Lesson Explainer: The Argument of a Complex Number | Nagwa Lesson Explainer: The Argument of a Complex Number | Nagwa

Lesson Explainer: The Argument of a Complex Number Mathematics • Third Year of Secondary School

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In this explainer, we will learn how to identify the argument of a complex number and how to calculate it.

When we plot complex numbers on an Argand diagram, we can see that complex numbers share many properties with vectors. For instance, adding and subtracting complex numbers is geometrically equivalent to the corresponding operations of vectors. We know that the characteristics of a vector are its direction and magnitude, so a complex number must have equivalent characteristics. We recall that the magnitude of a complex number is called its modulus. The direction of a complex number in the Argand diagram is the argument of the complex number.

Definition: Argument of a Complex Number

The argument of a complex number is the angle, in radians, between the positive real axis in an Argand diagram and the line segment between the origin and the complex number, measured counterclockwise. The argument is denoted arg(𝑧), or Arg(𝑧).

The argument 𝜃 of a complex number is, by convention, given in the range 𝜋<𝜃𝜋. However, we can also discuss a complex number with an argument greater than 𝜋 or less than 𝜋. The argument of a complex number within the range ]𝜋,𝜋] is called the principal argument. Other conventions use the range 0𝜃<2𝜋 for the principal argument, but this is less common.

If we are given the Cartesian form, 𝑎+𝑏𝑖, of a complex number, we can use right triangle trigonometry to find the argument of the complex number. For instance, consider the complex number given in the Argand diagram above. Since this complex number lies in the first quadrant, we can see that the argument of this complex number is an angle in the right triangle whose sides are the blue, green, and purple line segments. In this case, the tangent of this angle is the ratio oppositeadjacent; hence, tan𝜃=𝑏𝑎.

We can then compute 𝜃 by applying the inverse tangent function to both sides of this equation: 𝜃=𝑏𝑎.tan

This method can be used whenever a complex number lies in the first quadrant. In our first example, we will find the principal argument of a complex number in the first quadrant by using right triangle trigonometry.

Example 1: Finding the Argument of a Complex Number in Radians

Find the argument of the complex number 4+3𝑖 in radians. Give your answer correct to two decimal places.

Answer

Recall that the argument of a complex number is the angle, in radians, between the positive real axis of an Argand diagram and the line between the origin and the complex number, measured counterclockwise. We also remember that the argument of a complex number is, by convention, given in the range ]𝜋,𝜋].

We begin by plotting the complex number on an Argand diagram.

We have labeled the argument of the complex number in the Argand diagram above 𝜃. We can see that the argument of this complex number is an angle in the right triangle whose sides are the blue, green, and purple line segments. Applying right triangle trigonometry, we obtain tanoppositeadjacent𝜃==34.

We can then apply the inverse tangent function to both sides of this equation to find 𝜃=34=0.6435.arctanradians

Hence, argradians(4+3𝑖)=0.64 to two decimal places.

In the previous example, we were able to calculate the argument of a complex number, 𝑎+𝑏𝑖, by evaluating the inverse tangent of 𝑏𝑎. However, this is not the case for all complex numbers as the next example will demonstrate.

Example 2: Finding the Principal Argument of a Complex Number

Given that 𝑍=12+32𝑖, find the principal argument of 𝑍.

Answer

Recall that the argument of a complex number is the angle, in radians, between the positive real axis of an Argand diagram and the line between the origin and the complex number, measured counterclockwise. Additionally, we remember that the principal argument of a complex number is the argument that lies in the range ]𝜋,𝜋].

We begin by plotting the complex number on an Argand diagram as shown below.

We have labeled the argument of the complex number in the Argand diagram above 𝜃 and the supplementary angle 𝜙. We can see that 𝜙 is an angle in the right triangle whose sides are the blue, green, and purple line segments. Applying right triangle trigonometry, we obtain tanoppositeadjacent𝜙==.

We can then apply the inverse tangent function to both sides of this equation to find 𝜙==3=𝜋3.arctanarctanradians

We can then calculate the argument by subtracting 𝜙 from 𝜋: argradians(𝑍)=𝜋𝜙=𝜋𝜋3=2𝜋3.

We note that this argument lies in the range ]𝜋,𝜋]; hence, it is the principal argument.

We conclude that the principal argument of the given complex number is 2𝜋3.

In the previous example, we saw that the argument of a complex number, 𝑎+𝑏𝑖, is not always equal to the inverse tangent of 𝑏𝑎. In fact, if we had naively tried to calculate the argument of 𝑧 by evaluating 𝛼=,arctan we would have ended up with 𝛼=3=𝜋3.arctanradians

This argument represents a clockwise angle from the positive real axis of 𝜋3 radians, which would place the complex number in the fourth quadrant. It is apparent from the Argand diagram in the previous example that this is not the argument of the complex number. However, we can arrive at the correct value of arg(𝑧) by adding 𝜋 to 𝛼.

This effect demonstrates that we need to be careful when calculating the argument of a complex number that does not lie in the first quadrant. Also, we can see that there are different approaches to obtain arg(𝑧).

We outline two different methods of calculating the argument of a complex number. Whichever method we choose to use, plotting the number on an Argand diagram will be extremely useful and will help us avoid common errors in calculating the argument.

How To: Finding the Argument of a Complex Number Using the Inverse Tangent Function

To find the argument, arg(𝑧), of a complex number 𝑧=𝑎+𝑏𝑖, we need to consider which quadrant it lies in. The argument of a complex number 𝑧=𝑎+𝑏𝑖 can be obtained using the inverse tangent function in each quadrant as follows:

  • If 𝑧 lies in the first or the fourth quadrant, argarctan(𝑧)=𝑏𝑎.
  • If 𝑧 lies in the second quadrant, argarctan(𝑧)=𝑏𝑎+𝜋.
  • If 𝑧 lies in the third quadrant, argarctan(𝑧)=𝑏𝑎𝜋.

If the complex number does not lie on a quadrant, then it is either purely real or purely imaginary. If it is purely imaginary (𝑎=0), then argforargfor(𝑧)=𝜋2𝑏>0,(𝑧)=𝜋2𝑏<0.

If purely real (𝑏=0), then argforargfor(𝑧)=0𝑎>0,(𝑧)=𝜋𝑎<0.

Lastly, if 𝑎=𝑏=0, the argument is undefined.

These points are summarized in the following diagram.

The main benefit of the method described above is that we are given a formula to follow for each situation. However, this method also requires us either to memorize each rule or to have an available reference for the rules. An alternative method for finding the argument of a complex number is to use right triangle trigonometry to first identify the positive acute angle between the real axis and the line segment between the origin and the complex number in an Argand diagram. After finding the positive acute angle, we can find the argument of the complex number geometrically.

How To: Finding the Argument of a Complex Number Using Positive Acute Angles

We define the angle 𝜃 to be the positive acute angle between the line linking 𝑧 to the origin and the real axis as shown in the diagram.

We can then calculate the argument of 𝑧 in different quadrants as follows:

  • Quadrant 1: arg(𝑧)=𝜃
  • Quadrant 2: arg(𝑧)=𝜋𝜃
  • Quadrant 3: arg(𝑧)=𝜃𝜋
  • Quadrant 4: arg(𝑧)=𝜃

The two different methods for obtaining the argument of a complex number will lead to the same answer. The second method, which uses the positive acute angle, is more intuitive and requires less memorization. Using this method, we first compute the positive acute angle and then use it to find the argument of the complex number, which is the counterclockwise angle from the positive real axis, lying in the range ]𝜋,𝜋].

In the next example, we will apply this method to find the argument of a complex number lying in the third quadrant.

Example 3: The Relationship between the Complex Conjugate and the Argument

Consider the complex number 𝑧=45𝑖.

  1. Calculate arg(𝑧), giving your answer correct to two decimal places in an interval from 𝜋 to 𝜋.
  2. Calculate arg𝑧, giving your answer correct to two decimal places in an interval from 𝜋 to 𝜋.

Answer

Recall that the argument of a complex number is the angle, in radians, between the positive real axis of an Argand diagram and the line between the origin and the complex number, measured counterclockwise. We also remember that the argument of a complex number is, by convention, given in the range ]𝜋,𝜋].

Part 1

We begin by plotting the complex number on an Argand diagram as shown below.

We have labeled the acute angle 𝜙, which is related to the argument of the complex number 𝑧. If we can find angle 𝜙, the argument of this number can be obtained by adding 𝜋 to this angle. However, we can see that this argument will not lie in the range ]𝜋,𝜋]. We must then subtract the full revolution 2𝜋 from this resulting angle, which leads to the relation arg(𝑧)=(𝜙+𝜋)2𝜋=𝜙𝜋.

We can see that 𝜙 is an angle in the right triangle whose sides are the blue, green, and purple line segments. Applying right triangle trigonometry, we obtain tanoppositeadjacent𝜙==54.

We can then apply the inverse tangent function to both sides of this equation to find 𝜙=54=0.8960.arctanradians

Hence, to calculate arg(𝑧), we subtract 𝜋 from 𝜙, which gives argradiansroundedtodecimalplaces(𝑧)=𝜙𝜋=2.2455=2.25,2.

Part 2

Recall that the conjugate 𝑧 is obtained by switching the sign of the imaginary part of the complex number 𝑧. Hence, 𝑧=4+5𝑖. We now plot 𝑧 on an Argand diagram.

Similar to the previous part, we will find the argument of 𝑧 by first calculating 𝜙: 𝜙=54=0.8960.arctanradians

Since 𝜙 and arg𝑧 are supplementary, we can obtain arg𝑧 by subtracting 𝜙 from 𝜋: argradiansroundedtodecimalplaces𝑧=𝜋𝜙=2.2455=2.25,2.

In the previous example, we computed the arguments of a complex number and its conjugate. We can note that the argument of the complex conjugate in this example is the negative of the argument of the original complex number. This demonstrates a general rule of the argument.

Property: Argument of the Conjugate of a Complex Number

Given any nonzero complex number 𝑧 and its conjugate 𝑧 (also denoted 𝑧), argarg(𝑧)=𝑧.

In the next example, we will demonstrate how the multiplication and division of complex numbers is associated with the arguments of the complex numbers.

Example 4: Arguments of Products and Quotients

Consider the complex numbers 𝑧=1+3𝑖 and 𝑤=22𝑖.

  1. Find arg(𝑧) and arg(𝑤).
  2. Calculate arg(𝑧𝑤). How does this compare to arg(𝑧) and arg(𝑤)?
  3. Calculate arg𝑧𝑤. How does this compare to arg(𝑧) and arg(𝑤)?

Answer

Recall that the argument of a complex number is the angle, in radians, between the positive real axis of an Argand diagram and the line between the origin and the complex number, measured counterclockwise. We also remember that the argument of a complex number is, by convention, given in the range ]𝜋,𝜋].

Part 1

Let us start by plotting 𝑧 and 𝑤 on an Argand diagram.

Recall that the argument of a complex number 𝑧=𝑎+𝑏𝑖 lying in the first or the fourth quadrant is given by argarctan(𝑧)=𝑏𝑎.

Since 𝑧 and 𝑤 lie in the first and fourth quadrants, respectively, we can use the inverse tangent to find their arguments as follows: argarctanradians(𝑧)=31=𝜋3 and argarctanradians(𝑤)=22=𝜋4.

Part 2

We begin by calculating 𝑧𝑤 as follows: 𝑧𝑤=1+3𝑖(22𝑖).

Multiplying through the parenthesis, we obtain 𝑧𝑤=22𝑖+23𝑖2𝑖3.

Using 𝑖=1 and gathering real and imaginary terms, we obtain 𝑧𝑤=2+23+232𝑖.

Since both the real and the imaginary parts are positive, 𝑧𝑤 lies in the first quadrant of the Argand diagram and we can calculate the argument by evaluating the inverse tangent as follows: argarctan(𝑧𝑤)=2322+23.

Canceling the factor 2 from the top and the bottom, we have argarctan(𝑧𝑤)=311+3.

We can simplify the fraction by multiplying both the numerator and the denominator by the conjugate of the denominator: argarctan(𝑧𝑤)=31131+313.

Multiplying through the parenthesis, we obtain argarctanarctanarctanradians(𝑧𝑤)=1+23313=4+232=23=𝜋12.

Comparing this with arg(𝑧) and arg(𝑤), we find that argargarg(𝑧𝑤)=(𝑧)+(𝑤).

Part 3

We start by calculating 𝑧𝑤 as follows: 𝑧𝑤=1+3𝑖22𝑖.

To write this complex number in the Cartesian form, 𝑎+𝑏𝑖, we need to multiply both the numerator and the denominator by the conjugate of the denominator, which is 2+2𝑖: 𝑧𝑤=1+3𝑖(2+2𝑖)(22𝑖)(2+2𝑖).

Multiplying through the parenthesis, we have 𝑧𝑤=2+2𝑖+2𝑖3+2𝑖34+4.

Using 𝑖=1 and gathering real and imaginary terms, 𝑧𝑤=1413+141+3𝑖.

Since Re𝑧𝑤<0 and Im𝑧𝑤>0, the complex number 𝑧𝑤 lies in the second quadrant. Recall that if a complex number 𝑧=𝑎+𝑏𝑖 lies in the second quadrant, argarctan(𝑧)=𝑏𝑎+𝜋.

Hence, we have argarctan𝑧𝑤=1+313+𝜋.

Canceling the common factor 14, we have argarctan𝑧𝑤=1+313+𝜋.

Evaluating the inverse tangent, we obtain arg𝑧𝑤=5𝜋12+𝜋=7𝜋12.

Finally, comparing this with arg(𝑧) and arg(𝑤), we find that argargarg𝑧𝑤=(𝑧)(𝑤).

In the previous example, we observed the relationship between the multiplication/division of complex numbers and their arguments. This relationship shown in the example holds for general complex numbers.

Property: Arguments and Multiplication/Division of Complex Numbers

Given any nonzero complex numbers 𝑧 and 𝑧, argargargargargarg(𝑧𝑧)=(𝑧)+(𝑧),𝑧𝑧=(𝑧)(𝑧).

The next example will demonstrate how we can solve problems by applying the properties of the argument.

Example 5: Using Multiplication of Complex Numbers to Determine an Argument

A complex number is multiplied by another complex number 𝑧 and then by the complex conjugate 𝑧. How is the argument of the resulting complex number related to the argument of the original complex number?

Answer

Recall that the argument of the product of a pair of complex numbers is equal to the sum of the arguments of the two complex numbers.

We start with a complex number 𝑤; then, it is multiplied by 𝑧 and 𝑧. Hence, the result is 𝑤𝑧𝑧. We are asked how the argument of the resulting complex number is related to the argument of the original complex number. Therefore, we should consider arg𝑤𝑧𝑧. Using multiplicative properties of the argument, we can rewrite this as follows: argargargarg𝑤𝑧𝑧=(𝑤)+(𝑧)+𝑧.

We also know that the argument of a complex number equals the negative of the argument of its conjugate. Hence, we can replace arg𝑧 above with (𝑧)arg to write argargargargarg𝑤𝑧𝑧=(𝑤)+(𝑧)(𝑧)=(𝑤).

Therefore, the argument of the complex number after it is multiplied by another complex number 𝑧 and then by the complex conjugate 𝑧 is unchanged.

In our final example, we will consider the relationship between the argument and powers.

Example 6: Finding the Argument of Powers of Complex Numbers in Algebraic Form

Consider the complex number 𝑧=7+7𝑖.

  1. Find the argument of 𝑧.
  2. Hence, find the argument of 𝑧.

Answer

Recall that the argument of a complex number is the angle, in radians, between the positive real axis of an Argand diagram and the line between the origin and the complex number, measured counterclockwise. We also remember that the argument of a complex number is, by convention, given in the range ]𝜋,𝜋].

Part 1

Recall that the argument of a complex number 𝑧=𝑎+𝑏𝑖 lying in the first or the fourth quadrant is given by argarctan(𝑧)=𝑏𝑎.

Since the complex number lies in the first quadrant, we can calculate its argument by evaluating the inverse tangent of its imaginary part over its real part as follows: argarctanarctanradians(𝑧)=77=(1)=𝜋4.

Part 2

We recall that for any two nonzero complex numbers 𝑧 and 𝑧, argargarg(𝑧𝑧)=(𝑧)+(𝑧).

If the complex numbers are both equal to 𝑧, this means argarg𝑧=2(𝑧).

Using similar logic, we can find that argargargarg𝑧=3(𝑧),𝑧=4(𝑧).

Therefore, argargradians𝑧=4(𝑧)=4×𝜋4=𝜋.

In the previous example, we saw the relationship between the power of a complex number and its argument. Using similar logic as in that example, we can see that this relationship holds for general complex numbers for any positive integer power.

Property: Argument of the Power of a Complex Number

Given any nonzero complex number 𝑧 and a positive integer power 𝑛, the argument of 𝑧 is given by argarg(𝑧)=𝑛(𝑧).

In this explainer, we have considered how the argument of a complex number relates to the conjugates, multiplication, and division of complex numbers. However, we have intentionally left out the addition and subtraction of complex numbers since there is no simple relationship between these operations and the arguments of complex numbers. We will finish this explainer by illustrating in two different ways why we do not expect to find a simple relationship between addition/subtraction and the arguments of complex numbers.

Firstly, we recall that the addition and subtraction of complex numbers are geometrically equivalent to the corresponding vector operations and hence follow the triangle or parallelogram rules. In doing so, we can see that knowing only the arguments (angles) of the complex numbers will not be sufficient to find the argument of the resulting complex number. This is one way that we can see why there is no simple relationship between these operations and the arguments of complex numbers.

As an alternative way to see why no such simple relationship exists, let us consider the three complex numbers 𝑧=1+𝑖, 𝑧=2+3(1+𝑖), and 𝑧=1𝑖 plotted on the Argand diagram below.

We can see that argarg(𝑧)=(𝑧)=𝜋4 and that arg(𝑧)=𝜋4. Furthermore, 𝑧+𝑧=2, which has an argument of 0, whereas 𝑧+𝑧=3+3+1+3𝑖, whose argument is clearly not zero. We can, in fact, calculate the exact value of the argument as follows: argarctan(𝑧+𝑧)=1+33+3.

By multiplying both the numerator and the denominator by the conjugate of the denominator, we can simplify the fraction: argarctan(𝑧+𝑧)=1+3333+333.

Multiplying through the parenthesis, we obtain argarctan(𝑧+𝑧)=33+33333.

Finally, we can simplify and evaluate the inverse tangent to obtain argarctanarctanradians(𝑧+𝑧)=236=33=𝜋6.

To summarize what we have calculated here, recall that the complex numbers 𝑧 and 𝑧 have the same argument, 𝜋4. If a simple relationship between the arguments of complex numbers and the sum existed, the arguments of 𝑧+𝑧 and 𝑧+𝑧 would be the same. However, we have obtained argarg(𝑧+𝑧)=0,(𝑧+𝑧)=𝜋6.

This demonstrates that knowing the arguments of two complex numbers is not sufficient to be able to calculate the argument of their sum.

Let us finish by recapping a few important concepts from this explainer.

Key Points

  • The argument of a complex number 𝑧 is defined as the angle, in radians, between the positive real axis in an Argand diagram and the line segment from the origin to the complex number, measured counterclockwise.
  • The argument of a complex number 𝑧=𝑎+𝑏𝑖 can be obtained using the inverse tangent function in each quadrant as follows:
    • If 𝑧 lies in the first or the fourth quadrant, argarctan(𝑧)=𝑏𝑎.
    • If 𝑧 lies in the second quadrant, argarctan(𝑧)=𝑏𝑎+𝜋.
    • If 𝑧 lies in the third quadrant, argarctan(𝑧)=𝑏𝑎𝜋.
  • The argument has the following properties:
    • argarg(𝑧)=𝑧,
    • argargarg(𝑧𝑧)=(𝑧)+(𝑧),
    • argargarg𝑧𝑧=(𝑧)(𝑧),
    • argarg(𝑧)=𝑛(𝑧).
  • There is no simple relationship between the addition of complex numbers and their arguments.

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