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(If you are facing trouble to see the texts and mathematical expressions of this post, you can see the attached image below.)

$1.$ What is the informal (or intuitive) and formal definition of fraction and all kinds of fractions$?$

$2.$ Can we say that $“$a fraction is a number or expression of the form $\dfrac{a}{b}$ or $-\dfrac{a}{b}$ or ${\large c}\dfrac{{\small a}}{\small{b}}$ or $-{\large c}\dfrac{{\small a}}{{\small b}}$ or ${\small\circ}\dfrac{{\large a}_{\small 1}}{{\large b}_{\small 1}}\times{\small\circ}\dfrac{{\large a}_{\small 2}}{{\large b}_{\small 2}}\times{\small\circ}\dfrac{{\large a}_{\small 3}}{{\large b}_{\small 3}}\times\cdots\times{\small\circ}\dfrac{{\large a}_{\small n}}{{\large b}_{\small n}}$ where ${\large a},{\large b},{\large c},{\large a}_{\small 1},{\large a}_{\small 2},{\large a}_{\small 3},\dots,{\large a}_{\small n},{\large b}_{\small 1},{\large b}_{\small 2},{\large b}_{\small 3},\dots,{\large b}_{\small n}$ can be any number or expression but ${\large b},{\large b}_{\small 1},{\large b}_{\small 2},{\large b}_{\small 3},\dots,{\large b}_{\small n}$ can't be zero$,{\large c}\dfrac{{\small a}}{\small{b}}$ and $-{\large c}\dfrac{{\small a}}{{\small b}}$ are the forms of mixed fraction$,{\large n}\in{\mathbb{Z}}_{>0}$ and ${\small\circ}$ is the sign of any fraction of the form ${\small\circ}\dfrac{{\large a}_{\small n}}{{\large b}_{\small n}}$ which can be $+$ or $-"?$ Does fraction include the numbers like $-\dfrac{-6}{-3},$$\dfrac{1}{-2\sqrt{e}},$$-\dfrac{-\sqrt{5}}{-3},$$\dfrac{\pi}{e},$$\dfrac{-\pi e}{\sqrt{2}},$$-\dfrac{-\sqrt[5]{99}}{-\sqrt[7]{4}},\dfrac{\varphi^3-\varphi^6+7}{-\ln e^3},$$\dfrac{1.73}{-1000},$$-\dfrac{-0.66}{1.45},$$\dfrac{6i}{6i},$$\dfrac{-2i}{0.91},$$\dfrac{0}{-1+2i},$$-\dfrac{0-\sqrt[11]{9}i}{-\sqrt{7}+8i},$$-\dfrac{4-\sqrt[3]{ei}}{-\sqrt{-\pi i^2}+2.5i},$$-400\%,\varphi\%,$$-1.67\times10^{-5},$$\dfrac{-1}{-2}\times-\dfrac{4}{-7}\times\dfrac{-5}{8},$$\left(\dfrac{6-9.8i}{-\sqrt{8}+3i}\right)^2,$${\large 3}\dfrac{{\small -3}}{{\small 1}},$$-{\large 9.81}\dfrac{{\small \pi}}{{\small i}},$${\large 2}\dfrac{{\small 5}}{{\small 3}}\times\dfrac{1}{4},$$\dfrac{7}{9}\times\left(6.67\times10^{-11}\right),$$\dfrac{1}{3}\times50\%$ and the expressions like ${\large x}\dfrac{\small{\sqrt{x^3}}}{\small{5x^2}},$$\dfrac{x^\pi}{y^{i\varphi}},$$\dfrac{2x^y}{3y^x},$$\sqrt[4]{\dfrac{1-x^2}{1+x^2}},$$\dfrac{x^5}{y^2}\times\dfrac{z^9}{x-21},$$\dfrac{{\large\int} \dfrac{1}{y^2}\: \mathrm{d}y}{404},$$\dfrac{\sin x}{\log x},$$\dfrac{| x |}{x},$$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$?$ Which kind of fractions do they belong to$?$

$3.$ How can we express the set of all fractions in set-builder notation and roster notation$?$

$4.$ Is there any special symbol which is used to denote the set of all fractions$?$ Can I use the symbol $\mathbb{F}$ to denote this set$?$

$5.$ What was the historical background of fraction$?$

(Basically, I want to know the answer of my second and third question more precisely. To better understand my question, I have mentioned some confusing numbers and expressions in question no. $2$. I think if I make my second question more shorter, maybe I will fail to highlight the main point. Thank you.)

Jalal wrote: ↑

Sun May 23, 2021 11:24 am

$1.$ What is the informal (or intuitive) and formal definition of fraction and all kinds of fractions$?$

$2.$ Can we say that $“$a fraction is a number or expression of the form $\dfrac{a}{b}$ or $-\dfrac{a}{b}$ or ${\large c}\dfrac{{\small a}}{\small{b}}$ or $-{\large c}\dfrac{{\small a}}{{\small b}}$ or ${\small\circ}\dfrac{{\large a}_{\small 1}}{{\large b}_{\small 1}}\times{\small\circ}\dfrac{{\large a}_{\small 2}}{{\large b}_{\small 2}}\times{\small\circ}\dfrac{{\large a}_{\small 3}}{{\large b}_{\small 3}}\times\cdots\times{\small\circ}\dfrac{{\large a}_{\small n}}{{\large b}_{\small n}}$ where ${\large a},{\large b},{\large c},{\large a}_{\small 1},{\large a}_{\small 2},{\large a}_{\small 3},\dots,{\large a}_{\small n},{\large b}_{\small 1},{\large b}_{\small 2},{\large b}_{\small 3},\dots,{\large b}_{\small n}$ can be any number or expression but ${\large b},{\large b}_{\small 1},{\large b}_{\small 2},{\large b}_{\small 3},\dots,{\large b}_{\small n}$ can't be zero$,{\large c}\dfrac{{\small a}}{\small{b}}$ and $-{\large c}\dfrac{{\small a}}{{\small b}}$ are the forms of mixed fraction$,{\large n}\in{\mathbb{Z}}_{>0}$ and ${\small\circ}$ is the sign of any fraction of the form ${\small\circ}\dfrac{{\large a}_{\small n}}{{\large b}_{\small n}}$ which can be $+$ or $-"?$ Does fraction include the numbers like $-\dfrac{-6}{-3},$$\dfrac{1}{-2\sqrt{e}},$$-\dfrac{-\sqrt{5}}{-3},$$\dfrac{\pi}{e},$$\dfrac{-\pi e}{\sqrt{2}},$$-\dfrac{-\sqrt[5]{99}}{-\sqrt[7]{4}},\dfrac{\varphi^3-\varphi^6+7}{-\ln e^3},$$\dfrac{1.73}{-1000},$$-\dfrac{-0.66}{1.45},$$\dfrac{6i}{6i},$$\dfrac{-2i}{0.91},$$\dfrac{0}{-1+2i},$$-\dfrac{0-\sqrt[11]{9}i}{-\sqrt{7}+8i},$$-\dfrac{4-\sqrt[3]{ei}}{-\sqrt{-\pi i^2}+2.5i},$$-400\%,\varphi\%,$$-1.67\times10^{-5},$$\dfrac{-1}{-2}\times-\dfrac{4}{-7}\times\dfrac{-5}{8},$$\left(\dfrac{6-9.8i}{-\sqrt{8}+3i}\right)^2,$${\large 3}\dfrac{{\small -3}}{{\small 1}},$$-{\large 9.81}\dfrac{{\small \pi}}{{\small i}},$${\large 2}\dfrac{{\small 5}}{{\small 3}}\times\dfrac{1}{4},$$\dfrac{7}{9}\times\left(6.67\times10^{-11}\right),$$\dfrac{1}{3}\times50\%$ and the expressions like ${\large x}\dfrac{\small{\sqrt{x^3}}}{\small{5x^2}},$$\dfrac{x^\pi}{y^{i\varphi}},$$\dfrac{2x^y}{3y^x},$$\sqrt[4]{\dfrac{1-x^2}{1+x^2}},$$\dfrac{x^5}{y^2}\times\dfrac{z^9}{x-21},$$\dfrac{{\large\int} \dfrac{1}{y^2}\: \mathrm{d}y}{404},$$\dfrac{\sin x}{\log x},$$\dfrac{| x |}{x},$$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$?$ Which kind of fractions do they belong to$?$

$3.$ How can we express the set of all fractions in set-builder notation and roster notation$?$

$4.$ Is there any special symbol which is used to denote the set of all fractions$?$ Can I use the symbol $\mathbb{F}$ to denote this set$?$

$5.$ What was the historical background of fraction$?$

(Basically, I want to know the answer of my second and third question more precisely. To better understand my question, I have mentioned some confusing numbers and expressions in question no. $2$. I think if I make my second question more shorter, maybe I will fail to highlight the main point. Thank you.)

You should fix your latex It is unreadble.

Asif Hossain wrote: ↑

Sun May 23, 2021 9:36 pm

You should fix your latex It is unreadble.

I think my latex commands are correct. If you are still facing trouble, you can see the attached image above.

Last example is similar to continued fraction maybe

Dustan wrote: ↑

Mon May 24, 2021 11:12 pm

Last example is similar to continued fraction maybe

Yes, it is a continued fraction. What about the other examples?

1) I don't know the formal definition of fractions, but intuitively I know is that fraction is just a way to represent numbers.

2)I think saying that a fraction is $\frac{a}{b}, a,b \in \mathbb{R}$ or $\mathbb{C}$ and $b \neq 0$ is enough to say to make all the fractions possible.(Maybe)

3)In the set builders method I guess it should be the same as I said in 2. But in the rooster method, it is not possible for you to list all fraction, even if you take only one fraction for each number in $\mathbb{R}$[Meaning $2= \frac{4}{2} = \frac{2}{1} = \cdots $ but you take only one of them] Because the set of real number is an uncountable infinity, I guess you heard that. But we can list the set of rational fractions.

4)I have never heard of any notation for fractions, probably because you can get all the numbers with the set of real and set of complex numbers.[Again note I think fractions are just a way to represent numbers]

5)idk

In the end I might be wrong but this is what I think.