22 operation Properties of Infinity they will not teach you in class
The Real number system is so diverse that sometimes you will think it is the largest set of numbers in mathematics,but it is not,remember the real number is contained in the complex number, i.e $\mathbb{R}\subset\mathbb{C}$. But nevertheless the real number is truly large,so many things we will not be taught by our lecturers in the class, but this things require extra mile hard work for their discoveries, and as you continue to research and explore the world of mathematics,you will find it interesting and by that "a new mathematician is being groomed".
Today i will be posting 22 operation properties of the infinity $\infty$ of the real number system.
Today i will be posting 22 operation properties of the infinity $\infty$ of the real number system.
Given a number say $r\in\mathbb{R}$, then the following truly holds.The reader must understand that this operation properties are just conjectures and not established laws that has proofs and other logical modifications.
The properties include Addition laws which has 4 laws,multiplication law has 6 laws or axioms,division has 7,and the power laws has 5 laws.
Addition Law
(1) $\pm\infty+r=\pm\infty$ for $r\in\mathbb{R}$
(2) $+\infty+(+\infty)=+\infty$
(3) $-\infty+(-\infty)=-\infty$
(4) $+\infty+(-\infty)$ is undefined
(1) $\pm\infty+r=\pm\infty$ for $r\in\mathbb{R}$
(2) $+\infty+(+\infty)=+\infty$
(3) $-\infty+(-\infty)=-\infty$
(4) $+\infty+(-\infty)$ is undefined
Multiplication Law
(5) for $r\in\mathbb{R}$ and $r>0$, then $r(\pm\infty)=\pm\infty$
(6) for $r\in\mathbb{R}$ and $r<0$, then $r(\pm\infty)=\mp\infty$
(7) $0.(\pm\infty)=0$
(8) $\infty\times\infty=\infty$
(9) $-\infty\times{-}\infty=\infty$
(10) $\infty\times{-}\infty=-\infty$
(5) for $r\in\mathbb{R}$ and $r>0$, then $r(\pm\infty)=\pm\infty$
(6) for $r\in\mathbb{R}$ and $r<0$, then $r(\pm\infty)=\mp\infty$
(7) $0.(\pm\infty)=0$
(8) $\infty\times\infty=\infty$
(9) $-\infty\times{-}\infty=\infty$
(10) $\infty\times{-}\infty=-\infty$
Division Law
(11) for $r\in\mathbb{R}$ then $\frac{r}{\infty}=0$
(12) for $r\in\mathbb{R}$ then $-\frac{r}{\infty}=0$
(13) for $r\in\mathbb{R}$ and $r>0$, then $\frac{-\infty}{r}=-\infty$
(14) $\frac{\pm\infty}{\pm\infty}$=undefined
(15) $r\in\mathbb{R}$ and $r<0$ then, $\frac{-\infty}{r}=\infty$
(16) $\frac{\infty\text{(countable)}}{\infty\text{(uncountable)}}=0$
(17) $\frac{\infty\text{(countable)}}{\infty\text{(countable)}}=\infty$
(11) for $r\in\mathbb{R}$ then $\frac{r}{\infty}=0$
(12) for $r\in\mathbb{R}$ then $-\frac{r}{\infty}=0$
(13) for $r\in\mathbb{R}$ and $r>0$, then $\frac{-\infty}{r}=-\infty$
(14) $\frac{\pm\infty}{\pm\infty}$=undefined
(15) $r\in\mathbb{R}$ and $r<0$ then, $\frac{-\infty}{r}=\infty$
(16) $\frac{\infty\text{(countable)}}{\infty\text{(uncountable)}}=0$
(17) $\frac{\infty\text{(countable)}}{\infty\text{(countable)}}=\infty$
Power Law
(18) for $r\in\mathbb{R}$ and $0<r<1$, then $r^{\infty}=0$
(19) for $r\in\mathbb{R}$ and $r>0$, then $r^{\infty}=\infty$
(20) for $r\in\mathbb{R}$ and $r=1$, then $r^{\infty}=\text{undefined}$
(21) for $r\in\mathbb{R}$ and $r>0$, $\infty^r=\infty$
(22) for $r\in\mathbb{R}$ then, $\infty^{\infty}=\infty$
(18) for $r\in\mathbb{R}$ and $0<r<1$, then $r^{\infty}=0$
(19) for $r\in\mathbb{R}$ and $r>0$, then $r^{\infty}=\infty$
(20) for $r\in\mathbb{R}$ and $r=1$, then $r^{\infty}=\text{undefined}$
(21) for $r\in\mathbb{R}$ and $r>0$, $\infty^r=\infty$
(22) for $r\in\mathbb{R}$ then, $\infty^{\infty}=\infty$
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