Linked Questions

4
votes
3answers
331 views

Why don't we define division by zero as an arbritrary constant such as $j$? [duplicate]

Why don't we define $\frac 10$ as $j$ , $\frac 20$ as $2j$ , and so on? I know that by following the rules of math this eventually leads to $1=2$ , but we could make an exception and say that $j$ is ...
1
vote
2answers
2k views

Could 1/0 be an imaginary number? [duplicate]

There is no way to find the square root of a negative number. It just doesn't work. So the answer to the impossible question, "What number squared equals a negative number?" is just said to be $i$, an ...
6
votes
4answers
207 views

Could we “invent” a number $h$ such that $h = {{1}\over{0}}$, similarly to the way we “invented” $i=\sqrt{-1}$? [duplicate]

$\sqrt{-1}$ was completely undefined in the world before complex numbers. So we came up with $i$. $1\over0$ is completely undefined in today's world; is there a reason we haven't come up with a new ...
1
vote
1answer
71 views

If $i^2=-1$, at one point undefined, then why not define $\frac00$? [duplicate]

Would this even assist math in the way that $i$ did? Or is this just outright pointless and/or too exclusive to call for a definition?
0
votes
1answer
71 views

Is there a way of defining an “imaginary” number which is not i? [duplicate]

Assuming i does not necessarily have any other property but that it can represent (a, b) as a number in the form a + bi, how else could we define i? For instance, would it be appropriate to define a ...
3
votes
0answers
60 views

Why is there only one type of imaginary number? [duplicate]

We've defined the square root of -1 as an imaginary number i (or j, if you're a physicist). Is there any reason why we can't/haven't made other systems of imaginary numbers for other "impossible" ...
74
votes
5answers
5k views

A new imaginary number? $x^c = -x$

Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
34
votes
12answers
6k views

What allows us to use imaginary numbers?

What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers? In the beginning, when there were just reals, these operations were defined for them. Then,...
35
votes
5answers
6k views

Why should quaternions exist?

Why do quaternions exist? I want to believe they exist, but all I can think of are reasons they should not exist. These are my reasons. The quaternions are defined by the following equation: $$i^2 =...
46
votes
6answers
5k views

Why not to extend the set of natural numbers to make it closed under division by zero?

We add negative numbers and zero to natural sequence to make it closed under subtraction, the same thing happens with division (rational numbers) and root of -1 (complex numbers). Why this trick isn'...
21
votes
5answers
22k views

Is 'no solution' the same as 'undefined'?

Today in class my teacher wrote something along the lines of: $6^x = 0$ And proceed to heed a response from the class. A few people shouted undefined. So the teacher then writes: no solution $\...
9
votes
2answers
571 views

Could we invent a new number with $|p|=-1$?

We know that how a single definition $i^2=-1$ revolutionized our mathematics and solved many many problems. I wonder whether the definition $|p|=-1$ could have the potential of creating a new ...
6
votes
4answers
5k 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. ...
9
votes
3answers
897 views

difficulty of accepting $i^2 = -1$ for first timers [duplicate]

While teaching complex numbers for those who are encounter for the first time (usually 10th grader and 11th grader), I get the question like "Can even squared number give negative results? How ...
4
votes
4answers
169 views

Why don't we define $\frac00$ (or other undefined expressions) the same way we define $\sqrt{-1}$?

Why don't we define $\frac00$ (or other undefined expressions) the same way we define $\sqrt{-1}$? My guess, and my Calculus professor's opinion, is that by doing so we really do not achieve anything. ...

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