logic is that since $$
\sqrt[\cfrac{1}{2}]{x}=x^{1/{(\cfrac{1}{2}})}
$$
Which simplifies to $x^2$.
So as a general ...
0
votes
3answers
56 views
If 9.999… = 10, then is there a general proof for any number that has infinite trailing 9s?
I've read about $9.999...=10$, and I would say that I understand it. However, I am looking to apply that proof to all real numbers with trailing 9s. For example:
$72.999...=73$
I have the following ...
16
votes
1answer
315 views
Why is $\sum\limits_{n=1}^{\infty}e^{-(n/10)^2}$ almost equal to $5\sqrt\pi-\frac12$ (agreeing up to $427$ digits)?
The following I saw as an exercise in a text on modular forms. I am lacking the understanding for why the following is true, but it is nevertheless astonishing:
Why is the numerical value of
$$x:=\...
6
votes
3answers
142 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 ...
4
votes
1answer
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
$$\...
-2
votes
1answer
58 views
Calculating Square root of decimal number manually. [duplicate]
https://youtu.be/tRHLEWSUjrQ
In general, it will be difficult to compute the square root of a decimal number manually?
Examples :
50.73
71.21
156.45
1
vote
1answer
29 views
What does “d-” in decimal number mean
I'm trying to implement some functions over Amazon's Ion Value, while reading its document, I found an example of decimal number is 6.62607015d-34 what does ...
3
votes
3answers
98 views
Why is pi non-repeating?
Ok, I have just learnt the Pigeonhole Principle(PHP) and its application with decimal expansion.
To convey my question clearly, I need to convey my understanding of PHP with regards to decimal ...
5
votes
1answer
90 views
Do the digits of $\sum_{k=0}^n20^k$ repeat?
Consider the summation $\sum_{k=0}^n20^k$. Do the last digits of this always repeat? For example, with $n=54$ the summation is
$$\sum_{k=0}^{54} 20^k \\= 18\,962\,524\,746\,823\,141\,052\,631\,578\,...
0
votes
0answers
42 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,...
0
votes
1answer
59 views
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 ...
3
votes
3answers
49 views
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 ...
2
votes
2answers
55 views
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 ...
0
votes
0answers
35 views
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 ...
4
votes
3answers
89 views
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 ...
1
vote
1answer
39 views
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 ...
0
votes
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
$$a^{(n)} = a_0,a_1a_2\...
0
votes
1answer
17 views
Why does the the binary expansion of $x$ shift left two decimals for $1-x$?
I am working with ternary expansions for the first time and I do not understand why the expansion shifts left by two positions when one takes $1-x$ with $x \in (0,1)$. We write $x'$ for the ternary ...
4
votes
0answers
51 views
Infinitely many consecutive powers with no consecutive equal digits?
Let $P_k$ be the set of all $n\in\mathbb N$ such that $n^1,n^2,\dots,n^k$ have no consecutive equal digits.
In other words, their decimal expansion does not contain '$00$', '$11$' ,'$22$', $\dots$, '$...
0
votes
1answer
29 views
Given a number between 0 & 1, knowing that the decimal expansion terminates, how could you find out the number of decimal places?
Pretend someone hands you a real number between 0 and 1 (not including 0 or 1). All you know is that its decimal expansion terminates. What could you do to determine the number of decimal places in ...
1
vote
2answers
131 views
Is this a valid proof that $0.\overline{9} = 1$?
There
are
tens
of
posts
already
on
this
site about whether $0.\overline{9} = 1$.
This is something that intrigues me, and I have a question about this, including a "proof" which I have found ...
1
vote
2answers
67 views
Show cardinality between two sets $x=(0,x_1x_2x_3…)_{10}=\sum _{k=1}^{\infty }x_k 10^{-k}$
For every real number $x \in [0,1]$ can be written in decimal form:
$$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 ...
