For questions about decimal expansion, both practical and theoretical.

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6
votes
2answers
63 views

Can a change of basis modify irrationality/transcendence?

Fix a real number $x$. We can consider its binary expansion, for instance $x = (0.01101001100101101001011\ldots)_2$. Now we consider the real number $y = (0.01101001100101101001011\ldots)_{10}$ : we ...
0
votes
2answers
65 views

Determine all $n$-digit numbers that are divisible by the cyclic permutations of its digits

Given an integer $n \geq 2$, determine all $n$-digit numbers $M_0 = \overline{a_1a_2 \ldots a_n}$ $(a_i \neq 0, i = 1,2,\ldots,n)$ divisible by the numbers $M_1 = \overline{a_2a_3 \ldots a_na_1}$, $...
0
votes
0answers
11 views

Perfect powers by Oblath's result [duplicate]

What do you mean by this statement? Obl\'ath proved that the only perfect powers all of whose digits are equal to a fixed one $ a \neq 1$ in decimal representation are 4, 8 and 9. This is equivalent ...
0
votes
0answers
16 views

oblath's result in perfect powers

What do you mean by this statement? Obl\'ath proved that the only perfect powers all of whose digits are equal to a fixed one $ a \neq 1$ in decimal representation are 4, 8 and 9. This is equivalent ...
1
vote
2answers
31 views

Distinct digits in a combination of 6 digits

How many 6-digit numbers contain exactly 4 different digits? My approach is: For any 3 digis same and the remaining 3 different(aaabcd) 4*9*8*7*6 For any 2 duplicate digits(aabb) and the remaining ...
2
votes
2answers
66 views

How many 6-digit numbers contain exactly 4 different digits? [duplicate]

my solution is----> NUMBER can be 777210 this i denote by aaabcd ------ this can be done in ---> 10*1*1*9*8*7*[6!/3!] {1 for a thrice} NUMBER can be 772210 this i ...
5
votes
4answers
152 views

Are there any natural proofs of irrationality using the decimal characterization?

Mathematicians typically define rational number to mean quotient of two integers. It is not hard to show that a number is rational by that definition if and only if its decimal expansion terminates ...
1
vote
2answers
20 views

Cantor Set ternary representation

Let $I=[0,1]$ be an interval in $\mathbb{R}$ We represent all numbers in $I$ in the ternary form as $t=0_3.t_1t_2t_3...$ where $t_j=0,1$ or $2$ As per the usual construction of the Cantor set, we ...
0
votes
3answers
58 views

N is a four digit number. If the leftmost digit is removed, the resulting three digit number is 1/9th of N. How many such N are possible? [closed]

N is a four digit number. If the leftmost digit is removed, the resulting three digit number is 1/9th of N. How many such N are possible with solution?
1
vote
1answer
48 views

Real numbers $r$ and $1/r$ whose decimal representations contain the same digits

I was wondering idly this morning about real numbers $r$ with the property that the decimal representations of $r$ and $1/r$ both contain the same nonzero digits (not necessarily the same number of ...
3
votes
1answer
28 views

Returning a decimal number between 0 to 1 for showing how large each number in a set of number is

I hope the title does not go so far, I just want to describe what I want simply: I have a set of random numbers, and I want to return a decimal number from 0 to 1 to show how big (max) the number is. ...
4
votes
1answer
63 views

If $ a_n$ is increasingly divisible by $2$ and not a multiple of $10$ then the sum of its digits goes to infinity

Let $(a_n)_{n \geq 0}$ be a sequence of positive integers not divisible by 10 such that the number of factors 2 in $a_n$ tends to infinity for $n \to \infty$. Prove that the sum of the digits of an in ...
0
votes
2answers
59 views

Finding the value of abc

The ratio of the six-digit numbers $abcabc$ and $ababab$ is 55:54. Find the values of the digits $a$, $b$ and $c$.
1
vote
1answer
19 views

How do you convert a ternary to a novenary?

Say I have the ternary expansion $$0_3.t_1t_2...t_{2n-1}t_{2n}...$$ When converted into a novenary I am told it equals $$0_9.(3t_1 +t_2)(3t_3+t_4)...(3t_{2n-1}+t_{2n})...$$ I am not sure how to get ...
8
votes
1answer
235 views

Is the Fibonacci constant $0.11235813213455…$ a normal number?

Recall that a normal decimal number is an irrational number $\alpha \in \mathbb{R}$ such that each digit 0-9 appears with average frequency tending toward $\frac{1}{10}$, each pair of digits 00-99 ...
3
votes
5answers
144 views

How to find last two digits of $2^{2016}$

What should the 'efficient' way of finding the last two digits of $2^{2016}$ be? The way I found them was by multiplying the powers of $2$ because $2016=1024+512+256+128+64+32$. I heard that one way ...
0
votes
2answers
17 views

Writing a finite ternary as an infinite ternary with infinite number of $3^s$

A finite ternary can be written as an infinite ternary with finitely many trailing $3^s$. We can say $$0_3.t_1t_2t_3...t_nt=0_3.t_1t_2t_3...t_n(t-1)\bar{2}$$ where $t=1,2$. What does the $(t-1)$ bit ...
0
votes
1answer
15 views

Why does $\bar{t_n}-{T_n}=\frac{3}{4^{k_n}}$

We have that $$t=0_4.q_1q_2q_3... \in [0,1]$$ t is in Quaternary form. Let $$T_n=0_4.q_1q_2q_3... q_{k_{n-1}}0$$ and $$\bar{t_n}=0_4.q_1q_2q_3..q_{k_{n-1}}3$$ Why does $\bar{t_n}-{T_n}=\frac{3}{4^{...
1
vote
0answers
51 views

No square has a decimal expansion ending in 79

Show that no square number has a decimal ending in 79. More generally, find all possible two-digit endings for squares. Let any digit number ending at 79 be represented as $$a_nx^n+.....+7x+9$$ Plug ...
2
votes
3answers
77 views

Can fractions actually be converted to decimals?

I was working on a spreadsheet in Excel (I'm a plebe, I know), and I came across a fraction that actually equated to 33.3% of a total number. While looking at it, and looking at the number that went ...
5
votes
1answer
146 views

Is the infinite decimal fraction $1.23456…n$ irrational?

How to prove that the number $ 1.23456\dots n$ is an irrational number? The number consist, of course, of natural numbers in increasing sequence.
3
votes
1answer
49 views

Relationship between decimal length and Fibonacci number

There are 6 single digit Fibonacci numbers. For all other number of digits in the decimal system, there are either 4 or 5 Fibonacci numbers. For example, between 10000 and 99999 there are 5: 10946 ...
1
vote
3answers
53 views

Represent $\frac{2}{5}$ in quarternary form.

I have the following intervals in Quaternary representation $$\bigg[0,\frac{1}{4}\bigg]=[0_4.0, 0_4.1]$$ $$\bigg[\frac{1}{4},\frac{2}{4} \bigg]=[0_4.1, 0_4.2$$ $$\bigg[\frac{2}{4}, \frac{3}{4} \bigg]...
1
vote
2answers
77 views

Numbers divisible by $11$ [duplicate]

A number is divisible by $11$, when the difference between the sum of the digits in the odd positions counting from the left (the first, third, ....) and the sum of the remaining digits is either 0 or ...
0
votes
0answers
15 views

The representation of finite and infinite ternary's

In my notes I came across a sentence that says "every ternary may also be written as an infinite ternary with infinitely many trailing 3's". I dont understand this statement, what does it mean?
1
vote
0answers
33 views

Possible generalization of decimal expansion of $\frac{1}{7}$ on an ellipse

My question: Is there a generalization of the result below, either involving more digits or a fraction other than $\cfrac{1}{7}$? The Futility Closet has this surprising result: The One-Seventh ...
5
votes
4answers
890 views

Roman Numbers - Conversion to decimal number

I have read that if a smaller number is to the left of a larger number means that the smaller number has to be subtracted from the larger number. Ok I can understand quickly for below Roman Numbers : ...
1
vote
3answers
39 views

The digits of a positive

The digits of positive integer having $3$ digits are in A.P and their sum is $15$. The number obtained by reversing the digits is $594$ less than the original number. Find the number. My Attempt ; ...
0
votes
0answers
50 views

An exciting property of the decimal places of $\sqrt{n}$: A conjecture [duplicate]

Problem: Prove or disprove that for any finite string of digits (digits from from 0 to 9 ) of length $k $, there exists an $n \in \mathbb{N}$ such that at least the first $k$ digits to the right of ...
3
votes
6answers
151 views

Find the rightmost digit of: $1^n+2^n+3^n+4^n+5^n+6^n+7^n+8^n+9^n$

Find the rightmost digit of: $1^n+2^n+3^n+4^n+5^n+6^n+7^n+8^n+9^n(n$ arbitrary positive integer) First of all I checked a few cases for small $n$'s and in all cases the rightmost digit was $5$, so ...
4
votes
1answer
57 views

Irrationality of the concatenation of the rightmost nonzero digits in $n!$

Surfing the internet I bumped into a very interesting problem, which I tried to solve, but got no results. The problem is following: let $h_n$ be the most right non-zero digit of $n!$, for example, $...
-4
votes
1answer
76 views

Find the number of zeros at the end of $n!!$. [closed]

Can anyone give me a generalized way to find the number of zeroes at the end of $n!!$ ?
0
votes
1answer
41 views

Determine all three-digit numbers N having the property that N is divisible by 11, and N/11 is equal to the sum of the squares of the digits of N.

Determine all three-digit numbers N having the property that N is divisible by 11, and N/11 is equal to the sum of the squares of the digits of N.
-1
votes
1answer
38 views

Find the base 9 expansion of 1/6, using long division [closed]

Find the base 9 expansion of 1/6, using long division. Struggling with this question and finding next to no help on the internet or any books i have, any help is appreciated!
3
votes
0answers
117 views

Partitioning positive integers using digital rivers

I stumbled on a very simple computer science question from the British Informatics Olympiad for schools and colleges. Embedded in it is a very interesting numbers theory problem. Here is the ...
3
votes
3answers
29 views

Is there a binary fraction with finite decimal expansion that does not end in $5$?

I'm trying to come up with the finite decimal fraction not ending with $5$ which can be finitely expressed in binary. At the moment, I don't see how's that possible. Since decimal fractions can only ...
2
votes
3answers
358 views

WolframAlpha function that returns the 'decimal' part of a number [closed]

Is there a function or command in Wolfram Alpha for getting only the decimal part of a number? Something like this: DecimalPart(3.4231) = 0.4231 I will be using ...
4
votes
3answers
106 views

$1000$th decimal digit of $(8+\sqrt{63})^{2012}$

Find the digit at the $1000$th position at the right of the decimal point of the number $(8+\sqrt{63})^{2012}$ I took this problem from a Mexican Math Olympiad called Galois-Noether. It's the last ...
0
votes
0answers
12 views

Least upper bound property of decimal representation of reals

This is my attempt at a proof that real numbers represented by infinite decimals satisfy the least upper bound property, i.e. every upper bounded set has a least upper bound. I am not sure it is ...
2
votes
4answers
81 views

Is $1.0000…$ ( $1$ with infinite zeros) greater than $1.0$? [closed]

Given that $0.3333...$ is greater than $0.3$ and similarly $0.777...$ is greater than $0.7$, does it follow that the sum of $0.33...$ and $0.77...$ is greater than sum of $0.3$ and $0.7$?
0
votes
0answers
19 views

Get first digits of a very large quotient

Is there a method to get the first $n$ digits of a quotient (ex. a thousand digit number divided by a 5 digit number) without dividing all the way through? I suppose long division until $n$ digits are ...
3
votes
1answer
33 views

Theorem on Repeating Decimals

So I am wondering if anyone recognizes the following theorem: Given a prime $p$, and a base $b$ (natural number $>1$), the period of $\frac{1}{p}$ expressed in base $b$ is the unique $d$ that ...
0
votes
1answer
49 views

Last digit of a number

I was currently solving a question of permutations and in that I had to find the total ways of something. The answer was ${8\choose 4}$ which has last digit $0$ . A random thought that came to my ...
2
votes
1answer
40 views

Decimal expansions and topological connectedness

I'm a bit confused by the following footnote from Moschovakis's Notes on Set Theory, p. 135fn24 (in the note, $\mathcal{N}$ denotes the Baire space). The puzzling part is in bold: One may think of ...
0
votes
0answers
17 views

Is there any difference between fixed point and decimal point?

Source: Introduction to Computers' 1999 Ed.1999 Edition Fixed point number 774.3675 is just a decimal number with a decimal point to show a fractional part 3675/10000. I see no difference in the fixed ...
2
votes
1answer
112 views

How to prove $\limsup_{n \to \infty} |\sin(n)| = 1$?

Does decimal expansion of $\pi$ contain blocks of zeroes of any integer length? I.e. $0$, $00$, $000$, $\ldots$ I discovered this question, when trying to prove $$\limsup_{n \to \infty} |\sin(n)| = 1....
1
vote
1answer
60 views

a*b = a/b = b/a (what's this symmetry called?)

I was playing around with numbers the other day, and I found an interesting symmetry, that I would like to know if it has any specific name assigned to it. Let's assume the notation n:a to refer to ...
1
vote
2answers
129 views

How to prove that every power of 6 ends in 6?

Yesterday I had the traditional math matriculation exam, and in it there was a question "In what digit does the number $2016^{2016}$ end in?" After the test The Matriculation Examination Board ...
3
votes
2answers
66 views

Who is Petrick from Petrick's method?

I would like to ask your help. I think this is the best place for this. In my language -as well as English- I haven't found anything about Petrick yet. His method okay, but I would like to know about ...
4
votes
6answers
240 views

How can never ending decimal numbers represent finite lengths? e.g. pi(π), $\sqrt{2}$

Recently, I was in a discussion with a colleague that, whether the πd really can represent the accurate perimeter of a circle or not. To clarify that doubt, I came ...