Arithmetic Operations on Complex Numbers
We can perform various arithmetic operations such as addition, subtraction, multiplication, and division on complex numbers just as we can do on natural numbers. Here, we have to combine like terms while performing arithmetic operations; i.e., we must combine the real number with the real number and the imaginary number with the imaginary number.
Addition of Complex Numbers
When two complex numbers are added, the real part is added to the real part and the imaginary part is added to the imaginary part. Let us consider two complex numbers of the form z1 = a + ib and z2 = c + id. Now,
z1 + z2 = (a + c) + i(b + d).
Subtraction of Complex Numbers
When a complex number is subtracted from the other, the subtraction is separately performed for the real parts and then performed for the imaginary parts. Let us consider two complex numbers of the form z1 = a + ib and z2 = c + id. Now,
z1 – z2 = (a-c) +i(b – d)
Multiplication of Complex Numbers
The multiplication of two complex numbers is similar to the multiplication process of two binomials. Here, we use the formula i2 = -1. Let us consider two complex numbers of the form z1 = a + ib and z2 = c + id. Now,
z1 × z2= (a + ib) × (c + id) = (ac – bd) + i(ad + bc)
Division of Complex Numbers
While performing the division of two complex numbers, we make use of the reciprocal formula of a complex number. Let us consider two complex numbers of the form z1 = a + ib and z2 = c + id. Now,
z1/z2 = (a + ib) × 1/(c + id) = (a + ib) × (c – id)/(c2 + d2)
Is Every Real Number a Complex Number?
A complex number is referred to as the sum of a real number and an imaginary number. It is generally expressed as “z” and is written in the form of a + ib, where a and b are real numbers and i = √(-1). Here, “a” is a real part that is represented as Re(z) and “ib” is an imaginary part that is represented as Im(z). Some examples of complex numbers are 2 + 3i, 5–7i, 3 + i√4, etc. The imaginary number is generally expressed either as “i” or “j”, whose value is equal to √(-1). Hence, the square of an imaginary number gives us a negative value. The square root of negative numbers can be calculated using complex numbers. Some applications of complex numbers are in signal processing, fluid dynamics, quantum mechanics, electromagnetism, vibration analysis, and also many scientific research areas.
Real numbers are referred to as the union of the set of rational numbers and the set of irrational numbers, i.e., positive numbers, whole numbers, integers, rational numbers, irrational numbers, etc. are real numbers. Some examples of real numbers are -4, -7/11, 0, 9, √6, 3.8, etc.
A number that gives a negative value when squared is called an imaginary number. It is the product of a non-zero real number and the imaginary unit “i”, whose value is √(-1). An imaginary number can also be defined as the square root of negative numbers. Some examples of imaginary numbers are -2i, √5i, 3i, etc.