Complex Numbers

Question 1: What is a complex number?

Answer:

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.

Question 2: What is the standard form of a Complex Number?

Answer:

A complex number is generally expressed as “z” and is written in its standard form as z = a + ib. A complex number in its standard form has two parts: a real part and an imaginary part. Here, “a” is a real part that is represented as Re(z) and “ib” is an imaginary part that is represented as Im(z). 

Question 3: How to represent a complex number on a graph?

Answer:

A complex number of the form z = a + ib can be represented on the argand plane by considering its coordinates as (Re(z), Im(z)) = (a, ib). An Argand plane or a complex plane is a Euclidean plane concerning complex numbers where the real part of a complex number “a” is represented on the X-axis and the imaginary part “ib” is represented on the Y-axis.

Question 4: What is meant by modulus and argument in complex numbers?

Answer:

The modulus of the complex number (r) is the distance of the complex number represented as a point in the argand plane (a, ib), i.e., the linear distance between the origin (0, 0) and the point (a, ib).

r = √(a² + b²)

The argument of the complex number is the angle in the anticlockwise direction made by the line joining the geometric representation of the complex number and the origin, with the positive x-axis.

Argz (θ) = tan⁻¹(b/a)

Question 5: What is the polar form of a complex number?

Answer:

A complex number can also be represented and identified on the argand plane by using its polar form. To represent the complex number on the argand plane, the polar form makes use of the modulus and argument of the complex number. A complex number z = a + ib is expressed as z = r(cosθ + isinθ) in its polar form, where r is the modulus and θ is the argument of a complex number. Here, r is equal to √(a² + b²), whereas θ is equal to tan^-1(b/a).

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.