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Complex Numbers

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Published in: Mathematics
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Complex Numbers

Amanta B / Kochi

5 years of teaching experience

Qualification: 12th (Nirmala Higher Secondary School, Muvattupuzha - 2018), 10th (Presidency Central School, Mudavoor - 2016)

Teaches: Mathematics

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  1. COMPLEX NUMBERS • A complex number is a number expressed in the form z = a + ib. where a and b are real numbers and i = Ad-I (iota). The real part of z is given by Re(z) = a, and the imaginary part of z is given by Im(z) = b. The complex number is purely imaginary if Re(z) = 0. The complex number is purely real if Im(z) = O. • a +ib=O, if a = 0 and b = O. • For every positive real number a, •v/Za = • Two complex numbers are equal when their real parts are equal and their imaginary parts are equal, i.e., a +i x + iy iff a = x and b = Y• Algebraic Operations with Complex Numbers Addition: + + • Subtraction: (a+ib) — (C+id) = (a-c) + i(b-d) Multiplication: (a+ib) x (C+id) = (ac-bd) + i(ad+bc) (a + ib) (a + ib) (c — id) Division: x (c + id) — (c + id) (c — id)
  2. Properties of Algebraic Operations e) z +0 = z and z * 1 = z. (0 and 1 are the identity elements for addition and multiplication, respectively.) f) Additive inverse of z is -z. g) Multiplicative inverse of z is l/z. Conjugate of a Complex Number The conjugate of complex number z = a + ib is given by 2 = a - ib • If (Z) = Z then z is purely real 25 2
  3. Modulus of a Complex Number • Izl= 0, then z = O, i.e., Re(z)= Im(z) • Izl=läl . —Izl Re(z) I zl • —Izl Im(z) I zl z- 2 Izl n • Iznl=lzl • Izlllz21 I Zil IZ21 Polar form of a complex number The polar form of a complex number z = x + iy with coordinates (x, y) is given as z = r (cose + i sine). 2 6 = tan—I Y x • x = r Cose y = r Sine
  4. 1 st Quadrant 2nd Quadrant 3rd Quadrant 4th Quadrant e = 180 -a e = -(180-a) Square root of a complex number a + bi • a+ib=x —y2+ 2xyi 2 .61 = x —Y • b=2xy (x2 — Y2)2 + (2xy)2 = a2 + b2 • On solving x —Y' Cube roots of unity and x2 + Y2 will get x and y. The three cube roots of unity are: 1 , 2 2 The three cube roots may be denoted by 1 ,w,w .1 •vvvv . (vv3)n = I 2 2 then a ß = 1 2

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