Why this Algebra is not possible??

Question

We know that, I have calculated this, $$1^1=1,\; 1^2=1,\; 1^3=1,\; \ldots\; 1^n=1$$
and $$1^{-1},\; 1^{-2},\; 1^{-3},\;\ldots\; 1^{-n}=1$$

Now $$0^1=0,\; 0^2=0,\; 0^3=0,\;\ldots \; 0^n=0$$ What I have found on Math Stack Exchange and on Wikipedia

Then why, $$0^{-1}\ne 0,\; 0^{-2}\ne 0,\; 0^{-3}\ne 0 ,\; \ldots \;0^{-n}\ne 0$$

i.e. why is $$\frac 10 \ne 0 \; \text{and}\; \frac 10= \infty$$

and why we can't use our logic to solve this?

Also consider $\frac{1}{x}$ if x tends to $0$ then it will tends to infinity. If it is having a answer $\infty$ and -$\infty$ then it is a quadratic equation.

How can it be possible? Is it a quadratic equation? That have two possible solutions!!

1/0 is not "$\infty$". It is not anything: it is simply not defined. When one writes "1/0" one is writing something meaningless. Not every string of symbols that we can write has a meaning — if I write "/++/-23*" it does not mean anything, and similarly "1/0" does not mean anything. — Mariano Suárez-Álvarez, Mar 12 '17 at 03:24
Division is the inverse operation to multiplication. i.e. $\frac{6}{3}=2$ because $2 \cdot 3 = 6$. So if $\frac{1}{0} = x$ that would mean that $x \cdot 0 = 1$. We don't define any $x$ that does this. — wgrenard, Mar 12 '17 at 03:29
We could define 1/0 to be anything we like. We could define it to be 0, to be 14 or to be equal to the chair I am sitting on right now. We don't. Why? Because of various reasons, and you could ask what are those reasons (although I am sure someone has already asked that in this site, so please search before you ask) — Mariano Suárez-Álvarez, Mar 12 '17 at 03:30
Yes i know it is ∞ but why someone give the answer that it can vary from -∞ to ∞ ? — Creepy Creature, Mar 12 '17 at 03:33
As I said, it is not $\infty$. The string of symbols "1/0" does not mean anything at all. — Mariano Suárez-Álvarez, Mar 12 '17 at 03:36
Well, that is quite irrelevant! It is not a matter of opinion. — Mariano Suárez-Álvarez, Mar 12 '17 at 03:42
@CreepyCreature Take a look at the graph of $\tan x$ at $x=90^{\circ}$ or of $1/x$ at $x=0$. Based off of those what do you think that $1/0$ or $\tan 90^{\circ}$ should mean? — wgrenard, Mar 12 '17 at 03:46

score 3 · Answer 1 · answered Mar 12 '17 at 03:09

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How would logic dictate this? Yes, the $0$ and $1$ behave the same in some of the ways you indicate... but that does not mean that they should behave the same way in all those ways.

It's like saying: you and I both have two legs. You and I both have two arms. I have red hair. Therefore, you should have red hair.

answered Mar 12 '17 at 03:09

Bram28

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So 1/0 should give exact solution like ∞ then why it gives varying solution from -∞ to ∞? Is this thing is possible in maths – Creepy Creature Mar 12 '17 at 03:37
@CreepyCreature In general, $x/y$ is defined as 'that number z such that x = y*z'. But there is no such number for $1/0$. So, $1/0$ is undefined. It doesn't vary, and it is not $\infty$ .. It is simply not defined. – Bram28 Mar 12 '17 at 03:48
But as much I know in 1/x equation if x tends to 0 then it is obvious that it will tends to ∞. – Creepy Creature Mar 12 '17 at 03:51
In the definition of "a/b" there is no "tends" mentioned. That something tends to something is as relevant to the value of 1/0 as the color of the chair I am sitting on. The definition of x/y is precisely what Bram28 said, nothing more and nothing less. – Mariano Suárez-Álvarez Mar 12 '17 at 03:53

Why this Algebra is not possible??

1 Answers1