$i$ is defined as the square root of $-1$. I was wondering if number systems other than the complex numbers can be reached from the real numbers by a similar process. Like a number whose $\sin$ is $1.5$ or something.
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2While I don't know of anything quite like your example, its certainly possible to construct a system based around defining such a value and then see what results as a consequence. Something similar to the the complex numbers are the quaternions, which arise from defining $i^2=j^2=k^2=ijk=-1$. This non unique square root of -1 leads to the quaternions being noncommutative. – Tyberius Jul 31 '17 at 20:27
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4There actually are numbers $z \in \mathbb C$ such that $\sin z = 1.5$. – md2perpe Jul 31 '17 at 20:27
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Now you can ask your question about $e^x=0$. The answer you are getting don't apply in that case. – Hellen Jul 31 '17 at 20:32
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1Introduce a constant $\epsilon$ satisfying $\epsilon^2=0$ (but $\epsilon \neq 0$). Then you get hyperreals (or at least something like it). – md2perpe Jul 31 '17 at 20:35
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You may be interested in the Cayley-Dickson construction. – Noah Schweber Jul 31 '17 at 20:37
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3@md2perpe your construction leads to the dual numbers. The construction of the Hyperreals, which contain a huge variety of unlimited and infinitesimal numbers is much more complicated. – Hyperplane Jul 31 '17 at 20:47
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Thanks, @Hyperplane. I saw that the construction of the hyperreals is more complicated, but didn't find what this simple construction is really called. That's why I wrote "or at least something like it". – md2perpe Jul 31 '17 at 20:52
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I'm thinking a qualitative overview of hyperreal/surreal numbers is what's needed for this question – Eddy Jul 31 '17 at 21:49
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Related: https://math.stackexchange.com/questions/259584/why-dont-we-define-imaginary-numbers-for-every-impossibility – Hans Lundmark Aug 01 '17 at 07:51
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It should be noted that if you have two numbers $i$ and $j$ satisfying $i^2=j^2=-1$, and you want multiplication to commute and no zero divisors, then $j$ must either equal $i$ or $-i$. Proof: $(i-j)(i+j)=i^2+ij-ji-j^2=i^2-j^2=0$, thus either $i-j=0$ or $i+j=0$. Thus either $j=i$ or $j=-i$. – Akiva Weinberger Aug 01 '17 at 13:20
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@md2perpe Is the complex sin geometrically meaningful? Does it have a geometric instead of analytic construction? – Jacob Wakem Aug 01 '17 at 17:08
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Your question has two distinct sorts of answers.
The first answer is that the complex number field does a lot of things which you might naively think you need further extensions to do. For example, you don't need to go outside the complex numbers to find $\sqrt{i}$; that is merely $\frac{1+i}{\sqrt{2}}$.
In fact, $$ \sin\left( \frac{\pi}{2} - i\log\left( \frac{3-\sqrt{5}}{2}\right)\right)= \frac32 = 1.5 $$ to satisify your immediate question curiosity.
The other answer is that for various reasons, mathematicians do introduce fields beyond the complex numbers. A good example is the quaternions, which can be represented as linear combinations of four specially chosen $2\times 2$ matrices. The quaternions can be used to represent the effects of rotations on systems, for example. They are an example of a non-abelian field (that is, multiplication of quaternions does not commute).
Simply Beautiful Art
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Mark Fischler
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The standard terminology for "non-abelian field" is "division ring" or possibly "non-commutative division ring". – Daniel Schepler Jul 31 '17 at 20:49
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Others have pointed out that there are number systems extending the real numbers even further than the complex numbers do. If you want to generalize the construction of $i$, though, I suggest to step back a bit and consider number systems extending the rational numbers. There are quite many of these and the process used to construct them generalizes the construction of $\mathbb C$ out of $\mathbb R$ nicely. (The numbers you come up with this way can always be thought of as elements of $\mathbb C$, though.)
For example you might want to add a number $\xi$ to the set of rational numbers, such that $\xi^2=2$. If you still want addition and multiplication to work smoothly for all numbers, you will have to add further numbers to your set like $3\xi$ or $\frac1\xi$. In the end the set you constructed will consist of all formal sums $a + b\xi$ with $a$ and $b$ being rational numbers.
Similar constructions are possible for $\xi^2$ being any rational number! E.g. $\xi^2 = 3$ or $\xi^2 = 5$ will also yield specific number systems. In the case $\xi^2=-1$, we are back at the construction of $i$.
More generally you can construct number systems (fields) out of each other by adding roots of a polynomial: In the example above, $\xi$ has been a root of the polynomial $x^2-2$, while $i$ is a root of $x^2+1$.
The whole subject is studied more thoroughly in a dedicated branch of algebra, called the theory of field extensions.
kremerd
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I would add to the answers already here that if you're wondering specifically about whether or not any new number systems can be developed out of the real or complex numbers by extending some operation (like the square root) in a similar manner to how $i$ was, then you really can't.
This is because the complex numbers are "complete" in the sense that--with only a few exceptions--you can add, subtract, multiply, divide, exponentiate, take roots, logs, and trigonometric functions of any complex number(s). Within the real numbers, you may not be able to take the square root of anything; but within the complex numbers, you can; and the same goes for the other standard operations.
I say "anything", but there are a few exceptions. Even within the complex numbers, you still can't divide by zero, take the log of zero, raise zero to the zero, and more. As far as I know, you can't really extend these exceptions to make a new number system, because any such extension wouldn't behave like a normal number system (i.e. obeying the nice rules of algebra we're all used to that extend wonderfully and beautifully into the complex plane from the reals). But the point is that these are a relatively small set of exceptions, and so we may say that the complex numbers are "complete"---as opposed to the real numbers whose exceptions (in the case of square roots) amount to HALF the entire set!
WB-man
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