Addition of Complex Numbers

A complex number is the sum of real and imaginary numbers. They are represented by a+ib, where a is the real number, and ib is the imaginary number known. For instance, 23 + 4i is a complex number – 23 is a real number (Re), and 4i is an imaginary number (Im). 

You can perform all basic arithmetic operations on complex numbers. One of these fundamental operations applied to complex numbers is in addition.

Addition of Complex Numbers

The addition of complex numbers is one of the most basic operations. To add complex numbers, you must first rearrange the real and imaginary components of the numbers before operating. Group the similar terms and add them. 

The formula for adding complex numbers is z1 = e + ig and z2  = k + ih, where e, g, k, and h are all real integers:

z1  + z2 = e + ig + k + ih

z1  + z2 = (e + k) + (ig + ih)

z1  + z2 = (e + k) + i(g + h)

Hence, (e + ig ) + (k + ih ) = (e + k) + i(g + h)

Addition of Complex Numbers: Steps and Rules

The steps for the addition of complex numbers are as follows:

• Step 1: Separate complex numbers into real and imaginary components.
• Step 2: Add the real numbers of the complex number.
• Step 3: Add the imaginary numbers of the complex numbers.
• Step 4: To get the final answer in the correct form, arrange it in the form of a + ib. 

Some important notes on the addition of complex numbers:

• The addition of complex numbers is just like adding two binomials. You merely need to combine similar terms.
• The modulus and argument of a complex number are used to indicate the polar form of the number. To add complex numbers in polar form, you must convert them to rectangle form(z = a+bi) before proceeding. The final result is then converted to polar form.

Properties of Addition of Complex Numbers

The properties of the addition of complex numbers are listed below.

• Closure property

By definition, we know that the sum of any two complex numbers will always be a complex number. Hence the set of complex numbers is closed under addition.

• Commutative Law

With regard to addition, any two complex numbers, z1, z2 ∈ C, we can say their addition is commutative if z1 + z2 = z2 + z1.

Example for this property:

Consider the addition of complex numbers, z1 + z2

Where, z1= a1+ ib1 and z2= a2+ ib2

Therefore, z1 + z2 = (a1 + ib1) + (a2 + ib2)

Or, z1 + z2 = (a1 + a2)+i(b1 + b2), where a1, a2, b1, b2∈ R.

Now consider the addition of complex numbers, z2+ z1

By solving the addition of complex numbers as shown above, you will get

 z1 + z2= (a2 + a1) + i(b2 + b1)

 

Since the real numbers are commutative, a1+ a2= a2+ a1(by property of real numbers) and b1+ b2= b2+ b1, we can say (a2+a1) + i(b2 + b1) = (a1 + a2) + i(b1 + b2).

This proves that z1+ z2= z2+ z1.

• Associative Law

Complex numbers are associative under addition. Consider three complex numbers,

z1= a + ib , z2= c + id and z3= e + if

By the law of associativity we can say that ,(z1 + z2)+ z3= z1+(z2 + z3)

Or, (a + ib + c + id ) + (e + if) = (a + ib) + ( c +id + e + if)

Or, [(a + c) + i ( b +d)] + (e + if) = (a + ib) +[(c + e) + i( d +f)]

Or, (a + c + e ) + i (b + d + f ) = ( a + c + e) + i(b + d + f)

• Additive Identity

An additive identity is a number that gives the same number when added to any other complex number. For instance, for real numbers, 0 is the additive identity. Similarly, you can observe that 0 + i. 0 is the additive identity for complex numbers (You will denote it by 0).

• Additive Inverse

There is an inverse for any operation involving the addition of complex numbers, such that the addition of a complex number with it provides the additive identity. For every complex number z = a + ib, there exists a complex number – z = -a + i(-b) such that z + (-z) = 0 or the additive identity.

Conclusion

To summarise, the complex number is easily utilised to find the square root of a negative number. The addition, subtraction, multiplication, and division operations that can be done on natural numbers may similarly be performed on complex numbers. For the addition of complex numbers, the real element is added to the natural element, and the imaginary part is added to the imaginary part. There are different rules for subtraction, multiplication, and division of complex numbers.