Linked Questions

10
votes
6answers
2k views

Defining division by zero [duplicate]

I have looked through some of the previous questions posted on this topic, and I think mine is different. Is there a flaw in defining division by zero? For example, define $\frac{a}{0} = \infty_a$ ...
5
votes
4answers
3k views

Why does division by zero not have an imaginary number “option”? [duplicate]

In regular math, you cannot get the square root of a negative number. Likewise, you cannot divide by zero. Both dividing by zero and taking the square of a negative have no place in real life. ...
5
votes
2answers
254 views

Dividing a number by zero [duplicate]

Why can't you divide a number by zero? It is possible to say $\sqrt{-1}$ is an imaginary number $i$, but why can't you say $\frac{1}{0}$ is also an imaginary number $z$ (for example)?
4
votes
4answers
112 views

Treating $\frac{n}{0}$ as a constant? [duplicate]

Is there any way of treating $\frac{1}{0}$ without breaking maths? I tried just the variable $\lambda$, thinking that it would be easy to manipulate: $$\frac{2}{0} = 2\cdot\frac{1}{0} = 2\lambda$$ But ...
0
votes
2answers
99 views

A solution to the equation $\frac{1}{x}=0$ [duplicate]

The number $i$ is defined as a solution to the equation $x^2+1=0$. How come no one has yet defined a number $j$ as a solution to the equation $\frac{1}{x}=0$? The purpose of course is to be able to ...
1
vote
3answers
127 views

Can we make illegitimate operations with $0$ legitimate by adding a few more axioms? [duplicate]

Just out of curiosity, can we make illegitimate operations with $0$, say, division by $0$ legitimate simply by imposing additional axioms? If so, then what consequences may follow? If the answer is ...
0
votes
2answers
416 views

Undefined cases of 0/0 [duplicate]

Why 0/0= undefined? We always write $0/x\in\mathbb R$, where $x \in\mathbb{R}\setminus\{0\}$, and this $0/x = 0$. Again $x/0$ is always undefined. Now when it's about $0/0$ why we don't follow the ...
-6
votes
1answer
167 views

Does 0/0 = a new branch of numbers? Have I made a mistake in the equation? [duplicate]

So I thought... If $\frac00 = x$... then $0 = x\cdot0$... then $0x = 0$... then its technically possible to divide by $0$ again $\frac{0x}0 = \frac00$ ... since $\frac00 = x$ and $\frac{0x}0 = \...
0
votes
1answer
155 views

Why do we have a symbol for root of -1, but none for divide by zero? [duplicate]

Even though there is no real answer to the equation √-1, we give it the symbol i. What is the reason that there is a symbol for ...
-2
votes
1answer
85 views

Are there number systems that fix divide-by-zero? [duplicate]

Natural numbers are closed under addition and multiplication, but not subtraction. Fixed by... Integers are closed under subtraction, but not division. Fixed by... Rational numbers are closed under ...
1
vote
0answers
35 views

Changing the zero product property and defining division by zero [duplicate]

I know that defining division by zero is not possible because it violates the zero product property we define, that is, $0\times a=0$ for every $a$. I wonder whether we can somewhat circumvent and ...
-2
votes
1answer
26 views

Divison by Zero [duplicate]

I have seen multiple answers on the web, but I can't get my mind around why division by zero outputs an error and not zero. Can anyone explain this in laymen's terms?
423
votes
23answers
128k views

How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my math ...
303
votes
23answers
12k views

Why don't we define “imaginary” numbers for every “impossibility”?

Before the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had, so we said $i$ to be a new type of number. How come we don't do the ...
4
votes
2answers
3k views

Will Division by Zero be Defined Eventually? [duplicate]

Possible Duplicate: Division by $0$ I've always been inclined to believe that x/0 = NaN is a placeholder for a character or constant that no one has created ...

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