Refining my knowledge of the imaginary number

Solution 1:

The definition of complex number is given on page 1 of Churchill and Brown's book:

Complex numbers can be defined as ordered pairs $(x, y)$ of real numbers…

The definition of $i$ is given on page 2:

…let $i$ denote the pure imaginary number $(0,1)$…

So to answer your question, $i$ is not defined by the equation $i=\sqrt{-1}$, nor is it defined by the equation $i^2=-1$. Instead, it is defined as a particular ordered pair of real numbers, $i=(0,1)$. Then, given the definition of complex multiplication, one proves that $i^2=-1$.

Solution 2:

1. As you probably know, there are two solutions to $x^2+1=0$. We arbitrarily call one of them $i$ and the other $-i$, but this choice could have been made the other way (which is why complex conjugation is an automorphism of $\mathbb C$). So, both definitions are essentially correct, but of course, the first one is slightly less correct.

2. What is the difference between a "concept" and a "quantity"? This question seems more philosophical than mathematical...

3. Well, $i^i$ is usually defined as $e^{i\log i}$, where $e^{a+bi}=e^a(\cos b+i\sin b)$ and $\log(re^{i\theta})=\ln r+i(\theta+2n\pi)$, where $n$ is any integer (the logarithm is a multivalued function since the exponential function is not injective)

Solution 3:

it is easy to become the captives of our preconceptions. if you have honed a very efficient machine $\sqrt{}$ for extracting the square roots of positive real numbers you will rightly rub your head in puzzlement if asked to operate it on $-1$. but rather than focus on the genesis of $i$, celebrate the vistas it opens up, with its quantum leap through the sign barrier which constrains the arithmetic of real numbers. that one magical property $i^2=-1$ has far-reaching consequences.

take, for example, how $i$ interacts with another very basic mathematical reality - the circular functions and their remarkable combinatorial properties. it can be shown on the basis of geometry that: $$ \def\c{\cos}\def\s{\sin} \c(x+y)=\c(x)\c(y)-\s(x)\s(y) $$ and $$ \s(x+y)=\s(x)\c(y)+\s(y)\c(x) $$ using these identities it is easy to give an inductive proof that for any integer $n \ge 0$ we have: $$ (\c(x) + i \s(x))^n = \c(nx) + i \s(nx) $$ this is easily extended to rational values of $n$ and Lo! suddenly you have the infinite family of roots of unity at your disposal. want a number whose fifth power is $1$? well, you could always have $1$! but now there are also $e^{\frac{2 \pi i}{5}},e^{\frac{4 \pi i}{5}},e^{\frac{6 \pi i}{5}}$ and $e^{\frac{8 \pi i}{5}}$. of course in order to win the right to describe them as exponentials you have to plug $x=i\theta$ into the McLaurin expansion for $e^x$ to see how this gives rise to the identity: $$ e^{i\theta}=\cos\theta +i\sin\theta $$ in short, learn to work with $i$, enjoy it! you can of course explore various dusty definitions, but $i$ is such a special number that you cannot really assimilate it to any previous mode of mathematical thinking. look at the transformations it has wrought in physics over the last 150 years.

Solution 4:

I would like to add a bit to Chris Culter's excellent answer. It is possible that there is more than one possible correct definition for $i$. The definition Chris Culter quoted from your book is one such. One might ask if the definitions you quoted are also correct. The answer is that neither one is.

The expression “$\sqrt x$” is defined to mean the unique non-negative real number $y$ such that $y^2 = x$. This is a good definition when $x$ is a non-negative real number, but it is completely meaningless for other values of $x$, because for $x = -1$, say, there is no non-negative real number $y$ such that $y^2 = -1$. So an attempt to define $i$ by saying $i=\sqrt{-1}$ is an immediate failure; there is no such thing. The $\sqrt{}$ operator simply does not do that.

Going the other way, one can't define $i$ by simply saying that $i^2=-1$; that is not a definition. It describes the properties that we want $i$ to have, but it does not describe any object that actually has those properties, and there may not actually be one. It's very easy to assemble a list of properties that is not possessed by any object. For example, “the largest invisible purple water buffalo in Dubuque, Iowa” is a description of that sort; you can describe its properties, but there is no such thing. Or again, a recent question here asked for the volume of a polyhedron whose faces are five equilateral triangles. But there is no such polyhedron: every polyhedron with only triangular faces has an even number of them.

We have an idea of a number $i$ with $i^2=-1$, but in order to show that this idea is coherent, we must actually construct something that behaves that way. One way to do this is the construction in your book:

1. We take the set of pairs of real numbers $\langle a,b\rangle$
2. Define addition and multiplication operations on them
3. Show that these operations have the properties one usually expects, such as $a\cdot(b+c) = (a\cdot b ) + (a\cdot c)$.
4. Show that the resulting structure contains a proper substructure, namely the set of elements of the form $\langle a, 0\rangle$, that behaves just like the real numbers, so we can identify this substructure as the real numbers, and the element $\langle -1, 0\rangle$ is effectively the same as the real number $-1$
5. Show that this structure contains an element, namely $i=\langle 0, 1\rangle$ (or $i=\langle 0,-1\rangle$ if you prefer), which has the property that $i\cdot i = -1$.

This is not the only way to define $i$; there are many structures we could invent that would behave the way we want the complex numbers to behave. In advanced algebra courses, one uses the same strategy as in the previous paragraph, but one constructs the complex numbers as classes of polynomials instead of as ordered pairs:

1. Divide the polynomials into certain classes.
2. Define addition and multiplication of these classes
3. Show that these operations have the usual properties
4. Show that some of the classes behave the same way that the real numbers behave, and in particular that there is a class $[-1]$ that corresponds to the real number $-1$
5. Show that there is a certain class $[x]$ which has the property that $[x]\cdot[x] = [-1]$.

So there is more than one possible definition, but neither one of the suggestions you gave is enough to do it. Vladimirm's answer elsewhere in this thread shows yet another way to define $i$, again following the same basic strategy.