COMPLEX NUMBERS

3. COMPLEX NUMBERS

4. Consider the quadratic Equation X2 + 1 = 0 What is its solution ? X2 = -1 or x = But this number is not known to us.

6. What is i ? unreal i is an imaginary number complex Or a complex number imaginary Or an unreal number The terms are inter-changeable

7. Definition • A complex number is an expression of the form Z = a+ i b,wherea and b are real numbers and i is a mathematical symbol for which is called the imaginary unit. • For example, −3.5 + 2i is a complex number. • The real number a of the complex number z = a + bi is called the real partof z and the real number b is the imaginary part.

8. If b = 0, the number a + bi = a is a real number. Example: 5= 5+ i 0 If a = 0, the number a + bi is called an imaginary number Example: -2i= 0+ (-2)i

10. IRRATIONAL NUMBERS

11. Real numbers and imaginary numbers are subsets of the set of complex numbers. Complex Numbers Imaginary Numbers Real Numbers Irrational Rational

13. Practice Time!!!! • Simplify • Evaluate 3i x -4i

14. Addition of Complex Numbers • Let z1=a+iband z2=c+id be any two complex numbers. Then the sum of those two complex numbers is defined as : z1 +z2= (a+bi) + (c+di) = (a+c) + (b+d)i Addition of two complex numbers can be done geometrically by constructing a parallelogram

15. Practice Time!!!! Simplify • (2+3i ) + (4 -3i) • (-3+4i) + (-2- i10)

16. Properties of addition

17. Properties of addition

18. Difference of two complex numbers • Given any two complex numbers z1 and z2, the difference z1 - z2 is defined as follows : • z1 - z2= z1 +(-z2) Simplify (3i+2i) – ( -2 + i3)

19. Multiplication of two complex numbers • Multiplying complex numbers is similar to multiplying polynomials and combining like terms. • Let a+ib and c+id be any two complex numbers. Then the product of those two complex numbers is defined as follows: • (a+ib) (c+id) = (ac – bd) + i(ad + bc)

20. PROPERTIES OF MULTIPLICATION

21. Practice Time!!!! Simplify • (2+3i)(4-3i) • (-4+2i)(7-12i)

22. Division of two complex numbers • Given any two complex numbersaand b, where b≠ 0, the quotient is defined by • Simplify

23. The following identifies are true for complex numbers

24. POWERS OF i In general, for any integer k, i4k = 1, i4k+1 = i, i4k+2 = -1.

25. The modules and the conjugate of a complex number • Let z = a + ib be a complex number. Then the modulus of z, is denoted by IzI, is defined to be the non negative real number , i.e., IzI=and the conjugate of z, is denoted as , is the complex number = a- ib.

26. USEFUL RESULTS

27. USEFUL RESULTS

28. USEFUL RESULTS

30. IZI

31. y 3 2 1 x 2 3 1 2 + 3i We can represent complex numbers as a point.

32. COMPLEX NUMBERS Polar representation of a Complex number • Let the point P represent the non zero complex number z = x + iy. Let the directed line segment OP be the length r and Ө be the angle which OP makes with the positive direction of x-axis

33. COMPLEX NUMBERS • We may note that the point P is uniquely determined by the ordered pair of real numbers (r, Ө), called the polar coordinates of the point P. • We consider the origin as the pole and the positive direction of the x-axis as the initial line.

34. COMPLEX NUMBERS • We have , x = r cosӨ, y = r sin Ө and therefore , z = r(cosӨ + i sin Ө). The latter is said to be the polar form of the complex number. • Here is the modules of z and Ө is called the argument of z which is denoted by arg z.

35. For any complex number z ≠ 0, there corresponds only one value of Ө in 0 ≤ Ө < 2 However any other interval of length 2 for example - < Ө ≤ , can be such an interval. We shall take the value of Ө such that - < Ө ≤ , called principle argument of z and is denoted by arg z, unless specified otherwise. Figures in the next slide.

41. ENJOY COMPLEX NUMBERS & LIFE………