Introduction of Complex Numbers

The introduction of complex numbers plays a very important role in the theory of numbers.

The equations x\(^{2}\) + 5 = 0, x\(^{2}\) + 10 = 0, x\(^{2}\) = -1 are not solvable in the real number system i.e, these equations has no real roots.

For example, i is the solution of the equation x\(^{2}\) = -1 and it has two solutions i.e., x = ± i, where √-1.

The number i is called an imaginary number. Generally, the square root of any negative real number is called imaginary number.

The concept of imaginary numbers was first introduced by mathematician “Euler”. He was the one who introduced i (read as ‘iota’) to represent √-1. He also defined i\(^{2}\) = -1.

Definition of Complex number:

A complex number z is defined as an order pair of real numbers and is written as z = (a, b) or, z = a + ib, where a, b are real numbers and i = √-1.

In other words, in an ordered pair (a, b) of two real numbers a and b is represented by the symbol a + ib (where i = √-1) then the order pair (a, b) is called a complex number (or, an imaginary number).

Example of complex number:

3 + 2i, -1 + 5i, 7 – 2i, 2 + i√2, 1 + i, etc. are all complex numbers.


Real and imaginary part of a complex numbers:

According to the definition if the complex number (a, b) be denoted by z then z = (a, b) = a + ib (a, b ϵ R) where a is called the real part, denoted by Re(z) and b is called imaginary part, denoted by Im (z).

In other words, in z = a + ib (a, b ϵ R), if a = 0 and b = 1 then z = 0 + i ∙ 1 = i that is, i represents the unit of a complex quantity.

For this reason, the real number a is called the real part of the complex number z = a + ib and b is called its imaginary part.

In z = a + ib (a, b ϵ R), if b = 0 then z = (a, 0) = a + 0 ∙ i = a, (which is a real part) i.e., the complex number (a, 0) represents purely real number.

Again, in z = a + ib (a, b ϵ R), if a = 0 and b ≠ 0 then z = (0, b) = 0 + ib = ib which is called purely imaginary number

Therefore, a complex number z = a + ib (a, b ϵ R), reduces to a purely imaginary number when a = 0.

 

Equality of two complex numbers:

Two complex number z\(_{1}\) = a + ib and z\(_{2}\) = c + id

Two complex numbers z\(_{1}\) = (a, b) = a + ib and z\(_{2}\) = (c, d) = c + id are called equal, written as z\(_{1}\) = z\(_{2}\) if and only if a = c and b = d

In general, when real and imaginary parts of one of the complex number are respectively equal to the real and imaginary parts of the other complex number then they are equal.

For example, if the complex number z\(_{1}\) = x + iy and z\(_{2}\) = -8 + 3i are equal, then x = -8 and y = 3.


Note: Ordered pairs (a, b) and (b, a) represent two distinct complex numbers when a ≠ b.





11 and 12 Grade Math 

From Introduction of Complex Numbers to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. 2nd grade math Worksheets | Free Math Worksheets | By Grade and Topic

    Dec 04, 24 01:30 AM

    2nd Grade Math Worksheet
    2nd grade math worksheets is carefully planned and thoughtfully presented on mathematics for the students.

    Read More

  2. Time Duration |How to Calculate the Time Duration (in Hours & Minutes)

    Dec 04, 24 01:07 AM

    Time Duration Example
    Time duration tells us how long it takes for an activity to complete. We will learn how to calculate the time duration in minutes and in hours. Time Duration (in minutes) Ron and Clara play badminton…

    Read More

  3. Worksheet on Subtraction of Money | Real-life Word Problems | Answers

    Dec 04, 24 12:45 AM

    Worksheet on Subtraction of Money
    Practice the questions given in the worksheet on subtraction of money by using without conversion and by conversion method (without regrouping and with regrouping). Note: Arrange the amount of rupees…

    Read More

  4. Worksheet on Addition of Money | Questions on Adding Amount of Money

    Dec 04, 24 12:06 AM

    Worksheet on Addition of Money
    Practice the questions given in the worksheet on addition of money by using without conversion and by conversion method (without regrouping and with regrouping). Note: Arrange the amount of money in t…

    Read More

  5. Worksheet on Money | Conversion of Money from Rupees to Paisa

    Dec 03, 24 11:37 PM

    Worksheet on Money
    Practice the questions given in the worksheet on money. This sheet provides different types of questions where students need to express the amount of money in short form and long form

    Read More