The Algebraic Field C and Rectangular Form

The algebraic eld of complex numbers is a set of objects C called complex numbers to-gether with two binary operations, addition and multiplication, satisfying the eld axioms;

i.e. the operations are closed in C, they are associative and commutative, multiplica-tion distributes over addimultiplica-tion, both operamultiplica-tions have unique identity elements in C, every complex number has a unique additive inverse, and every complex number dierent from the additive identity has a unique multiplicative inverse. More concretely, elements of C are ordered pairs of real numbers (x, y). In fact, that complex numbers are generated by combining simpler parts is the reason for the identier complex. Addition is dened com-ponentwise as in the familiar addition of vectors in the plane R². So we have, for complex numbers z = (z1, z₂) and w = (w1, w₂),

z + w = (z₁, z₂) + (w₁, w₂) = (z₁+ w₁, z₂+ w₂) , but multiplication is dened according to

zw = (z₁, z₂) (w₁, w₂) = (z₁w₁− z₂w₂, z₁w₂+ z₂w₁) . We make the following observations

(0, 0) + (z₁, z₂) = (0 + z₁, 0 + z₂) = (z₁, z₂) = (z₁+ 0, z₂+ 0) = (z₁, z₂) + (0, 0)and (1, 0)(z1, z2) = (1z1− 0z2, 1z2+ 0z1) = (z1, z2) = (z11 − z20, z10 + z21) = (z1, z2)(1, 0)

and, also,

(z₁, z₂) + (−z₁, −z₂) = (z₁+ (−z₁), z₂+ (−z₂)) = (0, 0)

= ((−z1) + z1, (−z2) + z2) = (−z1, −z2) + (z1, z2) so that it is immediately apparent that the additive and multiplicative identities are (0, 0)and (1, 0), respectively, and to each element (z1, z₂) corresponds the additive inverse (−z₁, −z₂). The multiplicative inverse is more interesting and will be found subsequently.

Firstly, when we think about R², the standard basis, written as row vectors, comprises (1, 0) and (0, 1). We have found that, in the eld of complex numbers, the former is the multiplicative identity, but what role does the latter play? Consider the following decom-position

(z₁, 0)(1, 0) + (z₂, 0)(0, 1) = (z₁· 1 − 0 · 0, z₁· 0 + 0 · 1) + (z₂· 0 − 0 · 1, z₂· 1 + 0 · 0)

= (z1, 0) + (0, z2) = (z1, z2).

Furthermore, that for any product of the form (α, 0)(z1, z2)for α ∈ R, we have (αz1, αz2) allows the above decomposition to be written more compactly in the form of a linear combination of vectors in R² with real scalars z1 and z2,

z₁(1, 0) + z₂(0, 1) = (z₁, 0) + (0, z₂) = (z₁, z₂).

To introduce some additional notation, because (1, 0) is the multiplicative identity, it is naturally denoted by 1, and we have 1z = z1 = z for any z ∈ C. Also, if (0, 1) is denoted i, from the above decomposition, we may write any complex number (z1, z₂)as z1+ iz₂. This is called the rectangular form of the complex number (z1, z₂), the complex number i is called the imaginary number, z1 is called the real part of (z1, z₂), and z2 is called the imaginary part of (z1, z₂). Additionally, because (0, 0) is the additive identity, it is naturally denoted by 0, and we have 0 + z = z + 0 = z. Furthermore, with the rectangular form, addition and multiplication follow the approach taken in polynomial addition and multiplication with i playing the role of x with the additional possibility of reinterpreting x² as −1, e.g., (a + bx) + (c + dx) = (a + c) + x(b + d)and (a + bx)(c + dx) = ac + bdx²+ x(ad + bc) and, with x taken as i, these become

(a + ib) + (c + id) = (a + c) + i(b + d) and (a + ib)(c + id) = ac − bd + i(ad + bc).

These are precisely the forms expected when the denitions of addition and multiplication are cast in rectangular form.

It is also worth stating that (z1, z2)is often called the geometric image of the complex number z1+ iz₂ because it is a point in the plane corresponding to the complex number;

this is the source of the interpretation of constructs of planar geometry in terms of complex numbers and vice versa. Because the rectangular form is easier to manipulate, it, rather than the vectorial form, is used persistently. This is unless one wishes to emphasize or to

draw attention to the geometry, as is often desirable. We develop this idea further in the subsection on geometry to which this is building.

Complex numbers admit an interesting operation which eectively reverses the sign of the imaginary part. The conjugate, z, of a complex number, z = a + ib, is dened to be a − ib, so

a + ib = a − ib.

We also have an operation which measures complex numbers in some sense (but neverthe-less does not yield a total ordering of C) and, as we will see, has an obvious geometric interpretation. The modulus, |z|, of a complex number, z = a+ib, is dened to be√

a²+ b², so

|a + ib| =√

a²+ b².

We make the following observation for a complex number z = a + ib, zz = (a + ib) (a + ib) = (a + ib) (a − ib) = a²+ b² so

zz = |z|².

With these additional operations, we are now in a position to discuss the multiplicative inverses, z⁻¹, of non-zero elements z ∈ C, i.e. elements z = z1 + iz₂ with z1 and z2 not

is a suitable denition of the multiplicative inverse. Notice that it is perfectly well dened when the denominator is not vanishing, that is, when z is non-zero. In case the reader nds this to be dicult to remember, the process of dividing by a non-zero complex number in rectangular form may be achieved by choosing to multiply the numerator and denominator of the given quotient by the conjugate of the denominator as in

a + ib In particular, to obtain the multiplicative inverse of z = z1+ iz₂, multiply numerator and denominator by the conjugate to obtain

1

as expected. Now, prior to exploring yet a third form of complex numbers and the corre-sponding geometric interpretations, we introduce yet another important denition, func-tions which retrieve real and imaginary parts of complex numbers in terms of the number

and its conjugate. Observe the following for the complex number z = a + ib, z + z

2 = (a + ib) + (a + ib)

2 = (a + ib) + (a − ib)

2 = 2a

2 = a, the real part of z, and

z − z

2i = (a + ib) − (a + ib)

2i = (a + ib) − (a − ib)

2i = 2ib

2i = b,

the imaginary part of z. These calculations lead to natural denitions for the functions Re, Im : C → R dened by

Re(z) = z + z 2 and

Im(z) = z − z 2i

which recover the real and imaginary parts of z, so that, to reiterate, for complex number z = z₁ + iz₂, Re(z) = z1 and Im(z) = z2. It should also be apparent that a complex number z ∈ C is purely real if and only if z = Re z and that z ∈ C is purely imaginary, i.e. a non-zero real multiple of i, if and only if Re z = 0 but Im z 6= z.

We are now prepared to provide several more denitions, which are motivated by the geometric properties of complex numbers. We begin with the polar form, and interpret the algebraic properties and operations presented in this section in the geometric framework inherent in the denition of the complex number as ordered pair.