Complex no slides 476 kho tài liệu bách khoa

Complex Numbers General Introduction : Complete development of the number system can be summarised as N⊂W⊂I⊂Q⊂R⊂Z Every complex number z can be written as z = x + i y where x, y ∈ R and i = x is called the real part of z and y is the imaginary part of complex Note : only if atleast one of either a or b is non negative (If a and b are positive reals then ] Complex Plane z = a + ib Purely real Purely imaginary imaginary If b = if a = if b ≠ Hence + 0i is both a purely real as well as purely imaginary but not imaginary Note : (i) The symbol i combines itself and with real number as per the rule of algebra together with i2 = – ; i3 = –i ; i4 = 1, i2005 = i, ; i2006 = –1 Infact i4n = 1, n ∈ I note that (1 + i + i2 + … + i2006 = i)] (ii) Every real number can also be treated as complex with its imaginary part zero Hence there is one-one imapping between the set of complex numbers and the set of points are the complex plane Algebra of Complex Addition, substraction and multiplication of complex numbers are carried out like in ordinary algebra using i2 = –1, i3 = –i etc treating i as a polynomial Differences between algebra of complex and algebra of real number are (i) Inequality in complex numbers are never talked If a + i b > c + id has to be meaningful ⇒ b = d = Equalities however in complex numbers are meaningful Two complex numbers z1 and z2 are said to be equal if Re z1 = Re z2 and Im (z1) = Im (z2) (i.e they occupy the same position on complex plane) (ii) In real number system if a2 + b2 = ⇒ a = = b but if z1 and z2 are complex numbers then z12 + z22 = does not imply z1 = z2 = e.g z1 = + i and z2 = – i However if the product of two complex numbers is zero then at least one of them must be zero, same as in case of real numbers (iii) In case x is real then |x|= but in case of complex | z | altogether has a different meaning Q α1 α2 α3 ………α αn – = or –1 according as n is odd or even Q (w – α1) (w – α2) … (w – αn – 1) Q Sum of all the n, nth roots always vanishes Q (A) (C) i (B) –1 (D) –i Q If cos(α α – β) + cos (β β – γ) + cos(γγ – α) = –3/2 then prove that : (a) Σ cos 2α α = = Σ sin α (b) Σ sin (α α + β) = 0= Σ cos (α α + β) (c) Σ sin 2α α = Σ cos2 α = 3/2 (d) Σ sin 3α α = sin (α α + β + γ) (e) Σ cos 3α α = cos (α α + β + γ) (f) cos3 (θ θ + α) + cos3 (θ θ + β) + cos3 (θ θ + γ) = cos (θ θ + α) cos (θ θ + β) cos (θ θ + γ) where θ ∈ R Q Prove that all roots of the equation are collinear on the complex plane & lie on x = –1/2 Q If zr , r = 1, 2, 3, ……… 2m, m ∈ N are the roots of the equation Z2m + Z2m–1 + Z2m–2 + … + Z + = then prove that Complex numbers and binomial coefficients (i) (ii) (iii) (iv) (v) C0 + C5 + C8 + … C1 + C5 + C9 + … C2 + C6 + C10 + … C3 + C7 + C11 + … C0 + C3 + C6 + C9 + … Straight lines & Circles on Complex Plane (i) (ii) Equation of a line passing through z1 & z2 on argand plane z = z1 + λ(z2 – z1) (see vector equation of line) Circle | z – z0 | = r Q Find the area bounded by the curves Arg z = Arg & Arg (z – – complex plane ) = π on the Q Find all the points in the complex plane which satisfy the equations log5 (| z | + 3) – log || z | –1 | = and arg (z – – i) = Parametric Equation Of A Line z = z1 + λ (z2 – z1) where λ ∈ R which is the same as equation number i.e is purely real Reflection Points For A Line (Image of a point in a line) Use concept of straight line Write z = x + iy Equation of a circle described on the line joining z1 & z2 as diameter Note that the equation Arg (z + i) – Arg (z – i) = Does not represent a complete circle but only a semi circle described on the line segment joining (0, 1) & (0, –1) as diameter (in Ist and 4th quadrant) General locii on complex plane (a) (b) (c) (d) (e) | z – z1 | + | z – z2 | = constant (constant > | z1 – z2 | ) is an ellipse with its two foci at z1 and z2 | z – z1 | – | z – z2 | = constant (constant < | z1 – z2 | ) is a hyperbola with its foci as z1 and z2 | z – z1 |2 + | z – z2 |2 = | z1 – z2 |2 represent locus of a circle with z1 and z2 as its diameter (z – )2 + a (z + ) = represent a standard equation of parabola | z – z1 | + | z – z2 | = | z1 – z2 | represent a line segment