.

Complex Numbers:

Definition, Operations, and Applications

Complex numbers extend the concept of real numbers to include solutions to equations that do not have real solutions, such as the square root of negative numbers. They are essential in various fields of mathematics and engineering.

Definition

A  complex number is a number of the form:

   

where:

is the real part of the complex number,
is the imaginary part,
is the imaginary unit, defined as .

For example, in the complex number , is the real part and is the imaginary part.

Basic Operations

1. Addition: To add two complex numbers, add their real parts and imaginary parts separately.

   

2. Subtraction: To subtract two complex numbers, subtract their real parts and imaginary parts separately.

   

3. Multiplication: To multiply two complex numbers, use the distributive property and apply the fact that .

   

4. Division: To divide two complex numbers, multiply the numerator and the denominator by the conjugate of the denominator.

   

5. Conjugate: The conjugate of a complex number is . Conjugates are useful in division and finding magnitudes.

   

6. Magnitude: The magnitude (or modulus) of a complex number is given by:

   

7. Argument: The argument (or angle) of a complex number is the angle such that:

   

Polar Form of Complex Numbers

A complex number can be represented in the form

   

, where is the real part of the complex number,
is the imaginary part,
is the imaginary unit, defined as .

A complex number can be expressed in polar form as:

   

where is the magnitude (or modulus) of the complex number, given by

   

and is the argument (or angle) of the complex number, measured from the positive -axis, given by

   

Euler’s Form of Complex Numbers

Euler’s formula relates complex exponentials to trigonometric functions:

Alternatively, the polar form can also be written using Euler’s formula as:

   

This is a more compact and elegant way of expressing the polar form. Here,

   

represents the complex exponential, and is the argument of the complex number. This form is particularly useful for multiplication, division, and finding powers and roots of complex numbers.

Examples 1:

Convert

   

into Polar & Euler’s For

To convert into polar form:

Magnitude:

   

The argument is:

   

Thus, the polar form is:

   

In Euler’s form:

   

Examples 2:

Convert

   

into Polar & Euler’s For

To convert into polar form:

Magnitude:

   

Argument:

   

Polar form:

   

Euler’s Form

   

Derivation of Euler’s Form:

Euler’s form of a complex number provides a powerful and elegant way to represent complex numbers using exponential functions. Here’s a step-by-step explanation of Euler’s form and how it is derived from the polar form using the expansions of cosine and sine.

Euler’s Form of a Complex Number

Euler’s form of a complex number $z$ is given by:

   

where:

• is the magnitude of the complex number.
• is the argument (angle) of the complex number.
•  is the exponential function involving the imaginary unit $i$.

Derivation from Polar Form

Start with the Polar Form:The polar form of a complex number  is expressed as:

   

where:

• • is the magnitude of the complex number.
  • is the argument or angle.

Use Euler’s Formula:Euler’s formula states that:

   

is equal

   

 This is a fundamental result in complex analysis and connects exponential functions with trigonometric functions.

Substitute Euler’s Formula into the Polar Form:

To express the polar form in terms of Euler’s formula, substitute Euler’s formula into the polar form expression:

   

can be written as:

   

This substitution uses the fact that

   

can be expanded as

   

.

Expansion of Cosine and Sine

To see this more clearly, let’s expand

   

 using its Taylor series:

• • Taylor Series for

       

    :

   

​

Expanding

   

=

   

So:

   

​

• Separate Real and Imaginary Parts:

To separate the real and imaginary parts, recall that:

   

​Grouping terms:

Real Part:

   

Imaginary Part:

   

Hence, combining these:

   

WorkSheets On Complex Number

Convert Cartesian form into Polar & Euler’s Form

• Convert into polar form.
• Convert into Euler’s form.
• Find the polar form of .
• Convert to Euler’s form.
• Express in polar form.
• Find the Euler’s form of .
• Convert into polar form.
• Convert to Euler’s form.
• Express in polar form.
• Find the polar form of .
• Convert to Euler’s form.
• Express in polar form.
• Find the polar form of .
• Convert into Euler’s form.
• Express in polar form.

More Question:

• Convert to Euler’s form.
• Find the polar form of .
• Convert to Euler’s form.
• Express in polar form.
• Find the Euler’s form of .
• Convert to polar form.
• Convert into Euler’s form.
• Find the polar form of .
• Convert to Euler’s form.
• Express in polar form.
• Find the polar form of .
• Convert to Euler’s form.
• Express in polar form.
• Find the Euler’s form of .
• Convert to polar form.

Convert Polar & Euler’s form into   Cartesian Form

• , convert to Cartesian form.
• Convert into Cartesian form.
• Convert to Cartesian form.
• Convert ​ to Cartesian form.
• Convert  to Cartesian form.
• Convert ​ to Cartesian form.
• Convert  to Cartesian form.
• Convert  to Cartesian form.
• Convert to Cartesian form.
• Convert  to Cartesian form.
• Convert  to Cartesian form.
• Convert ​ to Cartesian form.
• Convert  to Cartesian form.
• Convert  to Cartesian form.
• Convert  to Cartesian form.

More Question

• Convert  to Cartesian form.
• Convert  to Cartesian form.
• Convert ​ to Cartesian form.
• Convert  to Cartesian form.
• Convert to Cartesian form.
• Convert  to Cartesian form.
• Convert ​ to Cartesian form.
• Convert to Cartesian form.
• Convert ​ to Cartesian form.
• Convert to Cartesian form.
• Convert ​ to Cartesian form.
• Convert  to Cartesian form.
• Convert
• Convert  to Cartesian form.
• Convert ​ to Cartesian form.

Applications

1. Engineering:

Complex numbers are used in electrical engineering to analyze AC circuits, where they represent impedance and phase relationships. They simplify the analysis of alternating current (AC) signals and circuits.

2. Control Systems:

In control theory, complex numbers are used to describe system dynamics and stability. The poles and zeros of transfer functions are complex numbers that help in designing and analyzing control systems.

3. Signal Processing:

Complex numbers are employed in signal processing to handle frequency domain analysis. Fourier transforms and filters often use complex numbers to analyze and manipulate signals.

4. Physics:

Complex numbers are essential in quantum mechanics, where they describe wave functions and probability amplitudes. The Schrödinger equation, a fundamental equation in quantum mechanics, uses complex numbers.

5. Mathematics:

Complex numbers are used in various branches of mathematics, including complex analysis, which studies functions of complex variables. They also appear in solutions to polynomial equations and many areas of applied mathematics.

6. Computer Graphics:

In computer graphics, complex numbers are used for transformations and rotations. They help in manipulating images and performing geometric transformations efficiently.

Complex numbers provide a powerful framework for solving problems that cannot be addressed with real numbers alone, making them an indispensable tool in mathematics and engineering.

Got Top
.

Introduction to Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable $x$, with the general form:

where , , and are constants, and . The term “quadratic” comes from “quad,” meaning square, since the highest exponent of the variable is .

Uses and Applications of Quadratic Equations

Quadratic equations are used in various fields, including:

Physics: To describe projectile motion, the trajectory of an object can be modeled using quadratic equations.

Engineering: In structural engineering, quadratic equations help calculate loads, tensions, and forces.

Economics: They are used in cost, profit, and revenue functions, particularly in finding maximum or minimum values.

Biology: Quadratic equations help model population growth, the spread of diseases, or other natural phenomena.
Geometry: These equations are used in determining areas, and for working with parabolas and ellipses.

Quadratic Expressions and Quadratic Equations

Method of Solving Quadratic Expressions ,

1.Factorisation Method:

The goal is to express the quadratic expression in the form of two binomials:

   

where and are constants, and the equation is solved by finding appropriate factors of .

2. Completing the Square Method:

To solve the quadratic expression, we can complete the square as follows:

   

The next step is to add and subtract a term to make a perfect square trinomial:

   

After simplifying, we can rewrite it as:

   

3. Quadratic Formula:

The quadratic formula is:

   

This method can be used for any quadratic equation to find its roots directly.

Factorisation Based on the Value of and the Sign of

In solving quadratic equations, it is important to consider the value of and the sign of .

CASE 1:

If , we check the sign of .

If is positive, both factors have the same sign as , and their sum equals .

Example:

   

First, find factors of :

   

Since , we split into and , so the equation becomes:

   

Now, factor:

   

Thus:

   

Setting each factor equal to zero:

   

So, the solutions are:

   

Similarly we can solve

Case II:

if is negative, factors subtracted and have different signs, the large factor keep the sign of .

for example , in this case we need to find the factors of , which when subtracted give us , so again the difference of and is , but we put sign with nd split  as . now the equation becomes

and

therefore the solution of the above equation are and .

If

we first multiply   and   and the factorize the value of , repeat the case I and case II accordingly.

Quadratic Equation Questions

1. Solve the quadratic equation: .
2. Solve the quadratic equation: .
3. Find the roots of .
4. Factor and solve: .
5. Solve for : .
6. What are the solutions to ?
7. Solve by completing the square: .
8. Use the quadratic formula to solve: .
9. Solve the equation: .
10. Solve by factoring.
11. Solve using the quadratic formula.
12. Find the roots of .
13. Solve by completing the square.
14. Factor and solve .
15. Solve using the quadratic formula.
16. Solve for : .
17. Find the solutions to .
18. Solve by factoring.
19. What are the roots of ?
20. Solve using the quadratic formula.

Got Top
.

Methods Of Solving System of Two Linear Equations

solving a system of two linear equations. There are three main methods to solve such systems:

• Substitution,
• Elimination, and
• Graphical Method.
• Matrix Method

Substitution Method

Explanation:
In the substitution method, we solve one equation for one variable and substitute that expression into the other equation to find the value of the second variable.

Steps:

Solve one equation for one variable in terms of the other (e.g., solve for
.
Substitute the expression into the other equation.
Solve for the second variable.
Use this value to find the first variable.

Example:
Solve the system of equations:

   

Step 1: Solve equation for :

   

Step 2: Substitute    into equation

   

Step 3: Solve for  :

   

Step 4: Put the value of in eq

   

The solution is:​

   

Substitution Method: 10 Practice Questions

   

   

   

   

   

   

   

   

   

   

   

   

   

Continue reading Mathematics Notes →