Mathematicians first approached complex numbers cautiously. Although it was clear that they were useful in solving certain problems at least formally (for example, they are needed to even make sense of the formulas we found in the previous lectures) what was not clear was that they made sense. Perhaps indiscriminate use of them would lead to contradictions.
Gauß solved this problem by realizing that one can define and its operations in terms of and its operations. As long as we are willing to accept that makes sense, then no contradictions will come up from the use of complex numbers.
Definition. The set of complex numbers is simply We define addition and multiplication as follows:
- If and then addition is just coordinatewise addition,
- With as above, we set
One easily checks that can be identified with the -axis One identifies with One easily verifies that and We say that the identification is a homomorphism.
Notice that complex addition is just addition of vectors, which has a clear geometric interpretation. Complex multiplication also has a geometric interpretation, as we will see below, but it is more subtle.
Of course, the point of introducing complex numbers is to make sense of square roots of negative numbers. Define Then one easily checks that (i.e., ). Due to these correspondences, one simply writes instead of
Theorem. with the operations as defined above is a field. This means that the following properties hold:
- (Commutativity of addition). For all complex numbers we have that
- (Commutativity of multiplication). Similarly, for all complex numbers
- (Associativity of addition). For all complex numbers we have that
- (Associativity of multiplication). Similarly, for all complex numbers
- (Distributivity). For all complex numbers one has that
- (Additive identity). There is a complex number such that for all complex numbers
- (Multiplicative identity). There is a complex number such that for all complex numbers
- (Additive inverses). With as above, for any complex number there is a complex number such that
- (Multiplicative inverses). For any there is a complex number such that Here, and are as above.
The proof of the theorem is more or less immediate. For example, one easily checks that we can take and (In fact, one checks that these are the only values that work.) Only item 9. requires some care. For this, assume that Then if and only if i.e., if and only if the system
in the two unknowns has a solution. It is easy to verify that, indeed, the system has a unique solution given by and In other words, it is indeed the case that whenever then there is an inverse given by
There is another way of proceeding here.
Definition. The complex conjugate of is the complex number
Note that so is a nonnegative complex number, and it is always positive, except when
Definition. The norm, or magnitude, or size, or modulus, or absolute value, of the complex number is
Notice that if is real, i.e., if for some then its absolute value as a real number coincides with the absolute value as a complex number.
Suppose now that Then, if the notation is to make any sense, we must have which coincides with the formula found before.
Definition. If then i.e.,
Conjugation is also a homomorphism:
Also, notice that
Definition. The argument of the complex number is the oriented angle between the -axis and the vector
The argument is only defined up to integer multiples of i.e., if is an argument of then so are etc. Years ago, people would say that the argument is a multivalued function or some such nonsense. Sometimes it is convenient to act as if the argument of 0 is defined. If so, we will simply say that any real is an argument of
An easy geometric argument shows the following:
Lemma. For any nonzero complex number , we have for any argument of If the equality holds for any The modulus of any number of the form is 1.
This representation is rather useful, as one easily verifies that
where is any argument of and is any argument of Thus, we have:
Lemma. The modulus of a product is the product of the moduli, and the argument of a product is the sum of the arguments (mod ).
This means that complex multiplication corresponds to a dilation followed by a rotation.
A particularly important particular case of the above is the following result:
Lemma. (De Moivre). For any positive integer and any complex number we have
Proof. There are several ways of presenting this result. Let’s try an argument by induction. First, the result is true for and there is nothing to show then. If the result holds for then where we are using the previous lemma. This is clearly equivalent to the case of the statement, and we are done.
We can define as and it is equally easy to check that De Moivre’s formula also holds for exponents that are negative integers.
The reason why one cares about De Moivre’s formula is that it allows us to find -th roots of any complex number.
Definition. Let be a positive integer. The complex number is an -th root of iff
Let’s say that and that where is an argument of and is an argument of That is an -th root of means that
- Since are nonnegative real numbers, there is only one possible value of namely what one usually denotes
- Either in which case as well, or else where the equality is of course up to an integer multiple of
Condition 2 means that for some integer Notice that if is a multiple of then the corresponding to and the corresponding to are the same (modulo ). On the other hand, if is not a multiple of then the values of we obtain are different, so the values of they correspond to are also different.
This means that there are exactly distinct complex -th roots of any nonzero complex number
Example. Take so and let the cubic roots of will be the numbers for These are, respectively, the numbers and Of course, these numbers coincide with the numbers we obtain when we solve the cubic equation