Complex Numbers

The sum of a real and an imaginary number is a complex number. A complex number is denoted by the letter z and has the form a + ib. Both a and b are genuine numbers in this case. The value ‘a’ is known as the real component and is indicated by Re(z), while ‘b’ is known as the imaginary part and is denoted by Im (z). ib is also known as an imaginary number and its value is √-1

Imaginary Numbers :

An imaginary number is a number that has a negative value when squared. An imaginary number is defined as the square root of a negative number that has no measurable value.

Power of i :

The letter i often known as the iota, is used to denote the imaginary component of a complex number. Additionally, the iota(i) can be used to find the square root of negative values. We know that i2 = -1, therefore we use that to calculate the value of √-4 = √i24 = +2i. The essential characteristic of a complex number is the value of i2 = -1. Let’s take a closer look at the expanding powers of i.

Graphical Representation :

Arithmetic Operations of Complex Numbers :

Combine comparable terms while executing complicated number arithmetic operations like addition and subtraction. It means that you should add real numbers to real numbers and imaginary numbers to imaginary numbers.

Addition : (a + ib) + (c + id) = (a + c) + i(b + d)

Subtraction : (a + ib) – (c + id) = (a – c) + i(b – d)

Multiplication : (a + ib). (c + id) = (ac – bd) + i(ad + bc)

Division : (a + ib) / (c + id) = (ac+bd)/ (c2 + d2) + i(bc – ad) / (c2 + d2)

Properties :

The following are the properties of complex numbers:

• When two conjugate complex numbers are added together, the result is a real number.

• A real number can be obtained by multiplying two conjugate complex numbers.

• If x and y are real numbers and x+yi = 0, then x and y must be equal.

• If p, q, r, and s are real numbers, then p+qi = r+si, p = r, and q=s.

• The commutative law of addition and multiplication applies to complex numbers.

• The associative law of addition and multiplication applies to complex numbers.

• The distributive law applies to complex numbers.

• If the sum of two complex numbers is real, and the product of two complex numbers is also real, these numbers are conjugate.

• The result of multiplying two complex numbers and their conjugate value should be another complex number with a positive value.

Polar Form of Complex Numbers :

A number is expressed in terms of an angle and its distance from the origin r in the polar form of a complex number. We use the same conversion methods to write a complex number in rectangle form represented as z=x+yi as we use to write it in trigonometric form:

x=r cos

y= r sin

r=x2+y2

Absolute Value  of Complex Functions :

A number’s absolute value (Modulus) is the number’s distance from zero. The modulus(|z|) is always used to indicate absolute value, and its value is always positive. As a result, the complex number

Z = a + ib has an absolute value of

|z| = √ (a2 + b2)

Examples :

(i) z = 3 + 4i

|z| = √(32 + 42)

= √(9 + 16)

= √25

= 5

(ii) z = 5 + 6i

|z| = √(52 + 62)

= √(25 + 36)

= √61

Conclusion :

• a + bi is a combination of a real and an imaginary number.

• i is the “unit imaginary number,” and a and b are real numbers.

• The values a and b are both possible to be zero.

• All of these numbers are complicated:

• a complex number with a=0 is called an imaginary number.

• a complex number with b=0 is a real number