Properties of Argument
Introduction
In polar form, complex numbers are expressed as
Properties of Argument
Two most important properties of argument is about operations of multiplication and division.
-
The argument of the product of two complex numbers is the sum of their arguments.
Proof:
Consider two complex numbers in polar form,
and . The product is: .Using trigonometric identities, this can be simplified to:
.Hence,
, which establishes the property: .Geometrical Interpretation of Multiplication
We may interpret this multiplication in a pretty useful way. Multiplication of
by has resulted in the rotation of by angle in the anticlockwise direction and its magnitude has changed from to .Principal Argument with respect to multiplication
If
and are principal arguments of two complex numbers and , where both arguments are within the interval , then adding and may result in an angle that does not lie within this principal range. Hence, while holds for arguments in a general sense, the sum may not be the principal argument of the product .To find the principal argument of
, one must ensure the result is within the principal range . If falls outside of this range, an adjustment of , for some appropriate , is necessary to correct it. Thus, the principal argument of the product, denoted simply as when referring to its principal value, is the adjusted sum that falls within the specified interval. -
The argument of the quotient of two complex numbers is the difference between the argument of the numerator and the argument of the denominator.
Proof:
Let
and be two complex numbers in polar form. The quotient is:Using trigonometric identities, we expand the numerator and use the fact that
to simplify to:Therefore, the argument of the quotient is the difference of the arguments:
which establishes the property:
-
If
is a real number, then the argument of , denoted as , is either or . The value depends on the sign of :- If
is a positive real number, then . - If
is a negative real number, then .
- If
-
If
is purely imaginary, then is or , depending on the sign of the imaginary part:- If the imaginary part is positive,
. - If the imaginary part is negative,
.
- If the imaginary part is positive,
-
The argument of the product of multiple complex numbers
is the sum of their individual arguments. -
The argument of a power of a complex number
raised to an integer exponent is times the argument of , modulo . This can be stated as:where
is an integer such that the argument lies within the principal branch if we are considering the principal value.Proof:
To prove that
, consider raised to the th power, which is equivalent to multiplied by itself times:The argument of a product of complex numbers is the sum of the arguments of the individual numbers. Hence,
where
is added times. Therefore,taking into account that the resulting argument may need to be adjusted by
to ensure it lies within the principal range if the principal argument is being considered.\ -
Proof:
To prove that
, consider the complex conjugate , which can be expressed as:since
.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
is:Since
is a non-negative real number, its argument is (or for any integer ), and the argument of is . Thus, we have:which establishes the property.