Algebra of Complex Numbers

A complex number is one that can be expressed as a+bi, where a and b are real numbers and I is the imaginary unit, satisfying the equation I=5+6i1  is a complex number, for example, where 5 is a real number and 6i is an imaginary number. As a result, the sum of the real and imaginary numbers is a complex number. Algebraic operations on complex numbers are represented in mathematics by four basic arithmetic operations: addition, subtraction, multiplication, and division. Complex number is the product of a real and an imaginary number.

Definition of complex numbers

Complex numbers are made up of two parts: a real number and an imaginary number. Complex numbers serve as the foundation for more advanced mathematics such as algebra. They can be used in a variety of real-world situations, particularly in electronics and electromagnetism.

Algebra of Complex Numbers

A Complex Number is an algebraic expression that contains the factor i= -1. These numbers are divided into two parts: the real part, denoted by Re(z), and the imaginary part, denoted by I(z). For the complex number represented by ‘z,’ the imaginary part is denoted by im(z).

Algebra of Complex Numbers Types

There are four kinds of algebraic complex numbers, which are listed below. The four complex number operations are as follows:

1. Addition 
2. Subtraction 
3. Multiplication 
4. Division

Complex Number Addition

To add two complex numbers, simply add their real and imaginary parts.

a+bi+c+di=a+c+(b+d)

Complex Number Subtraction

Simply subtract the corresponding real and imaginary parts of two complex numbers to subtract them.

                                             a+bi+c+di=a c+(b d)

Two Complex Numbers Multiplication

Multiplying two complex numbers is the same as multiplying two binomials. Assume that we need to multiply a + bi and c + di. We’ll add them up term by term.

a+bic+a+bidi=a+bic+a+bidi

= (ac+bci+a di+b d1)

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

Complex Numbers Division

The formula for the reciprocal of a complex number is applied in the division of complex numbers.

z1z2=ac+bdc2+d2+i(bc-adc2+d2)

Fundamental theorem

The theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree n a polynomial with complex coefficients has exactly n complex roots counted with multiplicity.

The use of successive polynomial division can be used to demonstrate the equivalence of the two statements.

Despite its name, there is no purely algebraic proof of the theorem because any proof must employ some form of analytic completeness of the real numbers, which is not an algebraic concept.

Furthermore, it is not fundamental to modern algebra; its name was given at a time when algebra was synonymous with equation theory.

Algebra of Complex Numbers problem

Solve this problem (4-5i)(12+11i)

Solution:

We already know how to multiply two polynomials, so we can also multiply two complex numbers. To obtain, we simply “foil” the two complex numbers.

4-5i12+11i=48+44i-60i-55i2 

=48-16i-55i24-5i12+11i=48+44i-60i-55i2=48-16i-55i2

i2=1i2=-1 then,

4-5i12+11i=48-16i-55-1

Conclusion

The algebraic methods define the algebraic operations on complex numbers. To explain the relationship between the numbers of operations, basic algebraic laws such as associative, commutative, and distributive law are used. The algebraic expressions are solved in a straightforward manner using these laws. Because algebra is a concept based on known and unknown values (variables), it has its own set of rules for solving problems.  It allows us to distinguish between the real and imaginary parts of any complex number. Furthermore, it is not only not distinguishable but also has some value. Complex numbers are most widely used in the field of electronics.