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Addition, subtraction, multiplication, and division are the four fundamental arithmetic operations that are always used in algebraic operations on complex numbers. A complex number is made up of two numbers: a real and an imagined one. The only way to express algebraic operations on complex numbers is through algebraic approaches. Simple algebraic laws such as associative, commutative, and distributive law are used to describe the link between the number of operations.
Let’s look at some examples of basic algebraic operations on complex numbers in this article.
Algebraic Operations on Complex Numbers
The addition, subtraction, multiplication, and division operations that are used with natural numbers can likewise be used with complex numbers. The following are the details of the various Algebraic operations on complex numbers.
1. Addition of Complex Numbers
Complex numbers can be added in the same way that natural numbers may. The real part is added to the real part, while the imaginary part is added to the imaginary part in complex numbers.
Take the two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2 as examples.
Then, as follows, the complex numbers z1 and z2 are added:
z1 + z2 = (a1 + a2) + i(b1 + b2)
Addition of Complex Numbers: Properties
1. Closure property: closure law states that adding complex numbers produces a complex number.
2. Additive identity: Additive identity is a complicated number that, when added, returns the original number. For real numbers, for example, 0 is the additive identity. Similarly, the additive identity for complex numbers is 0 + i.0 We will just denote it by 0.
3. Commutative property: When it comes to addition any two complex numbers commute, i.e. z1 + z2 = z2 + z1
4. Additive inverse property: There is an inverse for each operation involving the addition of complex numbers, such that the addition or subtraction of a complex number with it yields the additive identity. There is a complex number – z = -a + i(-b) such that z + (-z) = 0 or the additive identity for every complex number z = a + ib.
2. Subtraction of Complex Number
The process of subtracting complex numbers is identical to subtracting natural numbers. Subtraction is performed individually across the real part and then across the imaginary part for any two complex numbers. In the case of complex numbers.
z1 = a + ib , z2 = c + id
Then we have,
z1 – z2 = a – c + i(b – d)
3. Multiplication of Complex Number
In a few ways, multiplication of complex numbers differs from multiplication of natural numbers.
The product of the two complex numbers is
z1 = a + ib , z2 = c +id is z1 × z2 = ca – bd + i(ad + bc)
The polar form of multiplication for complex numbers differs slightly from the previously discussed version. To get the product of the complex numbers, multiply the absolute values of the two complex numbers and add their arguments.
If z1 = r1(cos cos θ1 + i sin sin θ1) and z2 = r2(cos cos θ2 + i sin sin θ2 )
z1 × z2 = r1 × r2(cos cos (θ1 + θ2) + i sin sin (θ1 + θ2 )
3. Multiplication of Complex Numbers: Properties
Closure property: Multiplying two complex numbers yields only a complex number
z1 × z2 = z
Commutative property: The order in which hangs are hung has no bearing on the outcome of their output
z1 × z2 = z2 × z1
Associative property : Rearranging complex numbers has no impact on the final product
z1[z2 × z3] = [z1 × z2]z3
Distributive property: When we multiply a complex number by the sum of two complex numbers, we get
z1[z2 + z3] = z1 × z2 + z1 × z3
4. Division of Complex Numbers
The reciprocal of a complex number is used in the division of complex numbers.
We divide the two complex numbers z1 = a + ib, z2 = c + id as follows:
Conclusion
In this article we conclude that, remember to combine “similar” terms when performing arithmetic operations on complex numbers, such as adding or subtracting. Check to see if the solution must be written in the simplest a+ bi form. Add the “real” and “imaginary” parts of the complex numbers together. Add the real part to the real part and the imaginary part to the imaginary part to add two complex numbers. Subtract the real component from the real part and the imaginary part from the imaginary part to subtract two complex numbers. Use the FOIL method to multiply two complex integers and combine like terms.
