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2
votes
1answer
58 views

Permuting digits in a power of $2$

Does there exist a natural number $N$ that is a power of $2$ whose digits (in the decimal representation) can be permuted to a different power of $2$? Thoughts: If such a number $N$ exists, then ...
0
votes
1answer
159 views

What is $\tau$ in base $12$?

I'm a big fan of both $\tau$ and the duodecimal system. And while I can find information for $\pi$ on both, I can't seem to find the number of $\tau$ in base $12$. $\tau$ is given as $\tau = 2\pi$. ...
8
votes
2answers
257 views

Computing the last non-zero digit of ${1027 \choose 41}$?

I am working on the following problem: Let $x_n$ be a sequence of positive odd numbers. If $N$ is the number of ordered pairs $(x_1, x_2, x_3, \dots, x_{42})$ such that $$x_1 + x_2 + x_3 + \dots + ...
0
votes
0answers
39 views

Metric for precision of a decimal number

I am working on cleansing a cities database and trying to increase the precision of the latitudes and longitudes of these cities. I am comparing what I have against an outside datasource and want to ...
2
votes
0answers
90 views

Appear the digits in the number $3^{3^{3^3}}$ approximately equally often?

Does the number $$3\uparrow \uparrow 4 = 3^{3^{3^3}}$$ contain the digits 0-9 approximately equally often ? With "approximately" I mean that a chi-squared test for equal distribution would produce ...
1
vote
0answers
86 views

Rudin: Supremum of Finite Decimals

I am reading Rudin. Note the following construction: Let $x>0$ be real. Let $n_0$ be the largest integer s.t. $n_0 \leq x$. Then, having chosen $n_0, \ldots, n_{k-1}$, let $n_k$ be the largest ...
1
vote
1answer
82 views

For which $n \in \mathbb{Z^+}$ is $(4n+9)/(2n^2+7n+6)$ a terminating decimal?

I saw this problem here. My approach: Let $A = (4n+9)/(2n^2+7n+6) = \frac{6}{2n + 3} - \frac{1}{n + 2}$ If $\frac{6}{2n + 3}$ and $\frac{1}{n + 2}$ are terminating decimals, then so is $A$. A ...
1
vote
0answers
114 views

Division of large numbers

Euclidean Algorithm Versus Horner Algorithm. I have come across this problem and I do not manage to find a good example for it. For all the numbers I pick there is no loss. Give an example of a ...
1
vote
3answers
67 views

How to round $ 0.03446$ to three decimal places?

What is $ 0.03446 $ rounded to three decimal places? I think it is $0.035$ but someone told me that it should be $0.034$. I think $0.035$ because the $6$ would change the $4$ into a $5$, which ...
2
votes
1answer
64 views

What Will Happen Without Decimal Expansion?

After a discussion on the complexity of decimal expansion (such as $0.\bar{9}=1$), some of my students (middle school) decided to throw away the decimal expansion of some numbers! Namely, the numbers ...
3
votes
1answer
187 views

$\displaystyle \frac{6}{2n-1} - \frac{1}{n} = \frac{p}{2^i5^j}$

$\displaystyle \frac{6}{2n-1} - \frac{1}{n} = \frac{p}{2^i5^j}$ For which $n$ is this expression true. $n$ and $p$ are integers. $i$ and $j$ are positive integers or zero.
4
votes
3answers
51 views

Graph of the last digit of $x^n$ - why is it symmetric when $n$ is even, and not when $n$ is odd?

I have discovered this fact: "The graph of the last digit of $x^n$ (where $x$ is positive) is asymmetrical if $n$ is odd, and symmetrical if $n$ is even." What is the logic behind this? For ...
0
votes
1answer
45 views

Finding number of primes of the form $1010\ldots 101$ (in base 10) [closed]

How many primes among the positive integers, written as usual in the base $10$, are such that their digits are alternating $1$’s and $0$’s, beginning and ending with $1$?
0
votes
3answers
55 views

How to do long division $1.068 / 40$ without using a calculator?

1.068 divided by 40. 1.068 / 40 = ? How would you do this problem by long division and any other methods to figure this out without using a calculator. Just pencil and paper.
-1
votes
2answers
38 views

11th form to 7th form

I can't solve this one: Consider an integer expressed in the 11-based form. In the 11-based form, digits 0 to 9 correspond to their decimal values, and A corresponds to 10. For example (15)_11 = (1 * ...
1
vote
1answer
120 views

What are the last 20 digits of mega?

What are the last 20 digts of the number mega, which is "2 in a pentagon" in steinhaus-moser-notation ? In contrary to power towers or tetration, the ending digits are not stable. I found out that ...
7
votes
0answers
84 views

Which digit occurs most often?

Is there any method to calculate, which digit occurs most often in the number $$4 \uparrow \uparrow \uparrow \uparrow 4\ ,$$ the fourth Ackermann-number ? Or would it be necessary to calculate the ...
0
votes
1answer
93 views

do you know another Magic Square with this property?

with the repeating digits of $\frac{1}{19} = 0.052631578947368421$ we can construct an exceptional magic square : The number 19 is a cyclic number with a period of 18 before the digits start to ...
4
votes
3answers
475 views

Dividing Decimals.. But remainders?

So, I understand how to do long division with decimals. So let's consider this problem: $10.5$ divided by $5.5$ (I chose this problem because it will OBVIOUSLY have a remainder) So we will look at ...
1
vote
1answer
325 views

How to calculate decimals of the fractional number 1/49?

I find this tricky one. How to calculate the first 50 digits/decimals of the fractional number 1/49? Two of my calculators and MatLab gives different answers so I'm curious, how this is calculated ...
4
votes
1answer
86 views

First digits of extremely large binomial coefficients

Can the first digits of a binomial coeffecient $$\binom{n}{k}$$ be calculated, if n and k are very large numbers ? For example Calculate the first ten digits of $$\binom{10^{85}}{10^{23}}$$ Any ...
2
votes
1answer
126 views

First digits of extremely large numbers (Generalization of “first digits of Graham's number”)

I found a question about the first digits of Graham's number and would like to generalize it : We want the first n digits of the number $a\uparrow^b c$. Which method is the most effective to do ...
8
votes
0answers
96 views

Biggest powers NOT containing all digits.

Let $m>1$ be a natural number with $m \not\equiv 0 \pmod{10}$ Consider the powers $m^n$ , for which there is at least one digit not occurring in the decimal representation. Is there a largest $n$ ...
5
votes
3answers
266 views

What is the smallest natural number n?

What is the smallest natural number n for which there is a natural k, such that, the lasts 2012 digit in the representation decimal of $n^k$ are equal to 1? I don't even know how to start with it ... ...
2
votes
2answers
102 views

Which real numbers have two representations?

Are they only numbers that end with 9999... and 0000... after the dot or some other too? If so, can you give an example?
0
votes
2answers
133 views

Decimal expansion of a Cauchy sequence

In one of the construction of $\mathbb{R}$ we make each real number an equivalence class of Cauchy sequences in $\mathbb{Q}$. More precisely, two Cauchy sequences $a_n$ and $b_n$ are equivalent iff ...
1
vote
0answers
80 views

Why are normal numbers important?

I'm studying normal numbers for my university dissertation - in particular whether or not algebraic numbers are normal. One thing I cannot find anywhere is why it matters. I'm not expecting a real ...
10
votes
3answers
507 views

Is a decimal with a predictable pattern a rational number?

I'm starting as a private Math tutor for a high school kid; in one of his Math Laboratories (that came with an answer sheet) I was stumped by an answer I encountered in the True or False section (I'm ...
2
votes
0answers
39 views

A four-digit square is of the form $aabb$. What's $a^2+b^2$? [duplicate]

The 5772 Ulpaniada included the following question: Consider a four digit square number (a number which is the square of a whole number).Its digit notation is $aabb$ (the thousands digit is $a$, ...
0
votes
3answers
160 views

Why does $0,\bar{9}$ equal $1$? [duplicate]

I am finding hard to understand why $0,99999..... = 1$ I have the following proof: Let $x$ be $0,9999...$ then $10x = 9,999...$ So $10x - x = 9,999 - 0,9999$ $9x = 9 \rightarrow x = 1$ From a ...
1
vote
4answers
61 views

How do I convert from binary base to decimal?

I have a homework problem and I don't understand it. Here is the problem: The base two number 11111(base 2) has the same digit in all places. The same number can be written in different bases. Find ...
2
votes
1answer
69 views

Calculating 6 decimal digits of $3^{\sqrt2}$ using a calculator.

How can we calculate $3^{\sqrt2}$ to 6 decimal digits, using only a semi-basic calculator (Which has the square root too) and a pen and paper? I asked this question from my teacher and he ...
1
vote
2answers
306 views

converting decimals to base negative-10

I have a decimal (base $10$) number, $44$, and would like to convert it to base $-10$. I know how to convert $$ 164_{-10} \mapsto 44_{10}, $$ but not the other way around.
2
votes
1answer
82 views

Fractions and long division. [duplicate]

$\frac{1}{9}=0.111\dots$ $9\times \frac{1}{9} = 0.999\dots$ $1=0.999\dots$ What is the problem here? Thanks for any help.
4
votes
0answers
96 views

Arrow notation and decimals.

We know how to add with decimals. We know how to multiply with decimals. We know how to exponentiate with decimals. Do we know how to work with decimals for power towers? for example, can we deal ...
1
vote
2answers
54 views

Divisibility by $9$

Suppose we have a natural number $N$ with decimal representation $A_kA_{k-1}\ldots A_0$. How do I prove that if the $\sum\limits_{i=0}^kA_i$ is divisible by $9$ then $N$ is divisible by $9$ too?
3
votes
0answers
66 views

Decimal system history

Today, "numbers" usually refer to real numbers and are most commonly conceptualized as consisting of all possible infinite decimal expansions (or binary expansions, etc). When did this way of thinking ...
2
votes
2answers
40 views

How to identify whether a fractional part of a number contains more that 2 digits.

EX. I want to accept numbers which have maximum of 2 digits after decimal points. i, e, 10.23 should be allowed and 10.233 should not be allowed. What mathematical operations can be done to ...
1
vote
1answer
152 views

About the sum of the first half and the latter half of the cyclic numbers of a repeating decimal

Let us call the sum of the first half and the latter half of the cyclic numbers of an irreducible fraction 'a division sum' when the period of a repeating decimal is even. Also, let $\lambda(l)$ be ...
20
votes
2answers
515 views

$\lfloor0.999\dots\rfloor= ?$ $0$ or $1$?

I think $\lfloor0.999\dots\rfloor= 1$, as $0.999\dots=1$,but I have doubt, as $\lfloor0.9\rfloor=0$,$\lfloor0.99\rfloor=0$,$\lfloor0.9999999\rfloor=0$, etc.
3
votes
1answer
94 views

Which Well-Known Numbers Are Alexandrian

A real number is said to be Decimal Alexandrian if its decimal representation contains every possible finite decimal sequence. It is a popular question whether $\pi$ is Decimal Alexandrian, or even in ...
1
vote
0answers
75 views

Prove that x has two base p decimal expansions

I am attempting to prove that if $x$ has a finite-length base $p$ decimal expansion, that it has precisely two base $p$ expansions. ($x = \frac{a_1}{p} + \dots +\frac{a_n}{p^n}$) My attempt: x can ...
2
votes
1answer
48 views

Cont'd Decimal Expansion, rational or not?

This is a follow up from this question. Since it's proven by Calvin Lin that $0.11235813213455...$ (Fibonacci Sequence), I'm not wondering if the sequence $$0.123456789101112131415...$$ (which is ...
0
votes
1answer
41 views

Representing a strange number as a fraction

Can this decimal with special patterns be expressed as a fraction? Is it a rational number? $$0.101001000100001000001...$$ Where the number of zeros after every 1 is increased by 1. Ty.
16
votes
2answers
700 views

Is the number $0.112358132134…$ rational or irrational?

Just out of curiosity, is the number $0.112358132134...$ a rational or irrational number? $...$ stands for Fibonacci sequence not repeating decimals!
0
votes
3answers
259 views

Is $0.9999…$ an integer? [duplicate]

Just out of curiosity, since $$\sum_{i>0}\frac{9\times10^{i-1}}{10^i}, \quad\text{ or }\quad 0.999\ldots=1,$$ Does that mean $0.999\ldots=1$, or in other words, that $0.999\ldots$ is an integer, ...
1
vote
2answers
109 views

CS problem, turned to mathematics

I am trying to solve some of the projecteuler problems using a much of a programmers approach. However, I would like to get more into the math, and therefore would try to do some mathematical ...
1
vote
3answers
90 views

Are there axioms or theorems about the decimal terminations of numbers?

I've got struck by curiosity: Are there axioms or theorems about the decimal termination of numbers? For example: $$\frac{1}{3}=0.3333333333333333\ldots$$ And ...
0
votes
3answers
131 views

Is this statement true: 1 = 0.99 [duplicate]

Now this question might sound a bit weird to some people, but the situation is this: Say I have the number $0.999..$ where there is an infinite number of 9's (much like $0.3333..$ with ...
1
vote
1answer
260 views

Period of a decimal expansion

Show that if n is a product of m distinct primes, then the period of the decimal expansion of 1/n is the lowest common multiple of the periods of 1/p over all primes p|n. I understand that the above ...