All About Fractions

For college and university level advanced Mathematics
Jalal
Posts:8
Joined:Mon Mar 22, 2021 10:39 am
All About Fractions

Unread post by Jalal » Sun May 23, 2021 11:24 am

(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.)
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Last edited by Jalal on Sun May 23, 2021 11:55 pm, edited 2 times in total.

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

Re: All About Fractions

Unread post by Asif Hossain » Sun May 23, 2021 9:36 pm

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. :(
Hmm..Hammer...Treat everything as nail

Jalal
Posts:8
Joined:Mon Mar 22, 2021 10:39 am

Re: All About Fractions

Unread post by Jalal » Mon May 24, 2021 12:04 am

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.

Dustan
Posts:71
Joined:Mon Aug 17, 2020 10:02 pm

Re: All About Fractions

Unread post by Dustan » Mon May 24, 2021 11:12 pm

Last example is similar to continued fraction maybe

Jalal
Posts:8
Joined:Mon Mar 22, 2021 10:39 am

Re: All About Fractions

Unread post by Jalal » Mon May 24, 2021 11:19 pm

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?

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Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm
Location:Dhaka, Bangladesh

Re: All About Fractions

Unread post by Mehrab4226 » Wed May 26, 2021 12:58 pm

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.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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