Complex Number and Quadratic Equations - Solutions and Study Material

For Video Tutorials on Complex Number and Quadratic Equations Visit Page No - 5

Complex Number : Let us denote \sqrt{-1} by the symbol i. Then ,we have i^{2} = -1. This means that I is a solution of the equation x^{2}+1=0

A number of the form a + ib, where a and b are real numbers, is defined to be a complex .

Complex Number: 2 + 3i, (-1)+i\sqrt{3} , 4+i\left(\frac{-1}{11}\right) are complex numbers.

For the Complex Number z = a + ib, a is called the real part, denoted by Re z and b is called the imaginary part denoted by Im z of the complex number z. For example, if z = 2 + i5, then Re z = 2 and Im z = 5.

Two complex numbers Z_{1} = a + ib and Z_{2} = c + id are equal if a = c and b = d

Example: If 4x + i(3x – y) = 3 + i (– 6), where x and y are real numbers, then find the values of x and y.

Solution: We have , 4x + i (3x – y) = 3 + i (–6) ... (1)

Equating the real and the imaginary parts of (1), we get ,4x = 3, 3x – y = – 6,

which, on solving simultaneously, give x = \frac{3}{4} , y = \frac{33}{4}

Algebra of Complex Numbers: In this Section, we shall develop the algebra of complex numbers

Addition of two complex numbers Let Z_{1} = a + ib and Z_{2} = c + id be any two complex numbers .Then , the Z_{1} is define as follows :

Z_{1} +Z_{2} = (a + c) + i(b + d) , which is again a complex number.

For Example (2 + 3i) +(-6 +5i) = (2-6) + I (3 + 5) = (-4 + 8 i)

The addition of complex number satisfy the following properties:

  • The closure law The sum of two complex numbers is a complex number, i.e., Z_{1} +Z_{2} is a complex number for all complex numbers Let Z_{1} and Let Z_{2}
  • The commutative law For any two complex numbers Z_{1} and Z_{2} ,
    Z_{1} +Z_{2} \:=\:Z_{2} +Z_{1}
  • The associative law For any three complex numbers Z_{1},Z_{2},Z_{3} ,
    Z_{1} +(Z_{2}+ Z_{3})\:=\: Z_{1}+( Z_{2}+ Z_{3})
  • The existence of additive identity There exists the complex number 0 + i 0 (denoted as 0), called the additive identity or the zero complex number, such that, for every complex number z, z + 0 = z
  • The existence of additive inverse To every complex number z = a + ib, we have the complex number – a + i(– b) (denoted as – z), called the additive inverse or negative of z. We observe that z + (–z) = 0 (the additive identity)

Latest Jobs

  1. Type
    Job Notification
    Job
    UPSC ENGINEERING SERVICES EXAMINATION, 2020 Notification.
    Location
    Anywhere
    Date Posted
    7 Oct 2019
  2. Type
    Job Notification
    Job
    IBPS Clerk Recruitment 2019 - Last Date 09-10-2019
    Location
    Anywhere
    Date Posted
    3 Oct 2019
  3. Type
    Job Notification
    Job
    National Institute of Technology Kurukshetra Recruitment of Professor - Last Date 31-10-2019
    Location
    KurukshetraHaryana, India
    Date Posted
    3 Oct 2019
  4. Type
    Admission
    Job
    National Archives of India - School of Archival Studies 79th short term certificate course in "Care & Conservation of Books, Mss. & Archives"
    Location
    Anywhere
    Date Posted
    3 Oct 2019
  5. Type
    Admission
    Job
    Orissa Maritime Academy admission into Pre-sea General Purpose Rating course (Jan -Jun 2020)
    Location
    ParadeepOdisha, India
    Date Posted
    1 Oct 2019
  6. Type
    Job Notification
    Job
    RITES Recruitment of Site Inspector - Civil, E&M and CAD Operator.
    Location
    GurugramHaryana, India
    Date Posted
    1 Oct 2019
  7. Type
    Job Notification
    Job
    NIT Manipur Recruitment of Registrar on Deputation.
    Location
    ImphalManipur, India
    Date Posted
    1 Oct 2019
  8. Type
    Job Notification
    Job
    ICAR - Indian Agricultural Research Institute invited for walk-in-Interview for filling the temporary post of JRF/SRF/Project Assistant.
    Location
    DelhiDelhi, India
    Date Posted
    30 Sep 2019
  9. Type
    Job Notification
    Job
    FCI Recruitment of GENERAL/ DEPOT/ MOVEMENT/ ACCOUNTS/ TECHNICAL/ ENGINEERING and More.
    Location
    Anywhere
    Date Posted
    30 Sep 2019
  10. Type
    Job Notification
    Job
    Bharati College Janak Puri Recruitment of Principal.
    Location
    DelhiDelhi, India
    Date Posted
    30 Sep 2019

Jobs by Category

Syllabus and Previous Papers

Leave a Reply

Your email address will not be published. Required fields are marked *