Now that my rant is out of the way, we can go back to the question I left hanging a couple of posts ago. What is the value of a negative number raised to a rational exponent? We can reduce this question down to the simple question, what is the square root of −1?
It's obviously not a real number, because the square of any real number is a positive number or 0. But that doesn't mean it doesn't exist. We can extend the real numbers to get the complex numbers. (Note that we are throwing out ∞ as a number and going back to the reals. Being a field is more important than having a greatest number.) By definition, i is a number such that i2=−1. Then the complex numbers are defined as numbers in the form a+bi, where a and b are both real numbers.
There is an alternative definition, which is equivalent. We can define z=(a,b), where (a,b) is an ordered pair of real numbers. The fact that it is an ordered pair means that (a,b)≠(b,a) if a≠b. We can then define addition and multiplication of the ordered pairs so that we get the same results as the previous definition. One advantage of this approach is that it encourages us to see complex numbers as points in a 2-dimensional plane.
The lecture notes begin with the definition as a+bi and then show the equivalence of the ordered pair (a,b). The book does the reverse. This is one subtle example of how the book and the lecture notes choose to emphasize different things, even when they cover the same material.
With either definition, the complex numbers are a field, and so retain all of the properties of the real numbers for addition and multiplication. In addition, for any complex number z and any real number x, except for z=x=0, zx is a complex number. However, the complex numbers are not ordered. One of the properties of ordered numbers is that x2>0 for any x≠0, and the whole point of the complex numbers is to define a number for which that is not true.
However, we can define the absolute value of a complex number. If z is a complex number, it can be written either as a+bi or as (a,b) where a and b are real numbers. Either way, the absolute value of z, |z|, is equal to √a²+b². Note that if we think of z as a point in a plane, then we can also think of z as a vector, in which case the absolute value of z is the length of the vector.
We can't speak of a complex number being greater or less than another complex number, but we can compare their absolute values. Note also that the real numbers are a subset of the complex numbers (any complex number where b=0 is a real number), and so the same definition for absolute value applies. In practice, for a real number a, if a>0 then |a|=a, and if a<0, then |a|=−a.
There are two useful inequalities that we can use with the absolute values of complex numbers. These inequalities hold for real numbers, but are almost too obvious to state, even though they are often useful. The first is the triangle inequality. Given two complex numbers z and w, |z+w|≤|z|+|w|. The name comes from a visual representation of z, w, and z+w as vectors in the plane. Visually, this is saying that with a triangle with sides of length z, w, and z+w, z+w must be less than or equal to the lengths of the other two sides. (They are only equal if all three vectors point in the same direction.) Of course, we can also rigorously prove the inequality by going back to the definitions for absolute value and for complex numbers.
The second inequality is essentially a restatement of the triangle inequality, but it is still useful as an independent statement. The inequality is |x-y|≤|x-z|+|z-y| for any three complex numbers x, y, and z. Again, since all real numbers are also complex numbers, these inequalities are true for the real numbers as well, and are more useful than they might seem based on how obvious they are.