Learn Decimals with Plainmath's In-Depth Explanations and Real-World Examples

How would you graph ${\displaystyle -\frac{5}{4}}$ on a number line?

Prove ${\displaystyle {10}^{\frac{P\left(n\right)}{2}}+1}$ is divisible by n

${\displaystyle \frac{3}{4}}$ of the books in the library are English books and the rest are Chinese books. There are 2849 more English books than Chinese books. What is the total number of books in the library?

S(n) properties
Recently, while reading a number theory textbook for Olympiads, i came across the following property:
${\displaystyle S(n_1+n_2)\le S\left(n_1\right)+S\left(n_2\right)}$

Prove that the difference between numbers with the same sum of digits is a multiple of 9.

Prove that ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{\mu \left(n\right)}{{10}^n}}$ is irrational

Repeating decimals in converted fractions, why these increments?
a) ${\displaystyle \frac{1}{31}=0.032258064516129\cdots }$ ., the repetend is 032258064516129 which can be broken into the numbers 032258, 064516, 129032 that are related by multiples of 2:
b) ${\displaystyle \frac{1}{13}=0.076923\cdots }$. , the repetend is 076923 which can be broken into 0769, 2307 that are related by multiplying by 3:
c) ${\displaystyle \frac{1}{17}=0.0588235294117647\cdots }$ , the repetend is 0588235294117647 which can be broken into the numbers 5882352941, 11764705882 that are related by multiplying by 2:
d) ${\displaystyle \frac{1}{19}=0.052631578947368421\cdots }$ , the repetend is 052631578947368421 which can be broken into the numbers 05263 , 15789 , 47367 , 142101 (think of the initial 1 in 142101 being removed from the 8 in the preceding 47368) that are related by multiples of 3:
e) ${\displaystyle \frac{1}{21}=0.047619\cdots }$ , the repetend is 047619 which can be broken into the numbers 0476, 1904 that are related by multiplying by 4:

Question which looks something similar to decimal expansion
Let
${\displaystyle A=\{\sum _{n=1}^{\mathrm{\infty }}\frac{a_i}{5^i}:a_i=0,1,2,3\text{ or }4\}}$
What is set A?

Prove that every real number ${\displaystyle x\in [0,1]}$ has a "certain" binary representation

Prove that for every odd integer n, exist a multiple m of n whose decimal representation entirely consists of odd digits.

Proving that 31123319 is the largest number with a self-accounting property is 10213223. What is the largest such number?

Determine, with proof, the minimum value of c such that ${\displaystyle c^n+2014}$ has all digits less than 5, where ${\displaystyle c,n\in \mathbb{N}}$ possible values of n. Note that ${\displaystyle \mathbb{N}}$ does not include 0. In other words, c,n shall not assume the value of 0 or anything lesser than 1 in general.

Convergence of ${\displaystyle \left(b_n\right)}$ if ${\displaystyle b=b_0.b_1{b}_2\cdots b_n\cdots }$

How come square root of 2 times itself equals 2

Let ${\displaystyle \alpha \in [0,9]}$. Is the set of real numbers such that the average of the digits of each of these numbers ${\displaystyle =\alpha }$, countable or uncountable?

Given a natural number ${\displaystyle n<{10}^9}$, find the maximum number of the multiple of 3, which differs by exactly one digit from the given one

What is the last non-zero digit of
${\displaystyle ((\cdots (((1!)!+2!)!+3!)!+\cdots )!+1992!)!}$?

What is the place value of 7 in 0.708?

What is the mistake in balancing the equation in a demonstration that ${\displaystyle 0.999\cdots =1}$?