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Properties of Argument

Introduction

In polar form, complex numbers are expressed as r(cos(θ)+isin(θ)), directly incorporating both magnitude and argument. This representation reveals the angular aspect of complex arithmetic, particularly the arguments' behavior under operations like multiplication and division. In this section and upcoming sections we are going to see some very important and surprising applications of complex numbers.

Properties of Argument

Two most important properties of argument is about operations of multiplication and division.

  1. The argument of the product of two complex numbers is the sum of their arguments.

    arg(z1z2)=arg(z1)+arg(z2)
    Proof:

    Consider two complex numbers in polar form, z1=r1(cosθ1+isinθ1) and z2=r2(cosθ2+isinθ2). The product z1z2 is:

    z1z2=r1r2[(cosθ1cosθ2sinθ1sinθ2)+i(cosθ1sinθ2+sinθ1cosθ2)].

    Using trigonometric identities, this can be simplified to:

    z1z2=r1r2[cos(θ1+θ2)+isin(θ1+θ2)].

    Hence, arg(z1z2)=θ1+θ2, which establishes the property:

    arg(z1z2)=arg(z1)+arg(z2).

    Geometrical Interpretation of Multiplication

    Multiplication of Two Complex Numbers

    We may interpret this multiplication in a pretty useful way. Multiplication of z2 by z1 has resulted in the rotation of z2 by angle θ1 in the anticlockwise direction and its magnitude has changed from r2 to r1r2.

    Principal Argument with respect to multiplication

    If θ1 and θ2 are principal arguments of two complex numbers z1 and z2, where both arguments are within the interval (π,π], then adding θ1 and θ2 may result in an angle that does not lie within this principal range. Hence, while arg(z1z2)=arg(z1)+arg(z2) holds for arguments in a general sense, the sum arg(z1)+arg(z2) may not be the principal argument of the product z1z2.

    To find the principal argument of z1z2, one must ensure the result is within the principal range (π,π]. If θ1+θ2 falls outside of this range, an adjustment of ±2kπ, for some appropriate kZ, is necessary to correct it. Thus, the principal argument of the product, denoted simply as arg(z1z2) when referring to its principal value, is the adjusted sum that falls within the specified interval.

  2. The argument of the quotient of two complex numbers is the difference between the argument of the numerator and the argument of the denominator.

    arg(z1z2)=arg(z1)arg(z2).
    Proof:

    Let z1=r1(cosθ1+isinθ1) and z2=r2(cosθ2+isinθ2) be two complex numbers in polar form. The quotient z1z2 is:

    z1z2=r1r2((cosθ1+isinθ1)(cosθ2isinθ2)cos2θ2+sin2θ2).

    Using trigonometric identities, we expand the numerator and use the fact that cos2θ2+sin2θ2=1 to simplify to:

    z1z2=r1r2(cos(θ1θ2)+isin(θ1θ2)).

    Therefore, the argument of the quotient is the difference of the arguments:

    arg(z1z2)=θ1θ2,

    which establishes the property:

    arg(z1z2)=arg(z1)arg(z2).
  3. If z is a real number, then the argument of z, denoted as arg(z), is either 0 or π. The value depends on the sign of z:

    • If z is a positive real number, then arg(z)=0.
    • If z is a negative real number, then arg(z)=π.
  4. If z is purely imaginary, then arg(z) is π2 or π2, depending on the sign of the imaginary part:

    • If the imaginary part is positive, arg(z)=π2.
    • If the imaginary part is negative, arg(z)=π2.
  5. The argument of the product of multiple complex numbers z1,z2,,zn is the sum of their individual arguments.

    arg(z1z2zn)=θ1+θ2++θn=argz1+argz2++argzn+2kπ,
  6. The argument of a power of a complex number z raised to an integer exponent n is n times the argument of z, modulo 2π. This can be stated as:

    arg(zn)=narg(z)+2kπ,

    where k is an integer such that the argument lies within the principal branch (π,π] if we are considering the principal value.

    Proof:

    To prove that arg(zn)=narg(z), consider z raised to the nth power, which is equivalent to z multiplied by itself n times:

    zn=zzz.

    The argument of a product of complex numbers is the sum of the arguments of the individual numbers. Hence,

    arg(zn)=arg(zzz)=arg(z)+arg(z)++arg(z),

    where arg(z) is added n times. Therefore,

    arg(zn)=narg(z),

    taking into account that the resulting argument may need to be adjusted by 2kπ to ensure it lies within the principal range if the principal argument is being considered.\

  7. arg(z)=arg(z)

    Proof:

    To prove that arg(z)=arg(z), consider the complex conjugate z, which can be expressed as:

    z=|z|2z,

    since |z|2=zz.

    The argument of a complex number is the angle it makes with the positive real axis. The argument of the product (or quotient) of two complex numbers is the sum (or difference) of their arguments.

    So, the argument of z is:

    arg(z)=arg(|z|2z).

    Since |z|2 is a non-negative real number, its argument is 0 (or 2kπ for any integer k), and the argument of z is arg(z). Thus, we have:

    arg(z)=arg(|z|2)arg(z)=0arg(z)=arg(z),

    which establishes the property.