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Consider the quadratic Equation X2 + 1 = 0 What is its solution ? X2 = - 1 or x = ± βπ But this number is not known to us. It is a tool to solve an equation. It has been used to solve equations for the last 200 years or so. It is defined to be i such that ; i2 = - 1 Or in other words; βπ = i What is i ? i is an imaginary number unreal complex Or a complex number Or an unreal number The terms are inter-changeable imaginary Definition β’ A complex number is an expression of the form Z = a+ i b, where a and b are real numbers and i is a mathematical symbol for βπ which is called the imaginary unit. β’ For example, β3.5 + 2i is a complex number. β’ The real number a of the complex number z = a + bi is called the real part of z and the real number b is the imaginary part. 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 A number such as 3i is a purely imaginary number A number such as 6 is a purely real number 6 + 3i is a complex number Z=x + iy is the general form of a complex number If x + iy = 6 β 4i then x = 6 and y = β 4 The βreal partβ of 6 β 4i is 6 The imaginary part of 6 - 4i is - 4 NATURAL NUMBERS INTEGERS RATIONAL NUMBERS IRRATIONAL NUMBERS REAL NUMBERS COMPLEX NUMBERS β’ Real numbers and imaginary numbers are subsets of the set of complex numbers. Imaginary Numbers Rational Real Irrational Numbers Complex Numbers Complex numbers do not have order! Practice Time!!!! βπ 1. Simplify 2. Evaluate 3i x -4i Addition of Complex Numbers β’ Let z1=a+ib and 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 Practice Time!!!! Simplify β’ (2+3i ) + (4 -3i) β’ (-3+4i) + (-2- i10) Properties of addition The closure law β’ The sum of two complex numbers is complex number The commutative law β’ For any two complex numbers a and b, a+b=b+a The associative law β’ For any three complex numbers a,b,c; (a+b)+c=a+(b+c) Properties of addition The existence of additive Identity β’ There exists the complex number 0+i 0(denoted as 0), called the additive identity or the zero complex number. β’ (2+i10 ) + 0 = 2+i10 The existence of additive Inverse β’ To every complex number z=a+ib, we have the complex number βa+i(-b) (denoted as βz), called the additive inverse of z. We observe that z+(-z)=0(the additive identity). Difference of two complex numbers β’ Given any two complex numbers z1 and z2, the difference z1 - z2 is defined as follows : β’ z1 - z2 = z1 +(-z2) Simplify (3i+2i) β ( -2 + i3) Multiplication of two complex numbers β’ Multiplying complex numbers is similar to multiplying polynomials and combining like terms. β’ Let a+ib and c+id be any two complex numbers. Then the product of those two complex numbers is defined as follows: β’ (a+ib) (c+id) = (ac β bd) + i(ad + bc) PROPERTIES OF MULTIPLICATION The closure law β’ The product of two complex numbers is a complex number The commutative law β’ For any two complex numbers a and b, ab=ba The associative law β’ For any three complex numbers a,b,c; (a b) c = a (b c). The existence of multiplicative identity The distributive law β’ There exists a complex number 1+i0 (denoted as 1), called the multiplicative identity such that a.1 =a, for every complex number a β’ For any three complex numbers a,b,c; β’ (1) a(b+c) = a.b+a.c β’ (2) (a+b)c = a.c+b.c β’ Practice Time!!!! Simplify β’ (2+3i)(4-3i) β’ (-4+2i)(7-12i) Division of two complex numbers β’ Given any two complex numbers a and b , π where bβ 0, the quotient is defined by π a 1 ο½ aο· b b β’ Simplify π+ππ πβππ β’ The following identifies are true for complex numbers POWERS OF i iο½i i ο½ 2 ο¨ ο1 ο© 2 ο½ ο1 i 3 ο½ ο1ο΄ i ο½ οi i 4 ο½ ο1ο΄ ο1 ο½ 1 In general, for any integer k, i4k = 1, i4k+1 = i, i4k+2 = -1. The modules and the conjugate of a complex number β’ 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 ππ + ππ , i.e., IzI = ππ + ππ and the conjugate of z, is denoted as z , is the complex number z = a- ib. β’ USEFUL RESULTS Consider z ο½ a ο« bi and z ο½ a ο bi (Conjugate) z ο« z ο½ 2a z ο z ο½ 2bi z ο½ (a ο« bi )(a ο« bi ) 2 ο½ a 2 ο« 2bi ο b 2 z ο½ (a ο bi )(a ο bi ) 2 ο½ a 2 ο 2abi ο b 2 β’ USEFUL RESULTS Consider z ο½ a ο« bi and z ο½ a ο bi (Conjugate) zz ο½ (a ο« bi )(a ο bi ) ο½ a 2 ο« b2 ο½ z 2 z (a ο« bi ) (a ο« bi ) ο½ ο΄ z (a ο bi ) (a ο« bi ) ο½ a 2 ο« 2abi ο b 2 a 2 ο« b2 USEFUL RESULTS Consider z1 ο½ a ο« bi and z2 ο½ c ο« di 1. 2. ο¨ z1 ο« z2 ο© ο½ z1 ο« z2 z1 z2 ο½ ο¨ z1 z2 ο© β’ An Argand diagram is a plot of complex numbers as points on a complex plane. Argand plane Argand plane β’ The complex plane use the x-axis as the real axis and y-axis as the imaginary axis. β’ The dashed circle represents the complex modulus |Z| of z and the angle theta represents its complex argument. z = x + iy IZI Argand (1806) is credited with the discovery, of the Argand diagram (also known as the Argand plane) . Historically, the geometric representation of a complex number as a point in the plane was important because it made the whole idea of a complex number more acceptable. In particular, this visualization helped "imaginary" and "complex" numbers become accepted in mainstream mathematics as a natural extension to negative numbers along the real line. y 2 + 3i 3 2 1 x 1 2 3 We can represent complex numbers as a point. Polar representation of a Complex number β’ Let the point P represent the non zero complex number z = x + iy. Let the directed line segment OP be the length r and Σ¨ be the angle which OP makes with the positive direction of xaxis β’ We may note that the point P is uniquely determined by the ordered pair of real numbers (r, Σ¨), called the polar coordinates of the point P. β’ We consider the origin as the pole and the positive direction of the x-axis as the initial line. β’ We have , x = r cos Σ¨, y = r sin Σ¨ and therefore , z = r(cos Σ¨ + i sin Σ¨). The latter is said to be the polar form of the complex number. β’ Here is the modules of z and Σ¨ is called the argument of z which is denoted by arg z. β’ For any complex number z β 0, there corresponds only one value of Σ¨ in 0 β€ Σ¨ < 2π However any other interval of length 2 π for example - π < Σ¨ β€ π , can be such an interval. We shall take the value of Σ¨ such that - π < Σ¨ β€ π , called principle argument of z and is denoted by arg z, unless specified otherwise. Figures in the next slide.