For questions about decimal expansion, both practical and theoretical.

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7
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3answers
91 views

Irrational Numbers : Show that $0.1248163264…$ is irrational

I was working through some basic Number Theory Problems in Rosen and came across the following problem : Show that the real number $0.1248163264...$ represented in ...
0
votes
0answers
31 views

Determining if a rational number has a terminating decimal expansion (proof)

Theorem: $x=\frac pq$ is any given rational number, $n$ and $m$ are any whole numbers (including zero) which you can choose. a) If $q=2^n5^m$ is possible, $x$ has a terminating decimal expansion. ...
6
votes
1answer
57 views

Finding all the zeroes in $100!$

Is there a way to find all the $0$s in $100!$? (Including zeroes that come between two non-zero numbers) I know that to find the $0$s at the end we can use the greatest integer method. I was just ...
9
votes
4answers
19 views

A 10-digit number whose $n$th digit gives the number of $(n-1)$s in it

There is a ten-digit number $X$ such that its first (left-most) digit is equal to the number of $0$s in $X$, the second digit gives the number of $1$s in $X$, and so on. The last (right-most) digit ...
1
vote
2answers
41 views

Do numbers higher than the base number have any meaning?

For instance, in base 5, does this have any meaning 6 + 7 + 8 = ? or in base 2 2 + 3 + 4 = ?`
2
votes
1answer
61 views

What are the bases $\beta$ such that a number with non-periodic expansion can be approximated with infinitely many numbers with periodic expansion?

Warning: This problem requires a bit of setting. Fix a finite set $A \subset \Bbb{Z}$ and consider an infinite non (ultimately) periodic sequence $\mathbf{a}=(a_i)_{i \geq 1}$ of elements of $A$ such ...
0
votes
0answers
17 views

Monograph about periodic representations of numbers in non-integer bases

I'm looking for a monograph (book, article, lecture notes, whatever) about the representation of numbers (real or complex) in non-integer bases. I am especially interested in results about algebraic ...
3
votes
2answers
81 views

Can a number have both a periodic an a non-periodic representation in a non-integer base?

Fix an algebraic number $\beta$ and consider a complex number $\alpha$ which admits multiple representations in base $\beta$. If one representation of $\alpha$ is ultimately periodic, must every other ...
2
votes
1answer
38 views

How to get the amount of digits after the decimal point

How can I easily find the amount of digits after the decimal point? For example: ...
6
votes
0answers
80 views

Does π start with two identical decimal sequences?

Let d$(x,y)$ be the sequence of decimals of π from the x:th one to the y:th one. My question: is there a number $n$ such that d$(1,n)$ = d$(n+1, 2n)$? I.e., does π start with two identical decimal ...
0
votes
2answers
33 views

Repeating Decimal in different base

I've come across the following question. Find $0.\overline{204}_6$ as a base ten fraction. I understand that is the question asked the repeating decimal in base $10$, I would then say that: $$x ...
2
votes
0answers
23 views

Question about repeating decimal?

For simple fraction, we can easily convert it to repeating decimal by calculator. Ex 1/3 = 0.33333..., 1/7=0.(142857),... But some fraction fraction like 10/29, 1/97,... The repeating part of them are ...
5
votes
1answer
71 views

Is there a $k$ such that $2^n$ has $6$ as one of its digits for all $n\ge k$?

It is true that every power of $2$ of the form $2^{6+10x}$, $x\in\mathbb{N}$, has $6$ as one of its digits. Something more is true, the last two digits are either $64$ or $36$. The OP suggests that ...
8
votes
3answers
118 views

nonzero digits in decimal representation of $\sqrt{2}$

let $1,d_1d_2d_3\dots$ be a decimal representation of $\sqrt{2}$. Prove that at least one $d_i$ with $10^{1999}<i<10^{2000}$ is nonzero. I have no idea how to solve it. I think that the given ...
1
vote
2answers
44 views

Find the fraction that creates a repeating decimal that repeats certain digits

Is there any way to find the fraction $x/y$ that, when converted to a decimal, repeats a series of digits $z$? For example: ${x}/{y} = z.zzzzzzzz...$ or with actual numbers, $x/y = 234.234234234...$ ...
3
votes
4answers
61 views

Proof that $2^{0.5}$ will not touch the $1.5$ mark on the number line when we try to mark it exactly?

I know this would be kind of silly, but then this has been troubling me for the past few days. All of us know $2^{\frac{1}{2}}$ is irrational. Let us try to mark this on the number line "exactly", as ...
1
vote
3answers
124 views

Why 6? Is this meant to be intuitive?

What is this question asking? Do you just simply count to number of 1's and 0's and then choose that integer, 6? $$y=10^k+10^{k+1}+10^{k+2}$$ In the equation above, $k$ is an integer. For ...
0
votes
3answers
70 views

Formal notation for representing decimal numbers

What is the formal mathematical notation for representing a decimal number using variables as it's digits? I am honestly surprised that this has not been asked yet on MSE. Let me clarify a ...
2
votes
1answer
136 views

What is the biggest fivedigit $abcde$ number, thats divisible by $bcde$, $cde$, $de$ and $e$?

What is the biggest fivedigit $abcde$ number, thats divisible by $bcde$, $cde$, $de$ and $e$? I was trying to find it but I couldnt. Can you help me with a step by step answer?
3
votes
2answers
53 views

Determine whether the decimal expansion of a rational number is infinite

This may be a naive question but I would like to know whether we can determine if a fraction (say $1/3$) will produce a rational number with an infinite number of digits after the decimal when ...
-2
votes
2answers
172 views

Convert decimal to Binary Floating Point - 8 Bit [closed]

I am trying to convert +3.5 to binary floating point, but im struggling to find the exponent. (8 Bit) Where 1st bit is the Sign, 3 bits for Exponent and 4 bits for Mantissa. Hope somebody can explain ...
1
vote
2answers
75 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
votes
1answer
27 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
votes
3answers
74 views

Number theory puzzle

If $(ABCD)÷(DCBA)=9$ 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
votes
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
vote
1answer
32 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
votes
1answer
28 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 ...
0
votes
0answers
16 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
votes
4answers
594 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
votes
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
votes
2answers
75 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
votes
1answer
96 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
votes
2answers
50 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
votes
1answer
23 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
votes
2answers
71 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
votes
1answer
27 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
21 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
votes
1answer
571 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
votes
1answer
30 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
votes
0answers
224 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
votes
2answers
79 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
votes
1answer
420 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
vote
4answers
199 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
votes
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
votes
1answer
42 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
vote
1answer
60 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
votes
1answer
36 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
votes
1answer
164 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
votes
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
votes
2answers
247 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 ...