Introduction to Complex Numbers

Introduction

Finding the square root of negative values is easier with complex numbers. When Hero of Alexandria, a Greek mathematician, attempted to discover the square root of a negative number in the first century, he came across the concept of complex numbers. But all he did was turn the negative into a positive and take the numeric root value.

Furthermore, Italian mathematician Gerolamo Cardano defined the real identity of a complex number in the 16th century while looking for the negative roots of cubic and quadratic polynomial formulas.

Signal processing, electromagnetic, fluid dynamics, quantum mechanics, and vibration analysis are just a few of the fields where complex numbers are used. We may learn about the definition, vocabulary, visualization, properties, and operations of complex numbers in this section.

What are Complex Numbers and How Do You Use Them?

It is denoted by the letter z and has the form a + ib. Both a and b are genuine numbers in this case. The value ‘a’ is known as the real component and is indicated by Re(z), while ‘b’ is known as the imaginary part and is denoted by I’m (z). ib is also known as an imaginary number.

Some of the complex number examples are 2+3i,−2−5i,12+i32,2+3i,−2−5i,12+i32, etc.

Graphing of Complex Numbers

The real and imaginary parts of a complex number can be thought of as an ordered pair (Re(z), I’m(z)) and can be represented as coordinate points in the euclidean plane. The complex plane, or Argand Plane, is the euclidean plane about complex numbers, named after Jean-Robert Argand. 

The real part – a, about the x-axis, and the imaginary part – ib, about the y-axis, are used to depict the complex number z = a + ib. Let’s try to grasp the meaning of two keywords related to the argand plane. The argument and modulus of a complex number.

• Modulus of the Complex Number

The modulus of a complex number is the distance between a point in the argand plane (a, ib) and the complex number. This distance is defined as r = |a2+b2a2+b2| and is measured from the origin (0, 0) to the point (a, ib). Furthermore, this may be deduced from Pythagoras’ theorem, in which the hypotenuse is represented by the modulus, the real component is the base, and the imaginary part is the altitude of the right-angled triangle.

• The argument of the Complex Number

The argument of the complex number is the anticlockwise angle formed by the line connecting the geometric representation of the complex number and the origin, with the positive x-axis. The inverse of the tan of the imaginary component divided by the real part of the complex number is the argument of the complex number.

Properties of a Complex Number

The features of complex numbers listed below can help you better comprehend complex numbers and execute various arithmetic operations on them.

• Conjugate of a Complex Number

By taking the identical real component of the complex number and altering the imaginary part of the complex number to its additive inverse, the conjugate of the complex number is generated. Conjugate complex numbers are those in which the sum and product of two complex numbers are both real numbers. The conjugate of a complex number z = a + ib is z = a – ib.

• Reciprocal of a Complex Number

In the process of dividing one complex number by another, the reciprocal of complex numbers comes in handy. The product of one complex number with the reciprocal of another complex number is the process of division of complex numbers.

• Equality of Complex Numbers

Complex numbers are equal in the same way that real numbers are equal. Two complex numbers  z1=  a1+ib1 and  z2=  a2+ib2 are said to be identical if their real portions a1=a2 and their imaginary parts b1=b2 are equal. In the polar form, two complex numbers are identical if and only if their magnitudes are the same and their argument (angle) differs by an integral multiple of 2.

• Operations on Complex Numbers

The addition, subtraction, multiplication, and division operations that can be done on natural numbers may likewise be performed when it comes to solving complex numbers. The following are the details of the various arithmetic operations on complex numbers.

• Addition of Complex Numbers

Complete numbers are added in the same way as natural numbers are added. The real part is added to the real part, while the imaginary part is added to the imaginary part in complex numbers. For two complex numbers of the form z1=  a+ib and  z2=c+id, the sum of complex numbers  z1+ z2=(a+c)+i(b+d).  All of the properties of addition apply to complex numbers.

• Subtraction of Complex Numbers

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.

• Multiplication of Complex Numbers

Multiplying complex numbers differs from the multiplication of natural numbers in a few ways. We must utilize the formula  i2 = -1 in this case. The product is

z1 z2= (ca – bd) + i(ad + bc) for the two complex numbers z1  = a + ib, z2 = c + id.

The polar form of multiplication for complex numbers differs slightly from the above-mentioned form of multiplication. To get the product of the complex numbers, multiply the absolute values of the two complex numbers and add their arguments.

• Division of Complex Numbers

The division of complex numbers makes use of the formula of reciprocal of a complex number. 

Conclusion 

We’ve studied the basics of complex numbers, solving complex numbers through different methods, as well as other topics.