Learn Argument of Complex Numbers with Vedantu
In Mathematics, complex planes play an extremely important role. We also call it a z-plane which consists of lines that are mutually perpendicular known as axes. The real numbers are represented by the horizontal line and are therefore known as the real axis whereas the imaginary numbers are represented by the vertical line and are therefore known as an imaginary axis. We basically use complex planes to represent a geometric interpretation of Complex Numbers. It is just like the Cartesian plane which has both the real as well as imaginary parts of a Complex Number along with the X and Y axes. Complex Numbers are branched into two basic concepts i.e., the magnitude and Argument. But for now we will only focus on the Argument of Complex Numbers and learn its definition, formulas and properties.
What is A Complex Number?
A Complex Number is written as a + ib, where “a” is a real number and “b” is an imaginary number. The Complex Number consists of a symbol “i” which satisfies the condition \[i^{2}\] = −1. Complex Numbers are referred to as the extension of one-dimensional number lines. In a complex plane, a Complex Number denoted by a + bi is usually represented in the form of the point (a, b). We have to note that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with absolutely no imaginary part is known as a real number.
What is the Argument of Complex Numbers?
The Argument of a Complex Number is an angle that is inclined from the real axis towards the direction of the Complex Number which is represented on the Complex plane. We can denote it by “θ” or “φ” and can be measured in standard units “radians”.
In the diagram above, the complex number is denoted by the point P. The length OP is the magnitude or modulus of the Number, and the angle at which OP is inclined from the positive real axis is known as the Argument of the point P.
How to Find Arguments of A Complex Number?
There are few steps that need to be followed if we want to find the Argument of a complex number. These steps are given below:
Step 1) First we have to find both real as well as imaginary parts from the Complex Number that is given to us and denote them x and y respectively.
Step 2) Then we have to use the formula θ = \[tan^{-1}\](y/x) to substitute the values.
Step 3) If by solving the formula we get a standard value then we have to find the value of θ or else we have to write it in the form of \[tan^{-1}\] itself.
Step 4) The final value along with the unit “radian” is the required value of the Complex Argument for the given Complex Number.
With this method you will now know how to find out the Argument of a Complex Number.
Argument of Complex Number Examples
1. Find the Argument of -1+i and 4-6i
Ans: We would first want to find the two Complex Numbers in the complex plane. This will make it easy for us to determine the quadrants where angels lie and get a rough idea of the size of each angle.
For, z= --+i
We can see that the Argument of z is a second quadrant angle and the tangent is the ratio of the imaginary part to the real part, in such a case −1. The tangent of the reference angle will thus be 1. Write the value of the second quadrant angle so that its reference angle can have a tangent equal to 1. If the reference angle contains a tangent which is equal to 1 then the value of reference angle will be π/4 and so the second quadrant angle is π − π/4 or 3π/4.
For z = 4 − 6i:
This time the Argument of z is a fourth quadrant angle. The reference angle has a tangent 6/4 or 3/2. None of the well known angles consist of tangents with value 3/2. Therefore, the reference angle is the inverse tangent of 3/2, i.e.\[tan^{-1}\](3/2). Hence the Argument being the fourth quadrant itself is 2π − \[tan^{-1}\](3/2).
In order to get a complete idea of the size of this Argument, we can use a calculator to compute 2π −\[tan^{-1}\](3/2) and see that it is approximately 5.3 (radians). In degrees this is about 303o.
Solved Example
1. Find the Argument of a complex number 2 + 2\[\sqrt{3i}\].
Ans: Let z = 2 + 2–\[\sqrt{3i}\].
The real part, x = 2 and the Imaginary part, y = 2\[\sqrt{3}\]
We already know the formula to find the Argument of a complex number. That is
arg (z) = \[tan^{-1}\](y/x)
arg (z) = \[tan^{-1}\](2\[\sqrt{3}\]/2)
arg (z) = \[tan^{-1}\](\[\sqrt{3}\])
arg (z) = \[tan^{-1}\](tan π/3)
arg (z) = π/3
Therefore, the Argument of the complex number is π/3 radian.
Complex Numbers
Complex Numbers are those numbers which are used in finding the square root of negative numbers. The Complex Numbers were first introduced by a Greek mathematician named Hero of Alexandria who tried to find the square root of negative numbers but wasn’t able to solve it. This was solved by an Italian mathematician called Gerolamo Cardano who found the negative roots of cubic and quadratic polynomial expressions using Complex Numbers. Complex Numbers have many uses in scientific research, fluid dynamics, quantum mechanics and signal processing.
A complex number can be defined as the sum of an imaginary number and a real number. It is written as a + ib which can be denoted by z. The a and b here are real numbers. The value of a is said to be the real part which is denoted by Re(z) and the b is said to be the imaginary part written as Im(z). The ib is also an imaginary number. For example 2 + 3i and -2 -5i are examples of Complex Numbers. The i written here is called iota which is used to represent the imaginary part of Complex Numbers. It is also helpful to find the square root of negative numbers.
What Does the Argument of Complex Numbers Mean?
When in geometrical representation, the line representing the complex number and origin makes an angle with the positive X-axis in an anticlockwise direction then this is said to be the argument of the complex numbers. This is represented by the inverse of tan of the imaginary number in the complex number which is divided by the real part of the complex number.
Argz(θ) = Tan−1(b/a)
FAQs on Argument of Complex Numbers
1. What is the difference between the general argument and the principal argument of a complex number?
The value of the principal argument is such that -π < θ =< π.
However, because θ is a periodic function having a period of 2π, we can also represent the argument as (2nπ + θ), where n is the integer. This is referred to as the general argument.
Now, consider that we have a complex number whose argument is 5π/2. This is a general argument which can also be represented as 2π + π/2. Here π/2 is the principal argument.
2. What are the properties of complex numbers?
The properties of complex number are listed below:
If a and b are the two real numbers and a + ib = 0 then a = 0, b = 0
When the real numbers are a, b and c; and a + ib = c + id then a = c and b = d.
A set of three complex numbers z1, z2, and z3 satisfy the commutative, associative and distributive laws.
The sum of two conjugate complex numbers is always real.
The product of two conjugate complex numbers is always real.
If both the sum and the product of two complex numbers are real then the complex numbers are conjugate to each other.
For two complex numbers z3 and z3 : |z1 + z2|≤ |z1| + |z2|
3. How is the graphing of complex numbers done?
We know that the complex number is composed of an imaginary part and a real part and the coordinates of these points can be represented on a plane called the Euclidean plane. This plane is in reference to the complex numbers which is called the complex plane or the Argand plane given by the scientist Jean-robert Argand. The complex number which is written as z = a + ib has the real part a and with reference to the X-axis. The imaginary part ib is with reference to the Y-axis.
4. Define modulus of complex numbers.
The complex numbers are graphed in the complex plane or Argand plane through which the distance can be measured, this is called the modulus of the complex number. The distance is linear and is measured from the origin (0,0) to the point (a + ib). It is measured as r = |\[\sqrt{a^2+b^2}\]| . Here, the use of Pythagoras theorem can be done where modulus is the hypotenuse, the base is the real part and the height is the imaginary part of the right-angled triangle.
5. What is equality of complex numbers?
The equality of complex numbers is similar to real numbers. z1=a1+ib1 and z2=a2+ib2 are two complex numbers which are said to be equal if the real part of the both the complex numbers are equal to. Also, the imaginary complex numbers are equal as well. In addition, the two complex numbers in the polar form have to be equal. This can happen only when they have the same magnitude and the argument of these complex numbers i.e. the angle differs by the integral multiple of 2 π.
6. How are complex numbers added?
The addition of complex numbers is similarly done like the natural numbers. The real part of the complex number is added to the real part and the imaginary part is added to the imaginary part. When there are two complex numbers z1=a + ib and z2 = c + id then they can be added as z1 + z2 = (a + c) + i(b + d). These follow certain laws some of which are given below:
Closure law in which the sum of two complex numbers is also a complex number.
Commutative law in which two complex numbers z1 and z2 can be summed as z1 +z2=z2+z1 Associative law is followed when there are 3 complex numbers, let's say z1, z2 and z3. These are added as z1 + (z2 + z3) = (z1 + z2) + z3 Other such laws are additive identity and additive inverse.
7. What is the process of subtracting complex numbers?
Like addition, subtraction of complex numbers is also similar to the subtraction of natural numbers. The subtraction process is done separately i.e. the real part of the complex number is subtracted with the real part and the imaginary parts are subtracted together. For example, let us consider the example of two complex numbers z1= a + ib and z2= c + id then it can be subtracted as z1- z2 = (a – c) + i(b –d).