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Addition, subtraction, multiplication, and division are the four fundamental arithmetic operations in mathematics. Algebraic operations on complex numbers are given by these four fundamental arithmetic operations. The combination of a real number with an imaginary number makes up what is known as a complex number.
The operations that can be performed on complex numbers using algebraic methods are defined entirely by those methods. The connection between the various numbers of operations can be understood by the application of several fundamental algebraic laws, such as the associative, commutative, and distributive laws. The solution to the algebraic equations can be found in a straightforward manner by applying these rules. As a result of the fact that algebra is an idea that is dependent on both known and unknown values (variables), its very own set of rules has been devised to solve difficulties.
Equality of complex numbers:
Assume that the two complex numbers are z1 and z2 respectively.
Here, z1 = a1 + ib1, and z2 = a2 + ib2.
If the two complex numbers z1 and z2 are equal to one another, denoted by the equation z1 = z2, then we can say that the real part of the first complex number is equal to the real part of the second complex number, while the imaginary part of the first complex number is equal to the imaginary part of the second complex number. In other words, the real part of one complex number equals the real part of the other complex number.(i.e)
Re(z1) = Re(z2) and Im(z1) = Im (z2)
Therefore, according to the equality of complex numbers, “that”
If a1 + ib1 = a2 + ib2, then a1 = a2 and b1 = b2.
Operations on complex numbers:
The following elementary algebraic operations on complex numbers are covered in this article:
- Combining Two Complicated Numbers in Addition
- Calculating the Difference Between Two Complicated Numbers
- Calculating the Product of Two Complicated Numbers
- Performing Division on Two Complicated Numbers.
Addition of two complex numbers:
It is well known that the formula for a complex number is z=a+ib, where a and b are two real values.Consider the following two complicated numbers: z1 = a1 + ib1 and z2 = a2 + ib2
After that, the definition of the sum of the complex numbers z1 and z2 is as follows:
z1+z2 =(a1+a2)+i(b1+b2)
It is clear to see that the real component of the resultant complex number is equal to the sum of the real components of each of the complex numbers, but the imaginary component of the resulting complex number is equivalent to the sum of the imaginary components of each of the complex numbers.
That is, Re(z1+z2)= Re(z1)+Re ( z1 )
Im(z1+z2)=Im(z1)+Im (z2)
When it comes to the complicated numbers,
z1 = a1+ib1
z2 = a2+ib2
z3 = a3+ib3
………..
………..
zn = an+ibn
a1+a2+a3+….+an = (a1+a2+a3+….+an)+i(b1+b2+b3+….+bn)
Multiplication of two complex numbers:
We are aware that the expansion of (a+b)(c+d) equals ac+ad+bc+bd.
Consider also the complex numbers z1 = a1+ib1 and z2 = a2+ib2 in a similar manner.
After that, the product of z1 and z2 can be described as the following:
z1 z2=(a1+ib1)(a2+ib2)
z1 z2 = a1 a2+a1 b2 i+b1 a2 i+b1 b2 i2
Because i2 = -1, hence,
z1 z2 = (a1 a2 – b1 b2) + i(a1 b2 + a2 b1)
Conclusion:
A complex number can be written in the form a + ib and is typically denoted by the letter z. Both a and b are considered real numbers in this context. Addition, subtraction, multiplication, and division are the four fundamental arithmetic operations in mathematics. Algebraic operations on complex numbers are given by these four fundamental arithmetic operations. The operations that can be performed on complex numbers using algebraic methods are defined entirely by those methods.
The solution to the algebraic equations can be found in a straightforward manner by applying these rules. If the two complex numbers z1 and z2 are equal to one another, denoted by the equation z1 = z2, then we can say that the real part of the first complex number is equal to the real part of the second complex number, while the imaginary part of the first complex number is equal to the imaginary part of the second complex number.
