Arguments of Complex Numbers: Formulas, Properties

In Mathematics, the complex plane represents a geometric interpretation of complex numbers. The plane illustrates real and imaginary components of a complex number as well the X and Y axes. The complex plane is otherwise known as the Argand plane. It is called so because it is made up of two mutually perpendicular axes. The horizontal line is known as the real axis as it represents real numbers. While the vertical line is known as the imaginary axis as it represents imaginary numbers. Complex numbers can be further mentioned as the extension of the one-dimensional number line.

Read Also: Complex Numbers and Quadratic Equations

Key Takeaways: Complex Number, real numbers, arguments of complex numbers, imaginary numbers, complex plane

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Meaning of Complex Number

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Complex numbers are the numbers written in the form of a + ib, where a represents real numbers and b represents imaginary numbers. A complex number consists of a symbol i.e ' i '. The symbol i represents an imaginary number whose square is - 1. 

\(i = \sqrt{-1}\). In the Complex plane, a complex number is indicated by a + bi. This is represented in the form of (a,b). A complex number with zero or no real part for example. -i, 5i, etc., are called purely imaginary. A complex number with zero or no imaginary part is known as a real number.

Complex Numbers

An imaginary number is represented by 'i' or 'j' equal to \(-\sqrt{-1}\), as a consequence, the square of the imaginary number shows a negative value.

The video below explains this:

Complex Numbers Detailed Video Explanation:

Read More: Algebraic Operations on Complex Numbers

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Arguments of Complex Numbers

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The angle incline from the real axis in the direction of the complex number that is represented on the complex plane is defined as the arguments of complex numbers. It is indicated by '\(\Theta\)'. The unit of measurement is called Radians.

Arguments of Complex Numbers

This diagram shows the complex number is indicated by the point P. The length OP is the magnitude of the number and the angle at which OP is inclined from the real axis can be called the argument of the point P.

The equation for a complex number is represented as (cos\(\Theta\)+i sin\(\Theta\)), here\(\Theta\) is the argument. The argument function is denoted - arg(z), z indicates the complex number i.e “z=x+iy.

How to Find Arguments of Complex Numbers

Steps to find arguments of complex numbers:

1. Find both real as well imaginary parts from the complex number given. Then denote them as X and Y.
2. Use the formula \(\Theta\)= tan −1(y/x) to substitute the values.
3. While solving, if you get a standard value then find the value of \(\Theta\) or write in the form of tan −1.
4. Final value along with the unit of measurement i.e “radian” is the required value of the complex argument for the given complex number.

With the help of these steps, you can find the arguments of complex numbers.

Check Important Notes for Vectors

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Formulas of Complex Number

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Conjugate of a Complex Number

Let's consider, a complex number z= a+ ib,

Conjugate is written as z¯

Value defined as z¯ = a - ib.

Algebra of Complex Number

Z1= a+ib and Z2= c+I'd. 

Complex numbers a, b,c,d €R and i = \(\sqrt{-1}\)

• Addition:

Z1+Z2 = (a+bi)+(c+di)= (a+c) + (b+d)i

• Subtraction:

Z1- Z2 = (a+bi)-(c+di)= (a-c)+(b-d)

• Multiplication:

Z1•Z2 = (a+bi)(c+di)

= a(c+di)+bi( c+di)

= ac +adi+bci + bdi²

= ac-bd+(ad+bc)i

∴ i² = -1

• Dividing:

z₁/z₂=(a₁+b₁)/ (a₂+ib₂)

= (a₁a₂ – b₁b₂) + i(a₁b₂+a₂b₁) / a²₁+a²₂

= (a₁a₂-b₁b₂)+ (a₁b₂+a₂b₁)a₁²+a₂²

= 2

≠ 0

Check Also: Coordinate Geometry

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Properties of Complex Number

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Some important properties are:

1. z+z¯=2Re(z)
2. z-z¯=2iIm(z)
3. |z| =0⇒z=0
4. z . z¯=|z|²
5. |z¯|=|z|=|—z|
6. The two conjugate complex numbers sum is always real.
7. The product of two conjugate complex numbers is always real.

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Summary of Complex Numbers

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Complex numbers, their formula, and examples play a crucial role in the world of mathematics. Complex numbers serve the importance of maths in science as it is a powerful tool kit for solving challenging issues. A complex number allows us to solve any polynomial equation. For example – x² - 2 x + 5 = 0 = x² -2x +5= 0x, squared, -2,x+5 =0 doesn't have any imaginary solution.

The video below explains this:

Quadratic Equations Detailed Video Explanation:

Check More: Binomial Theorem

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Points to Remember

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Following are some important points:

• A complex number is a representation of two numbers that are real and imaginary numbers.
• One part of a complex number is purely real and the other part is purely imaginary.
• Real numbers are the ones that are present in a number system like positive, negative and zero rational or irrational fractions, integers, etc. Example- 12,-45,√5, the, etc.
• Real number is denoted as a Re().
• The graphical representation of an argand plane is also known as an argand diagram or complex plane.

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