# Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

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### Difference between fraction, decimal and percentage.

Converting between decimals ,percentages and fractions are treated to very trivial. However what I do not understand is the meaning for each of the operations, a slightly detailed reply to each of ...
1answer
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### Taylor's series theorem expansion examples [closed]

Obtain the Taylor's series expansion for the following 1. $\log(z+1)$ $1/z^2$ I've worked out some samples but couldn't get through with the combination
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### More or less rigorous proof of the digital root formula

Could someone give me a more or less rigorous proof of the digital root formula? I saw this question. It asks only about intuition and so, the answers there are not helpful for me to understand the ...
0answers
14 views

### Decimal rounding of division

I have integers 3<=a<100 and 3<=b<10 billion. (The maximums are arbitrary upper bounds) I also have some positive number c expressed as a decimal with no more than 40 digits after the ...
0answers
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### Given a natural number $n< 10^{9},$ find the maximum number of the multiple of $3,$ which differs by exactly one digit from the given one

I have an algorithm to solve this problem * given a natural number $n< 10^{9},$ find the maximum number of a multiple of $3,$ which differs by exactly one digit from the given one." My ...
1answer
44 views

### Proof that the set of Pochhammer numbers satisfies Benford's law

Consider the set $S_x$ of the following Pochhammer numbers: $$(x)_n := \frac{\Gamma(x+n)}{\Gamma(x)}\,, \tag{1}$$ with the gamma function: $$\Gamma(n) := (n-1)!\,. \tag{2}$$ From "experiment"...
2answers
45 views

### S(n) properties

Recently, while reading a number theory textbook for Olympiads, i came across the following property; $S(n_1+n_2) \le S(n_1) + S(n_2)$ Where S(A) is the sum of digits of A in base 10. In my textbook, ...
1answer
97 views

### Prove that $\sum_{n=1}^{\infty} \frac{\mu(n)}{10^n}$ is irrational

First of all, I'm aware that this question has been previously asked, (see: show that $\sum \frac {\mu(n)}{10^n}$ is irrational) however I did not find the solutions there particularly useful. In ...
1answer
27 views

### Curious short pattern in least common multiple of binomial coefficients

$$f(n) = \text{lcm}\Bigg(\binom n 1, \binom n 2, \dots,\binom n n\Bigg)$$ If we list $f(n) =\; $$\text{A002944}$$(n)$ it starts of kind of boring, but at $n = 14$ we see a curious pattern in base $10$...
1answer
35 views

### Can fractional/decimal radicals/roots exist?

For questions like "What is the 1/2th root of x would the answer be $x^2$? My logic is that since $$\sqrt[\cfrac{1}{2}]{x}=x^{1/{(\cfrac{1}{2}})}$$ Which simplifies to $x^2$. So as a general ...
3answers
62 views

### If 9.999… = 10, then is there a general proof for any number that has infinite trailing 9s?

I've read about $9.999...=10$, and I would say that I understand it. However, I am looking to apply that proof to all real numbers with trailing 9s. For example: $72.999...=73$ I have the following ...
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### Calculating Square root of decimal number manually. [duplicate]

https://youtu.be/tRHLEWSUjrQ In general, it will be difficult to compute the square root of a decimal number manually? Examples : 50.73 71.21 156.45
1answer
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### What does “d-” in decimal number mean

I'm trying to implement some functions over Amazon's Ion Value, while reading its document, I found an example of decimal number is 6.62607015d-34 what does ...
3answers
105 views

### Why is pi non-repeating?

Ok, I have just learnt the Pigeonhole Principle(PHP) and its application with decimal expansion. To convey my question clearly, I need to convey my understanding of PHP with regards to decimal ...
1answer
90 views

### Do the digits of $\sum_{k=0}^n20^k$ repeat?

Consider the summation $\sum_{k=0}^n20^k$. Do the last digits of this always repeat? For example, with $n=54$ the summation is \sum_{k=0}^{54} 20^k \\= 18\,962\,524\,746\,823\,141\,052\,631\,578\,...
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### Define an injection $(0,1)^2\rightarrow(0,1)$

Define an injection $(0,1)^2\rightarrow(0,1)$. Is your function surjective? Explain. Hint: use decimal expansions. I am so confused. What does $(0,1)^2$ mean? It's not cartesian product, right? Is (0,...