Table of Contents
What are Complex Numbers?
Complex numbers are numbers that can be represented in the form a+bi, where a and b are real numbers and i is the imaginary unit. Complex numbers are used in mathematics and physics to solve problems that cannot be solved using only real numbers.
Define Complex Numbers
A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of negative one. Complex numbers are used in mathematics and physics to describe quantities that cannot be expressed as a simple real number.
Equality of Complex Numbers
Equality of complex numbers states that two complex numbers are equal if and only if they have the same real part and the same imaginary part.
Purely Real Complex Numbers
If we define a complex number as a number that can be expressed in the form a + bi, where a and b are real numbers and i is the square root of -1, then all of the following numbers are complex:
2, -3.5, 0.5, -i
However, the following numbers are not complex:
1, -2, 3, 0
Purely Imaginary Complex Number
A purely imaginary complex number is a number that can be expressed in the form \(a + bi\) where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.
Zero Complex Numbers
A complex number is a number that can be represented in the form a+bi, where a and b are real numbers and i is the imaginary unit. A complex number can be written in polar form as r(cosθ+isinθ), where r is the magnitude and θ is the angle.
A complex number can be thought of as a point in the complex plane. The real part of a complex number is the point’s x-coordinate, and the imaginary part is the point’s y-coordinate.
Graphical Representation of Complex Numbers
Complex numbers can be represented on a graph with the real number line on the horizontal axis and the imaginary number line on the vertical axis. Points on the graph represent complex numbers. The real part of a complex number is shown by the position of the point on the real number line, and the imaginary part of the complex number is shown by the position of the point on the imaginary number line.
Mis-conception of Complex Number
A complex number is a number that can be expressed in the form a+bi, where a and b are real numbers and i is the imaginary unit.
Powers of Iota (i)
The power of iota is to create change. With a simple symbol, it can alter the course of history. It can inspire people to take action and make a difference in the world.
Addition of Complex Numbers
To add two complex numbers, add the real parts and add the imaginary parts.
For example,
3 + 5i
= 8
Subtraction of Complex Numbers
To subtract two complex numbers, we first subtract the real parts and then subtract the imaginary parts.
Product of Complex Numbers
The product of two complex numbers is found by multiplying the real parts and multiplying the imaginary parts.
Conjugate of a Complex Number
The conjugate of a complex number is the number with the sign of the imaginary part reversed.
Division of Complex Numbers
The division of complex numbers is very similar to the division of real numbers. The dividend is the complex number that is being divided, and the divisor is the number that is doing the dividing. The quotient is the result of the division, and the remainder is the number that is left over after the division is complete.
To divide two complex numbers, use the following steps:
1. Write the dividend and the divisor next to each other with a dividing line between them.
2. Write the quotient above the line and the remainder below the line.
3. Perform the division using the steps shown in the section on division of real numbers.
4. Write the answer in standard form.
Here is an example of how to divide two complex numbers:
3 + 4i ÷ 2i
3 + 4i ÷ 2i = 1 + 2i
Here is another example:
(3 − 2i) ÷ (1 + i)
(3 − 2i) ÷ (1 + i) = (3 + 2i) ÷ (1 − i)
Here is the final answer:
(3 + 2i) ÷ (1 − i) = (3 − 2i) ÷ (1 + i)
