For questions about decimal expansion, both practical and theoretical.

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2answers
81 views

How to check quickly $\frac{2}{3}=.101010… $ holds?

Every $x \in [0, 1]$ can be expressed in the form $\dfrac{a_1}{2}+\dfrac{a_2}{2^2}+\dots + \dfrac{a_m}{2^m}+\dots$ , where each $a_i$ equals either $0$ or $1$. For such $x$, we have the binary ...
-2
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1answer
28 views

Light bulb decimal question

The average lifetime for 12 bulbs is 187.5 hours.another light bulb gave a lifetime of 203 hours .what would the mean lifetime be if this resuot was included? Answer to 1 decimal place
0
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3answers
81 views

Number theory puzzle

If $(ABCD)รท(DCBA)=9$ where $A,B,C$ and $D$ are distinct and all them belong to ${0,1,2,3,4,5,6,7,8,9}$ but $A$ and $D $are not equal to zero then find $A,B ,C$ and $D$. I tried with the decimal ...
11
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4answers
1k views

Are there any bases which represent all rationals in a finite number of digits?

In base 10, 1/3 cannot be represented in a finite number of digits. Examples exist in many other bases (notably base 2, as it's relevant to computing). I'm wondering: does there exist any base in ...
1
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1answer
37 views

The name for numbers with a certain digit sum.

What is the term for a number that has a certain digit sum? For instance 12 is the "digit sum" of 84, 138, 525 and so on. But what kind of number is 84, 138 and 525 to the number 12? Is there a term ...
0
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1answer
53 views

How does (r-1) complement for subtraction work?

My instructor gave an algorithm for doing subtraction with (r-1)'s complement. For subtracting M - N, it goes like the following. 1) Find the (r-1)'s complement of N by using formula r^n - r^m - N. n ...
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0answers
26 views

Diminished radix complement of a fractionated number?

If I have 5623.34 in base 10, how would I find the diminished radix complement of this number. I know for a non-fractionated number, all you have to do is use the formula ...
30
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4answers
625 views

What is the *middle* digit of $3^{100000}$?

The decimal representation of $3^{100000}$ has $47713$ digits. What is the $23857^{th}$ digit - i.e. the one in the $10^{23856}$'s place? There are lots of questions on this site asking for the ...
0
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1answer
38 views

Converting from twos complement to decimal?

I am currently reading a textbook and I can't seem to understand what the examples in the book did. I do believe it is an error with the book, but if not can someone explain? How come there is no ...
2
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2answers
180 views

Why must the decimal representation of a rational number in any base always either terminate or repeat?

Wikipedia makes the following statement about rational numbers. The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same ...
3
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1answer
100 views

Length of the non-periodic portion of the decimal expansion of $\frac 1n$

The following question was asked in the Indian National Mathematics Olympiad (INMO) 2015. For any natural number $n>1$,write the infinite decimal expansion of $\frac 1n$. Determine the length ...
0
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2answers
51 views

How many numbers end with $0,2,9$?

The question is very simple: How many positive integers from $900$ and down end with $0,2$ and $9$? I think it is either $270$ numbers or $271$ numbers, but I am not sure which one.
0
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1answer
28 views

Cantor Set and Base 3 Decimal Expansions

I'm trying to show that every point in the Cantor Set (obtained by "middle-thirds" removal, starting with $[0,1]$) has a base 3 decimal expansion consisting of only zeros and twos. I think the proof ...
3
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2answers
79 views

Constructing A Space Filling Curve that fills the Unit Square

I'm reading Neal Carothers' Real Analysis, and he constructs a curve defined over $[0,1]$ that fills the unit square as follows: Let $f$ be a real-valued function over $[0,1]$ such that $f$ is $0$ ...
0
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1answer
44 views

Converting decimal dates to calendar dates?

I have the decimal number 2005.067 which represents the date January 25, 2005. The math equation below shows how the decimal number is created 2005 + (25 - 0.5)/365. I want to be able to grab 2005.067 ...
2
votes
2answers
23 views

How many numbers are in a numbering system with the basis 15 and 4 digits, where the digit sum equals 15

First of all, my question is very similiar to this one: How many numbers between $100$ and $900$ have sum of their digits equal to $15$? but i didn't quite understand how to adapt it to my problem, so ...
35
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1answer
607 views

Numbers $n$ such that the digit sums of $n, n^2,\cdots,n^k$ coincide.

Let $S(n)$ be the digit sum of $n\in\mathbb N$ in the decimal system. When I was playing with numbers, I noticed the followings : ...
0
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1answer
37 views

Multiplying numbers splitting the number into 4 digit numbers

Doing some programming exercise how to sum big numbers,I split the numbers into $n$ numbers of $4$-digit numbers $1240135981395813958$ I split into $1240$,$1359$,$8139$,$5813$,$958$ and summing with ...
14
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0answers
271 views

Numbers $n$ such that the digit sum of $n^2$ is a square

Let $S(n)$ be the digit sum of $n\in\mathbb N$ in the decimal system. About a month ago, a friend of mine taught me the followings : $$S\left(9\color{red}{^2}\right)=S(81)=8+1=3\color{red}{^2}$$ ...
4
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2answers
80 views

Prove: for all $n$ there's a $m$ such that the sum of digits in $mn$ is equal to $n$. [closed]

In the following $n,m$ are natural numbers. I need to prove that for all $n$ there's a $m$ such that the sum of digits in $mn$ is equal to $n$. Any ideas? Thanks.
25
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1answer
434 views

Does my “Prime Factor Look-and-Say” sequence always end?

I'm trying to create a challenge for PP&CG where the object will be to find the longest sequence in a given time, but I'm worried that there may be an infinite sequence that will ruin things. The ...
1
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4answers
325 views

How many digits are there in 100!? [closed]

How many digits are there in 100 factorial? How does one calculate the number of digits?
0
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0answers
24 views

Having problems understanding this algebraic expression

So I have a solution where $B = 0.1959552$ and $A = 0.00048515$. The problem asks, $A$ is $10$ times more likely than $B$. The teacher wrote (i.e. $A = 10 \cdot B$). Is this the right notation or ...
0
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1answer
44 views

$Pi(n) = Pi(n-2) + x\times [Pi(n-1)]$ for all convergent numerators and denominators. True?

Where $x = A001203$, $Pi = A002486$, $A002485$ $Pi(n) = Pi(n-2) + x\times [Pi(n-1)]$ for all $Pi > n+1 $ Hypothesis: This relation evaluates true for all $A002486$ and $A002485$. Lemma: All ...
1
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1answer
61 views

Given $x,y\in\mathbb R$ is there a “formulaic” way to obtain a $q\in\mathbb Q$ with $a<q<b?$

Is there an assignment of reals $x,y$ to a rational number $q(x,y)$ for which $$\forall_{\mathbb R} x.\forall_{\mathbb R}(x<y).\left(x<q(x,y)<y\right)\hspace{.2cm}?$$ For computable reals, ...
0
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1answer
48 views

How to round percents

Suppose there is election such that $n$ votes are given to $m$ candidates. I would like to express the results of elections in two decimal places, like ...
10
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1answer
200 views

Why do divisions like 1/98 and 1/998 give us numbers continuously being multiplied by two each time in decimal form?

For example, when I divided $1$ by $98$, I got an amazing result of $0.0102040816326530612244897...$ where so many numbers get multiplied by two every time in the right pattern with some carrying. ...
5
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5answers
2k views

How many even 3 digit numbers contain at least one 7.

How many even 3 digit numbers contain at least one 7. I got 126, but it was not an answer choice for the problem. Can anyone help?
2
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2answers
265 views

Real numbers as decimals

I'm looking for a book that develops the theory of real numbers in a rigorous way in terms of their decimal expansions. The exposition should be concrete and preferably aimed at mathematically ...
0
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2answers
109 views

Prepend a 9 or append a 0?

Given a positive integer $x$, will $x$ always be larger if one prepends a 9 in comparison to appending a 0? For x = 1, prepending is largest because $91 > 10$ For x = 9, prepending is largest ...
2
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1answer
51 views

Irrationality of Decimal Expansion of Primes

I've heard the proof that this number is irrational is accessible to even a novice to number theory: $\alpha = 0.2 \ 3 \ 5 \ 7 \ 11 \ 13 \ 17 \ldots$ The proof may utilize that a number is ...
0
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1answer
25 views

Decimal expansion__Represent numbers as $x = \sum_{k=1}^{\infty} \frac{a_k}{b^k}$?

If $b>1$ is an integer, is well know that the numbers $x\in (0,1]$, can be written as $$x = \sum_{k=1}^{\infty} \frac{a_k}{b^k}$$ for some integers $a_k \in \{0,1,\ldots ,b-1\} $. My problem is ...
4
votes
3answers
192 views

Subtraction of two repeating decimals

When I was looking at the proof that every repeating decimal is rational, I came across this example: $x=5.33333333\ldots$ ($3$ repeat indefinitely) $10x=53.3333333\ldots$ ($3$ repeat indefinitely) ...
0
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3answers
130 views

How do irrational numbers lie on the number line?

If we construct a square with side length 1, take its diagonal length : $\sqrt{2}$ However I still don't understand HOW it can lie on the number line. Imagine another irrational number $\pi = ...
0
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2answers
72 views
29
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9answers
4k views

Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist?

Does $1.0000000000\cdots 1$ (with an infinite number of $0$ in it) exist?
1
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1answer
33 views

Does decimal fraction has hex value?/can hex be fraction?

I was wondering if a decimal fraction could be converted into a hexadecimal fraction? I have seen it many times ? but I have been also told that decimal or binary fraction has no meaning in hex. ...
1
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3answers
83 views

How would I find the decimal expansion of $1/99^2$

I want to find the repeating decimal expansion of $1/99^2$. All I know is that $1/99 = 0.010101\cdots$. How would I continue?
53
votes
10answers
6k views

Can a number have infinitely many digits before the decimal point?

I asked my teacher if a number can have infinitely many digits before the decimal point. He said that this isn't possible, even though there are numbers with infinitely many digits after the decimal ...
1
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1answer
80 views

The digit 3 and 2 digit number question

The digit 3 is written at the right of a certain 2-digit number forming a 3-digit number. The new number is 372 more than the original 2-digit number. What is the sum of the digits of the original ...
4
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1answer
82 views

Perfect squares formed by two perfect squares like $49$ and $169$.

Let $a$ be a perfect square number whose decimal representation is the concatenation of two perfect squares, for example $49$ (from $4$ and $9$), $169$ (from $16$ and $9$), ... and $4900$, $490000$ ...
0
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1answer
71 views

Usage of decimal expansion

I learned about the rigorous construction of rationals as a set of equivalence classes of ordered integers with operations defined on this set. I understand that the decimal expansion is another way ...
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4answers
126 views

Computing $0.0625^{-2.25}$ without a calculator

It is quite easy to see that $0.0625^{-2.25} = 512$ by plugging this into a calculator. Of course, mathematics existed for millennia before the invention of the calculator; is there a way to compute ...
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3answers
40 views

How to interpret fractional number of bits of precision

In double-precision floating-point format there're effective $53$ bits of mantissa stored. This lets us estimate maximum number of decimal digits of precision available: ...
5
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1answer
41 views

Numbers $n$ such that there is some digit occuring in each power of $n$

If a positive integer $n$ is congruent to $0$, $1$, $5$ or $6$ modulo $10$, there is some digit occuring in each of the powers of $n$. If the decimal expansion of $n$ ends in a $0$, $1$, $5$ or $6$ ...
4
votes
2answers
103 views

Find the 1005th digit after the decimal point expansion of the square root of N.

Let $N$ be the positive integer with $2008$ decimal digits, all of them $1$. That is, $N=1111...1111$, with $2008$ occurrences of the digit $1$. Find the $1005th$ digit after the decimal point ...
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2answers
521 views

Calculating decimal range for two's complement

Given this question : ...
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1answer
85 views

How to be unambiguous about a number's base's base?

Say you want to note down a number to another person, and want it to be unambiguous (perhaps the other person is an alien and has more than 10 fingers or something). So if you say 12345, base 42 ...
0
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1answer
62 views

Is there a 10-digit emirp?

Does a 10-digit emirp exist? Unfortunately, the lists of emirps I could find on the Web are quite small and my programming skills aren't good enough to write a program to check all the primes up to ...
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4answers
67 views

Find all whole numbers such that the number increased by the sum of its digits equals 73.

I'm really lost on how to figure this out. Work shown would help.