Algebra of Complex Number

Introduction

An algebraic statement with the component √ i =- 1 (square root) is called a complex number. The algebra of a complex number has two parts: the real part, denoted by Re(z), and the imaginary part, which is characterized by I(z). For the complex number represented by ‘z’, the imaginary portion is denoted by Im(z).

Positive, fraction, negative, rational, irrational, integer, decimal, or even zero can be used for the real or imaginary portion. These complex numbers are known as ‘Purely Imaginary Numbers’ since only the real part of any complex number ‘z’ is zero, i.e., Re(z) = 0. If the imaginary portion of any complex number ‘z’ is zero. When I’m(z) = 0, these are referred to as ‘Purely Real Numbers’.

What are Imaginary Numbers?

Real numbers multiplied by the imaginary unit ‘i’ are called imaginary numbers. In math, the letter I (or ‘j’ in some publications) denotes the imaginary portion of any complex integer. It aids in distinguishing between the real and imaginary parts of any complex number. 

i = √-1

The discipline of electronics is where complex numbers are most commonly used. Because I is already reserved for current electronics, they began using ‘j’ instead of I for the imaginary part.

Explain Algebra of Complex Numbers

Equality of Complex Numbers

Two complex numbers z₁and z₂ are likely to be similar if

Condition 1) Re (z₁) = Re (z₂)

Condition 2) Im (z₁) = Im (z₂)

So If, z₁ = x + 3i and z₂ = -2 + yi are equal, then as per above conditions,

Re (z₁) = x and Re (z₂) = -2, so x = -2

And similarly

Im (z₁) is equal to 3 and Im (z₂) = y, so y is equal to 3

Addition of Complex Numbers

Let z₁ = a + ib and z₂ = c + id, then the sum of these two complex numbers in standard form that is z₁ + z₂ calculated as:

(z₁ + z₂) = (a + ib) + (c + id)

= (a + c) + i (b + d)

Therefore, 

z₁ + z₂ = Re (z₁+ z₂) + Im (z₁+ z₂)

The addition of complex numbers can be another complex number.

The following qualities are contained in the addition of complex numbers.

• Closure Law: The Closure Law states that the sum of two complex numbers equals another complex number. If z₁ + z₂ is a complex number, then z will be a complex number as well
• Commutative Law: According to commutative law, z₁ + z₂=z₂ + z₁ for any two complex numbers z₁ and z₂
• Associative Law: (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃) for any three complex algebra numbers
• Existence of Additive Identity: Additive identity, also known as zero complex number, is indicated as 0 (or 0 + i0), implying that z + 0 = z for any complex number z
• Additive Inverse Existence: Any complex numbers additive inverse or negative. The real and imaginary parts of the complex number z have the opposite sign. The symbols for it are –z and z + (-z) = 0

Multiplication of Two Complex Numbers

• The Law of Closure: In other words, the product of any two complex numbers is another complex number. So, for example, if z = z₁ – z₂ and z₁ and z₂ are both complex numbers in standard form, then z will be a complex number

• The Law of Commutation: According to the commutative rule, Z₁ and Z₂, Z₁-Z₂= Z₂- Z₁
• Identity Multiplication: Multiplicative Identity is symbolized by 1 (or 1 + i0), which means that z.1 = z for every complex integer z
• Inverse Multiplication: For any non-zero complex number z,1/z, or z^-1 is called as the multiplicative inverse as z,1/z=1 if z= x+ iy then

Difference of Two Complex Numbers

If z₁ = a + ib and z₂ = c + id are both complex numbers, then the difference between them is. The formula for z₁ – z₂ is:

(a + ib) – (c + id) =z₁ – z₂

= i (b – d) + (a – c)

Therefore,

Re (z₁ – z₂) + Im =z₁ – z₂ (z₁ – z₂)

A complex number’s difference can be another complex number.

Significance of Asterisk in Complex Numbers?

In complex numbers, an asterisk (*) denotes the complex conjugate of any complex number.

If each complex number is represented by z₁ = x + it, then its complex conjugate is expressed by

The complex conjugate of any complex number can alternatively be defined as having the same real part and magnitude of the imaginary part as the provided complex number but with the opposite sign.

Conclusion 

Here, we’ve learned about the algebra of complex numbers, the difference between two complex numbers, imaginary numbers, and more. 

There’s no arguing with the fact that the subject is difficult to grasp and demands a significant amount of time and effort.