For questions about decimal expansion, both practical and theoretical.

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2
votes
3answers
74 views

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

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
0answers
6 views

The peano curve mapping- finite and infinite ternary's.

Peano defined a map $f_p$ from a unit interval to the unit square in terms of the operator $$kt_j=2-t_j$$ where $(t_j=0,1,2) $ as $$f_p(0_3.t_1t_2t_3t_4...)=\left(\begin{array}{c} 0_3.t_1(k^{t_{2}}t_3)...
0
votes
2answers
11 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
14 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^{...
-2
votes
3answers
660 views

Getting 2016.20162016…to infinity after dividing [on hold]

Dividing 2240000/1111 seems to give 2016.20162016... to infinity. That is, 2016 keeps repeating. Can someone please tell me what the chances of this surprising effect is? Thank you!
1
vote
0answers
50 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
75 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 ...
3
votes
1answer
47 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
52 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
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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
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0answers
32 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
864 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
38 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
49 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
147 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
73 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
31 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
115 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
28 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
103 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
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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
32 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
46 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
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0answers
16 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
111 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
127 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
62 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
227 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 ...
0
votes
4answers
32 views

Why does dividing a number with $n$ digits by $n$ $9$'s lead to repeated decimals?

For example, $\frac{1563}{9999} = 0.\overline{1563}$. Why does that make sense from the way the number system works? I can vaguely see that since the number $b$ with $n$ $9$'s is always greater ...
0
votes
0answers
26 views

How to find if $n$th root of $m$ is rational?

I know that if $m$ is an integer, then the $n$th root of $m$ must be an integer, else it is irrational. But what if $n$ and $m$ both are decimals - is there a way of easily telling if the $n$th root ...
2
votes
2answers
63 views

Show that every rational number $q\in\mathbb{Q}, q\in [0,1]$ has an eventually repeating ternary expansion.

Show that every rational number $q\in\mathbb{Q}, q\in [0,1]$ has an eventually repeating ternary expansion. Recall that $q$ is a rational number provided it can be written as $q=\frac{m}{n}$ where $m\...
3
votes
3answers
74 views

The last two digits of $13^{1010}$.

$13^{1010}$ $13^{\phi(100)} \equiv 1 \mod 100$ $13^{40} \equiv 1 \mod 100$ $(13^{40})^{25} \equiv 1^{25} \mod 100$ $13^{1000} \equiv 1 \mod 100$ $13^{1010} \equiv 13^{10} \mod 100$ That's all I ...
5
votes
2answers
70 views

Last $m$ digits of a sum

What is an efficent way (not using any computer programs and such) to find last $m$ digits of some terrible looking sum, for example I don't know $$1^{1000}+2^{1000}+3^{1000}+\ldots+(10^{1000})^{1000}?...
1
vote
2answers
56 views

In a given range how can i find how many times a two digit number appears ?

I want find how many times a two digit number appears in a given large range , Range is 10^500 . Example : I want to find 21 in given range and the range is 15 to 240 , there are total of 12 numbers ...
0
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0answers
39 views

Proof Regarding Decimal Expansions

Just as a precursor, I'm not a mathematics major (though I've had some experience with abstract algebra and number theory), so I have absolutely no idea how to approach this problem. Hints, not ...
5
votes
4answers
1k views

Factorial question: number of trailing zeroes in 125! [duplicate]

How many zeros are after the last nonzero digit of 125! ? The answer is 31, but how do you solve it?
2
votes
2answers
70 views

prove that $\sqrt{2}$ is not periodic.

If I am asked to show that $\sqrt{2}$ does not have a periodic decimal expansion. Can I just prove that $\sqrt{2}$ is irrational , and since irrational numbers are don't have periodic decimal ...
14
votes
1answer
212 views

Is $0.248163264128…$ a transcendental number?

My question is in the title: Is $a=0.248163264128…$ a transcendental number? The number $a$ is defined by concatenating the powers of $2$ (in base $10$). It is possible to express $a$ as a ...
9
votes
0answers
134 views

Swapping the digits of an algebraic number (e.g. $\sqrt 2$)

Let an algebraic number, say $ a=\sqrt 2 = 1.41421356237309504880...$, and define $$b=f(a)=1.14243165323790058408...$$ by swapping the digits $a_{2i+1}$ and $a_{2i+2}$ for $i≥0$, corresponding to ...
0
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0answers
26 views

Is $\sigma(2^r)$ a palindrome (in base $10$) for some $r > 2$, where $\sigma$ is the sum-of-divisors function?

(Note: This post is a bit related to this earlier MSE question.) The title says it all. Is $\sigma(2^r)$ a palindrome (in base $10$) for some $r > 2$, where $\sigma$ is the sum-of-divisors ...
25
votes
1answer
351 views

Is there any palindromic power of $2$?

My question is in the title: Is it possible to find $n≥4$ such that $2^n$ is a palindromic number (in base $10$)? A palindromic number is a number which is the same, independently from which ...
1
vote
2answers
28 views

How do i find three successive natural odd numbers for which the sum of their squares can be written in decimal system as :$\overline{xxxx} $? [closed]

let $a, b,c$ be a successive natural odd numbers, I would like to know how do i find three successive natural odd numbers for which the sum of their squares : $a²+b²+c²$ can be written in decimal ...
8
votes
4answers
422 views

Decimals of the square root of $n$.

Let $a_1, \ldots, a_k$ be any sequence of digits (i.e., each $a_i$ is between 0 and 9). Prove that there exists an integer $n$ such that $\sqrt{n}$ has its first $k$ decimals after the decimal point ...