For questions about decimal expansion, both practical and theoretical.

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15
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1answer
109 views

Are there more than 2 digits that occur infinitely often in the decimal expansion of $\sqrt{2}$?

The other day I got to thinking about the decimal expansion of $\sqrt{2}$, and I stumbled upon a somewhat embarrassing problem. There cannot be only one digit that occurs infinitely often in the ...
1
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2answers
91 views

A number with 6 distinct digits which get multiplied by 5 if we move the last digit to front [closed]

There is a number with 6 different digits, if we pick the last digit of that number and place before that number we got $5$ times our number. How to find such a number?
1
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0answers
45 views

Representing negative numbers with an infinite number?

Motivation We all know that: $$ .\bar{9} =.999 \dots= 1$$ I was wondering if the following (obviously not rigorous) statement could be defined on the same footing? Question $$ x = \bar{9} $$ $$ ...
7
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5answers
667 views

Math induction problem with large numbers

I am trying to figure out how to prove $17^{200} - 1$ is a multiple of $10$. I am talking simple algebra stuff once everything is set in place. I have to use mathematical induction. I figure I need ...
0
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0answers
17 views

Normal numbers and equidistribution

Does anyone know how to prove the following criterion? (Due, 1949) $x \in (0,1)$ is normal for base $10$ $\iff \{10^nx\}_n$ is equidistributed. "$\Leftarrow$" is quite easy using definition of ...
5
votes
1answer
42 views

Does every finite digit-sequence appear in some factorial?

Suppose, some finite digit-sequence is given. Can we prove or disprove, that there is always some number $n$, such that the digit-sequence appears in the decimal-expansion of the number $n!$ ? If ...
-1
votes
2answers
80 views

Irrational numbers are non-terminating/non-repeating decimals [closed]

Why is it true that all irrational numbers are non-terminating/non-repeating decimals? By definition, an irrational number is one that can't be expressed as a ratio of integers.
10
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6answers
772 views

Can I guess an irrational number formula from its digits?

Let us say I have 10,000 digits started from some point (lets say the 16th digit) of the decimal expansion square root of some arbitrary number, like 13. Is there any way I can get back the original ...
0
votes
1answer
34 views

Conversion base $r$ to base $10$ (decimal) Algorithm

There's another algorithm for converting from base r to base 10? The only one I know is the following one: For example 20 (base 5) to base 10 is: $2X5^1 + 0x5^0 = 10.$
2
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0answers
64 views

Can it be proven that infinite many primes can be formed only using two distinct digits?

It seems obvious that infinite many primes can be formed only using two distinct digits. $67776767776667777777$ is an example for such a prime. Even if we allow only the digits $0$ and $1$, there ...
1
vote
1answer
32 views

Average of decimal digits

I define the function $d_{\mathrm{avg}} : [0, 1]\to [0, 1]$ such that for $0.x_1x_2x_3\cdots$ the decimal expansion of $x$ (defined such that $\nexists N : x_k = 9$ for all $k \geq N$), ...
2
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2answers
89 views

If $n$ is a square, can $n$ consist of only odd digits?

The question is: If $n$ is a square, can $n$ consist of only odd digits? I have a feeling that the answer is no, with the only exceptions being $n=1,9$. I am not sure how to go about proving this ...
6
votes
5answers
405 views

Prove that in every sequence of 79 consecutive positive numbers written in decimal system there is a number whose sum of the digits is divisible by 13

Prove that in every sequence of $79$ consecutive positive numbers written in decimal notation there is a number the sum of whose digits is divisible by $13$. I tried to take one by one sets of ...
3
votes
0answers
76 views

Is there any kind of known pattern to $\sqrt 2$ in base 2?

Is there any kind of known pattern to $\sqrt 2$ in base 2? Is there any classification categories for decimal digits of numbers that for example would put $\sqrt 2, \sqrt 3 \cdots \sqrt n$ into ...
1
vote
2answers
71 views

Maximum length of repeating digits in a decimal

Total math novice here. I'm wondering if (as I think should be the case) there is no maximum length of a repeating decimal period, for example: ...
1
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2answers
41 views

Are there methods to recursively calculate the decimal expansion of real numbers?

Using the concept of self-similarity, it's possible to encode the decimal expansion of a number as a sort of 'fractal' object. For instance, consider the sequence, $$(1) \quad C_0=0.1, \ C_1=0.101, \ ...
1
vote
0answers
34 views

What formulas are available to find the nth digit of a number?

Imagine that'd I'd like to investigate the digits of $\sqrt{2}$, or of any real number. If I want a formula for the nth digit of a real number $x$, we have, $$(1) \quad \operatorname{d_n}(x)=\lfloor ...
6
votes
2answers
134 views

Where, if ever, does the decimal representation of $\pi$ repeat its initial segment?

I was wondering at which decimal place $\pi$ first repeats itself exactly once. So if $\pi$ went $3.143141592...$, it would be the thousandth place, where the second $3$ is. To clarify, this ...
5
votes
2answers
67 views

Repeating decimals linked to reciprocals of primes

Now this question is base dependent, I will assume base 10 but feel free to generalize. I was noticing that for small primes that are not factors of the base ($2$ and $5$ terminate) the reciprocal of ...
0
votes
1answer
22 views

Series of digits in a number

Given a certain number $\mathcal{N}$ with infinite digits, for instance $\pi$ or $\sqrt2$, it's possible to know if this number has a L-string of repeated numbers, but without actually computing the ...
3
votes
2answers
112 views

Write $0.2154154\overline{154}$ as a fraction

Let $x = 0.2154154\overline{154}$ , I have to prove that it is a rational number just by writing it as a fraction with the proper steps. I note that the repeating part, $154$, is composed by 3 ...
1
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0answers
57 views

Should we say $0.\bar9$ *can* equal $1$ also in the hyperreals?

Consider the sequence $R_n$ of repunits, defined as $\displaystyle\frac{10^n-1}{9}$. We have $$\frac{R_{n+1}}{R_n}=\frac{9}{9} \frac{10^{n+1}-1}{10^n -1}=\frac{10^{n+1}-1}{10^n ...
1
vote
2answers
64 views

Calculating last digit of a number using binomial theorem

How to calculate the last digit of a number say like $$\large 3^{4^{5}}$$ using binomial theorem? P.S:I know how to solve it using modular arithmetic. I saw this one but its not of much use in ...
8
votes
1answer
106 views

Finding the $1000$-th decimal of $\sqrt{1111…111}$

As I was cleaning up my desk, I found my Calculus exam from almost a year ago. I remember there was only a bonus task that required either a tad more wit, either a bit more time. It goes like this : ...
0
votes
1answer
59 views

How can i count number of digits in tetrated numbers?

As you read in the title, I need a technique for counting number of digits in tetrated numbers. For example: ${3^{(3)}}^3 = 7625597484987 $(13 digits) ${7^{(7)}}^7 = $How many digits ...
1
vote
2answers
36 views

creating a fraction or decimal using only addition or subtraction

how do i create a decimal or a fraction by only using addition or subtract? I have the numbers 1 and 2, and I want to end up with .5 -- have been stuck on this for quite a bit! I cannot just do 1/2, I ...
0
votes
1answer
26 views

Significant figures.

Calculate how many gram of p-nitrophenol you require to prepare 250 mL of a 11 mM solution. (Answer to 3 significant figures) I worked it out as $$250/1000 L * (11*10^{-3}\text{ M})$$ = $$2.75\times ...
0
votes
3answers
172 views

Find the number of zeros after decimal point in $0.2^{25}$ [closed]

Find the number of zeroes immediately after decimal point in $(0.2)^{25}$,given that $\log 2=0.30101$ My attempt: I found the answer as $17.\dots$ Should we add $1$ as $17$ is the characteristic ...
0
votes
3answers
64 views

Show that number of units of $2^{2^n}+1$ is always $7$

Given that $n$ is a natural number such that $n\geq2$, show that number of units of $x=2^{2^n}+1$ is always $7$ For $n=2$ we have $x=17$. For $n=3$ we have $x=257$ We can show that the number of ...
3
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2answers
124 views

Length of digits before the period in decimal expansion for rational numbers

I'm a newbie with number theory and I've been reading this page and trying to figure out how to calculate the length of the digits before the period and digits of the period of a rational number of ...
0
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0answers
25 views

How to know if addition of factorials changes the amount of trailing zeroes?

In this specific puzzle, it is $603!+604!+605!$. If it was $603!*604!*605!$ instead, I would simply count all the occurrences of multiples of 5, 25 and 125, which would be $148$ for $603!$ and ...
4
votes
3answers
59 views

Understanding expansion on base $k$

There are some times when one might need to use expansion of real numbers on some base $k$. One example is when dealing with Cantor's set, one uses expansion of the numbers inside $[0,1]$ on base $3$. ...
15
votes
1answer
147 views

Multiplication of doubly-infinite decimal numbers

Summary of the question: Doubly-infinite decimal numbers are "numbers" whose decimal expansions are allowed to extend infinitely both to the left and right of the decimal point, for example ...
1
vote
3answers
59 views

Show that $n-m$ is a multiple of 9 when $n$ and $m$ have same digits

I have just proved the divisibility rule for 3 and 9. Let $n\in\mathbb{N}$. Let $m$ be a number that appears when you shuffle the digits in $n$. Show that $n-m$ is a multiple of 9. Can anyone offer ...
1
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0answers
47 views

Numerical Mathematics and Computing - 6 digit rounding arithmetic

This is a homework problem on a mathwebsite (webwork) which states: For the function f(x)=sqrt(x+4)−2, find an alternative formula that can accurately evaluate its value for small x. To which I ...
0
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1answer
25 views

Isolating Decimals

I'm in need of isolating the decimal part of a number using maths only, no excel functions or anything like that, but it's proving to be much harder than I thought it would be. For example, I have ...
0
votes
0answers
30 views

Operations on binary numbers, that have equal numbers of two available digits, that preserve this property?

I'm looking for operations on binary numbers that have the same number of the available digits, $0$ and $1$, such as $100011$, $100101$ and $10$. These could be integers as the examples given are, or ...
0
votes
1answer
24 views

Sequences of consecutive digits of arbitrary length in irrational ternary numbers (take two).

Suppose we have an irrational number represented in base $3$ such that there can only be a maximum of $n$ consecutive $1$'s or $2$'s in the ternary expansion. Furthermore, suppose the only digit ...
5
votes
5answers
766 views

Does every irrational number contain arbitrarily long sequences of some digit?

Suppose we have an irrational number represented in base $3$ such that there can only be a maximum of $n$ consecutive $1$'s or $2$'s in the ternary expansion. Does this imply there are arbitrarily ...
0
votes
1answer
84 views

Decimal expansion of $x\in [0,1]$

This is an exercise from Royden Real Analysis: Let $p$ be a natural number greater than 1, and $x$ a real number, $0 \leq x \leq 1$. Show that there is a sequence $\{a_n\}$ of integers with $0 \leq ...
4
votes
1answer
62 views

Properties of the Digit Product + Digit Sum of a number

The other day I started messing around with some properties and noticed a pattern emerging when the digit product and digit sum of a number were added together. For example, 15. (1+5)+(1*5) = 11. If ...
7
votes
1answer
2k views

Is 4 the second or third digit of pi

If someone says that they know 10 digits of pi, does that mean that they know ten digits starting with the 3 in 3.14 or with the 1 in 3.14?
5
votes
2answers
82 views

Proving the irrationality of the concatenation of the $n$th powers of primes

Note: The apostrophes are meant to separate different groups of digits. Like, $0.{1^2}'{2^2}'{3^2}'{4^2}'\cdots=0.14916\cdots$. I wasn't able to come up with something better. It is easy to show ...
1
vote
1answer
43 views

Sum of digits of polynomial smaller than of factorial

I'm trying to prove this : Let $f \in Z[X]$ then for sufficiently large $n$ we have $$s(f(n))<s(n!)$$ where $s$ is the sum of digits function. What I have so far : I thought this must be true ...
1
vote
2answers
25 views

Fixed points of iterates of a certain map $\Bbb N \to \Bbb N$

I have stumbled onto chains of numbers that are interesting in that, when they are split up into their digits, summed, squared, and repeating some number of times, yield the original number. This ...
4
votes
0answers
77 views

What's the order of growth of the 'double-and-rearrange' numbers?

This question asks about the reachability of some specific numbers via a procedure that starts from the number 1 and where a valid step is to either double the current number to yield a new number, or ...
20
votes
1answer
272 views

Integers $n$ for which the digit sum of $n$ exceeds the digit sum of $n^5$

This question is strongly inspired by The smallest integer whose digit sum is larger than that of its cube? by Bernardo Recamán Santos. The number $n=124499$ has digit sum $1+2+4+4+9+9=29$ while its ...
4
votes
2answers
72 views

Squares of a number yields a palindrome?

I was doing my statistics homework when I observed an interesting pattern: $ 11^2 = 121 $ $ 111^2 = 12321$ $ 1111^2 = 1234321 $ $ 11111^2 = 123454321 $ $ 111111^2 = 1.234565432 \times 10^{10} $ ...
5
votes
1answer
114 views

The smallest integer whose digit sum is larger than that of its cube?

79 is an example of a number whose digital sum is greater than that of its square (6241). Which is the least number, if any, whose digital sum is greater than that of its cube?
3
votes
2answers
175 views

Is the number $0.1234567891011121314\ldots$ a rational or irrational number? [duplicate]

Is the number $0.1234567891011121314\ldots$ a rational or irrational number? The number has a very clear pattern but however in order for the number to be a rational number it would have to be ...