# 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 number i such that $$2^i=-1$$ instead of a number i such that $$i^2=-1$$ Are there any such numbers that are well established or some similar concept (i.e. a number which is essentially an ordered pair or, more generally, a list)?

I ask this primarily because I want to know if there is any reason not to define i in some other way, apart from the fact that there are some often used equations like Euler's formula which make use of the definition of i.

• There is already such a complex number: Indeed if $$j=\frac{i \pi}{\ln(2)}$$ then $2^j=-1$. – Maximilian Janisch Feb 15 at 0:50
• Does this answer your question? Why don't we define "imaginary" numbers for every "impossibility"? – Brian Feb 15 at 1:00
• What do you want $i$ to DO? You could define $i$ as $7$ as $(a,b)$ is a number represented as $a+bi = a+7b$ for all we care. So I don't really understand what you are asking.... By the way, as Maximillian Janisch points $2^{variable} = -1$ does have a complex solution of $variable = \frac {imaginaryunit\pi}{\ln 2}$ where $imaginaryunit$ is the imaginary unit where $imaginaryunit^2 = -1$. – fleablood Feb 15 at 1:12
• Here's an interesting but related tidbit. Did you know you can define a $j$ such that $j^2=-1$ but $j\ne i$ ? Look up quaternions, a four-dimensional number space. You can look at any complex subspace $a+bi$, $a+bj$ and $a+bk$ and observe purely complex properties within them. It isnt until you mix and match and look at the entire space $a+bi+cj+dk$ that you get different properties. So, in fact, there is no dilemma that needs resolution between the $i$ used by mathematicians and the $j$ used by engineers; they dont even have to be the same imaginary axis. – SquishyRhode Feb 15 at 1:50

Interestingly enough, the example you wrote of having some number (let's call it $$j$$ as to not confuse it with the imaginary unit $$i$$) such that $$2^j=-1$$ introduces the idea of taking logarithms of negative numbers.
The quaternions are given by elements $$a+bi+cj+dk$$ where $$a,b,c,d\in\mathbb{R}$$ and $$i^2=j^2=k^2=ijk=-1$$.
To be clear, we can define all sorts of "exotic numbers", but having different definitions, mean they will behave differently. So it doesn't really make sense to ask for "a different definition of $$i$$", the symbol $$i$$ specifically refers to a solution to $$x^2+1=0$$. Your final question might be better phrased as why are we interested in the imaginary number $$i$$, as opposed to other exotic numbers? As expressed above, we do consider other kinds of numbers, and sometimes they bring us back to $$i$$ anyways!