1 / 11

Complex Numbers

Complex Numbers. An Introduction. Complex Number. Any number of the form x + iy where x,y are Real and i =√-1, i.e., i 2 = -1 is called a complex number. For example, 7 + i10, -5 -4i are examples of complex numbers.

ismael
Download Presentation

Complex Numbers

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Complex Numbers An Introduction

  2. Complex Number Any number of the form x + iy where x,y are Real and i=√-1, i.e., i2 = -1 is called a complex number. For example, 7 + i10, -5 -4i are examples of complex numbers. The set or complex numbers is denoted by C, i.e., C={x+iy: x,y belongs to R(set of Real number) and i=√-1}. The complex number x+yi is usually denoted by z; x is called real part of z and is written as Re(z) and y is called imaginary part or z and is written as Im(z). So, for every z belongs to C. Z=Re(z) + iIm(z). For example, if z= 3 +5i, then Re(z)=3 and Im (z)=5.

  3. Algebra of Complex Numbers Addition of complex numbers Let z1=x1+iy1 and z2=x2+iy2 be any two complex numbers, then the sum or addition of z1 and z2 is defined as (x1+x2) + i(y1+y2), and it is denoted by z1+z2 . The sum or two or more complex numbers is also a complex number. Negative of a complex number Let z=x+iy be any complex number, then the number –x-iy is called negative of z and is denoted by –z. Negative of a complex number is again a complex number.

  4. Algebra of Complex Numbers Difference of Complex numbers Let z1=x1+iy1 and z2=x2+iy2 be any two complex numbers, then the difference of z2 from z1 is defined as (x1-x2) + i(y1-y2), and it is denoted by z1-z2 . The difference of two complex numbers is also a complex number.

  5. Algebra of Complex Numbers Multiplication of complex numbers. Let z1=x1+iy1 and z2=x2+iy2 be any two complex numbers, then the product or multiplication of z1 and z2 is defined as (x1x2 - y1y2) + i(x1y2 + y1x2) and it is denoted by z1z2 . For example: z1 = 5+3i and z2 = 2+4i then z1z2= (10-12) +i(20+6)=-2+26i.

  6. Square roots of a complex number Lets understand the entire process with the help of an example. Find the square roots of 3-4i Solution: let x+iy be a square root of 3-4i Then (x+iy)2 =3-4i x2-y2 +2xyi=3-4i ( (a+b)2 =a2+b2+2ab) => x2–y2=3 ………………..(i) And 2xy=-4, i.e., xy=-2………..(ii) We have (x2+y2)2= (x2-y2)2+4x2y2 32+4(4)=9+16=25 x2+y2=+5 but as x2+y2 =5………………..(iii) On adding (i) and (iii), we get 2x2 =8 => x2 =4 => x=+2 Substituting in (ii), we get x=2, y=-1 and x=-2, y=1. Therefore, the two square roots of 3-4i are 2-i and -2+i.

  7. Cube Roots of Unity Let x be a cube root of unity, then x3=1. => x3-1=0=> (x-1)(x2 +x+1)=0 => either x-1=0 or x2 +x+1=0 Either x=1or x=(-1+√-3)/2 Thus, the three cube roots of unity are 1, (-1+i√3)/2 and (-1-i√3)/2.

  8. Some properties of cube roots of unity Either of the two non-real cube roots of unity is the square of the other. If one of the non-real cube root of unity is denoted by w(read it as omega), then the other is w2. Further, to avoid any possible confusion, we shall take w= (-1+i√3)/2 Thus, the three cube roots of unity are 1,w and w2. Sum of the three cube roots of unity is zero. Thus 1+w+w2=0 Product of the cube roots of unity is one. i.e. 1.w.w2 =w3=1 Either of the two non-real cube roots of unity is reciprocal of the other. Since w3=1, therefore, w.w2=1=> w and w2 are reciprocals of each other.

  9. De-Moivre’s Theorem When we write the complex number in polar form, then z=r(cosφ + i sin φ) Then according to De-Moivre’s Theorem zn = [r(cosφ + i sin φ)]n = rn(cosnφ + i sin nφ)

  10. Roots of complex number If the Discriminant of any quadratic equation <0 i.e. –ve number. The roots of that quadratic equation are not real, that means the roots of such quadratic equation are complex number. If d<0 where d=b2-4ac Then roots are (–b+√b2-4ac)/2a.

  11. Roots of complex number Solve the equation 3x2+7=0 Here, the discriminant=02-4x3x7=-84 Therefore x=(-0+√-84)/2x3 =(+2(√21)i)/6 => (+(√21)i)/3

More Related