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COMPLEX NUMBERS

COMPLEX NUMBERS. Consider the quadratic Equation X 2 + 1 = 0 What is its solution ?. X 2 = - 1 or x =. But this number is not known to us. What is i ?. unreal. i is an imaginary number. complex. Or a complex number. imaginary. Or an unreal number. The terms are inter-changeable.

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COMPLEX NUMBERS

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  1. COMPLEX NUMBERS

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

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

  4. 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.

  5. 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

  6. IRRATIONAL NUMBERS

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

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

  9. 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

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

  11. Properties of addition

  12. Properties of addition

  13. 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)

  14. 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)

  15. PROPERTIES OF MULTIPLICATION

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

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

  18. The following identifies are true for complex numbers

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

  20. 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.

  21. USEFUL RESULTS

  22. USEFUL RESULTS

  23. USEFUL RESULTS

  24. IZI

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

  26. 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

  27. 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.

  28. 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.

  29. 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.

  30. ENJOY COMPLEX NUMBERS & LIFE………

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