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MB1105 QR MB1105 QR Presentation Transcript

  • In this document you can review important • Definitions • Facts • Formulae • Procedures Class – XI, CBSE MB1105: Complex Number and Quadratic Equations Learn Math for Free. Visit www.ganitgurooz.com
  • What is iota? Class – XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com
  • What is iota? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Complex Numbers 2 W e denote 1 by the sym bol . T herefore, w e have 1. W e call the sym bol as iota. i i i Learn Math for Free. Visit www.ganitgurooz.com View slide
  • What are Complex Numbers? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com View slide
  • What are Complex Numbers? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Complex Numbers Any num ber of the form is called a com plex num ber, if and both are real num bers. (a) is called as the real part of , denoted by R e( ) (b) is called as the im aginary part of , denote z a ib a b a z z b z d by Im ( ). T he set of com plex num bers is denoted by . If 0 and 0, the com plex num ber becom es 0 0 0, w hich is called the zero com plex num ber. z C a b i Learn Math for Free. Visit www.ganitgurooz.com
  • What is Equality of Complex Numbers? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com
  • What is Equality of Complex Numbers? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Complex Numbers T w o com plex num bers and , are said to be equal, if and . a ib c id a c b d Learn Math for Free. Visit www.ganitgurooz.com
  • What is addition of Complex Numbers? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Operations on Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com
  • What is addition of Complex Numbers? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Operations on Complex Numbers 1 2 1 2 T w o com plex num bers are added by adding their respective real and im aginary parts. T hu s, if and are tw o com plex num bers, then z , w hich is again a com plex num ber. W e can al z a ib z c id z a c i b d so visualize the addition of com plex num bers in the com plex plane as follow s: Learn Math for Free. Visit www.ganitgurooz.com
  • What is difference of two complex numbers? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Operations on Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com
  • What is difference of two complex numbers? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Operations on Complex Numbers 1 2 1 2 1 2 1 2 G iven any tw o com plex num bers and , the difference is defined as follow s: . z z z z z z z z Learn Math for Free. Visit www.ganitgurooz.com
  • What is multiplication of complex numbers? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Operations on Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com
  • What is multiplication of complex numbers? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Operations on Complex Numbers 1 2 1 2 2 2 1 2 1 12 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 W e can m ultiply tw o com plex num bers usin g the distributive property as follow s: . T hus, . O n the com plex plane, th z z a ib a ib a a a ib a ib ib ib a a ia b ib a b b z z a a b b i a b a b e product of tw o com plex num bers is tough to determ ine till w e learn about the polar representation of com plex num bers. But, for now , here is a visual that can help you understand this a bit. Learn Math for Free. Visit www.ganitgurooz.com
  • What is division of two complex numbers? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Operations on Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com
  • What is division of two complex numbers? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Operations on Complex Numbers 1 2 2 1 2 1 T o divide tw o com plex num bers, w e m ultiply the dividend by the m ultiplicative inverse of the divisor. T herefore, given any tw o com plex num bers and , w here 0, the quotient is defined as z z z z z z z 1 2 2 1 .z z Learn Math for Free. Visit www.ganitgurooz.com
  • What is conjugate of a complex number? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Conjugate of Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com
  • What is conjugate of a complex number? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Conjugate of Complex Numbers T he conjugate of a com plex num ber is defined by . O n the com plex planes these num bers are the reflection of each other w ith respect to the real (horizontal) ax is. z x iy z x iy Learn Math for Free. Visit www.ganitgurooz.com
  • What is modulus of a complex number? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Modulus of Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com
  • What is modulus of a complex number? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Modulus of Complex Numbers 2 2 M odulus of a com plex num ber is denoted by m od or and is defined as , w here R e and Im . W e call as the absolute value of . W e also note that 0. z a ib z z z a b a z b z z z z Learn Math for Free. Visit www.ganitgurooz.com
  • What is Complex Plane? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Complex Plane Learn Math for Free. Visit www.ganitgurooz.com
  • What is Complex Plane? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Complex Plane A com plex plane or Argand plane provides a visual representation of com plex num bers . It is a plane having a com plex num ber assig ned to each of its point. It consists of a h orizontal axis called the real axis and a vertical axis called the im aginary axis. T hus, a com plex plane is a m odified C artesian plane, w ith the real part of a com plex num ber represente d along the -axis, and the im aginary partx along the -axis. T hus, the com plex num ber corresponds to the point ( , ) in the com plex plane. y z x iy x y Learn Math for Free. Visit www.ganitgurooz.com If a com plex num ber is purely real, then im aginary part is zero. T herefore, a purely real num ber is represented by a p oint on -axis. A purely im aginary com plex num ber is represented by a point on -a x y xis. T hat is w hy -axis is know n as the real axis and -axis as the im aginary axis. x y
  • What is Complex Plane? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Complex Plane Learn Math for Free. Visit www.ganitgurooz.com In the Argand plane, the modulus of the complex number is the distance between the point to the origin as is shown in the figure. The points on the -axis corresponds to the complex numbers of the form 0 and the points on the -axis corresponds to the complex numbers of the form 0 . The representation of a complex number x a i y ib z and its conjugate in the Argand plane are, respectively, the points , and , . x iy z x iy P x y Q x y Geometrically, the point , is the mirror image of the point , on the real axis as is shown in the figure. x y x y
  • What is polar representation of complex number? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Complex Plane Learn Math for Free. Visit www.ganitgurooz.com
  • What is polar representation of complex number? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Polar Representation of Complex Number Learn Math for Free. Visit www.ganitgurooz.com The polar coordinate system consists of concentric circles centered at origin. A ny point ( , ) can be specified as a set of coordinates ( , ) where is the distance of the point from the origin and a b r r is the angle that the radius to the point makes with the positive direction of the horizontal axis. (See image) Any complex number can be represented in its polar form by writing cos and sin . The following image explains how we get the above relations. x iy x r y
  • What is polar representation of complex number? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Polar Representation of Complex Number Learn Math for Free. Visit www.ganitgurooz.com 2 2 W e have cos sin . T he latter is said to be the polar form of the com plex num ber . H ere is the m odulus of , cos , sin , and is called the argum ent (or am plitud e) of and i z x iy r i z x y r x y z z r r z s denoted by arg . For any non zero com plex num ber there is only one value of in 0 2 . H ow ever, any other interval o f length 2 can also be taken as such an interval, for exam ple . T he uniq z z ue value of such that for w hich cos and sin , is know n as the principle value of the argum ent of . T he general value of the argum ent is 2 , is an integer and is the principle v x r y r z n n alue of arg .z
  • What is polar representation of complex number? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Polar Representation of Complex Number Learn Math for Free. Visit www.ganitgurooz.com The following figures shows some of the possible arguments of a complex number such that 0 2 and . For 0 2 , we have the following: z For , we have the following:
  • What is Quadratic Equation? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Quadratic Equations Learn Math for Free. Visit www.ganitgurooz.com
  • What is Quadratic Equation? Class - XI MB1105: Complex Number and Quadratic Equations Important Definitions Topic: Quadratic Equations Learn Math for Free. Visit www.ganitgurooz.com 2 An equation of the form 0 where 0 and , , are real numbers, is called a quadratic equation. The number , , are called the coefficients of the quadratic equation. A root of the above quad ax bx c a a b c a b c 2 2 ratic equation is a number (real or com plex) such that 0. The roots of the above quadratic equation are given by 2 where the quantity 4 is known as the discriminant of the equat a b c b D x a D D b ac 1 2 ion. If 0, then quadratic equation has non real but complex roots, given by and . ( Since 0, thus, and 2 2 he D b i D b i D x x D a a 1 2 nce ). Clearly, , are complex conjugate of each other. D i D x x
  • Fact # 1 Every real number is a complex number so the system of complex numbers includes the system of real numbers. Class - XI MB1105: Complex Number and Quadratic Equations Important Facts Topic: Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com
  • Fact # 2 0 is both purely real and purely imaginary number. Class - XI MB1105: Complex Number and Quadratic Equations Important Facts Topic: Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com
  • Fact # 3 A complex number is an imaginary number if and only if its imaginary part is non zero. Here real part may or may not be zero 4 + 3i is an imaginary number, but not purely imaginary. Class - XI MB1105: Complex Number and Quadratic Equations Important Facts Topic: Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com
  • Fact # 4 All purely imaginary number except zero are imaginary numbers, but an imaginary number may or may not be purely imaginary. Class - XI MB1105: Complex Number and Quadratic Equations Important Facts Topic: Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com
  • Fact # 5 Complex numbers have 2 real dimensions, whereas the real numbers have only one dimension. Thus, while dealing with real numbers, we usually move on the real line, whereas while dealing with complex numbers, we move on a plane (a 2-D object). Class - XI MB1105: Complex Number and Quadratic Equations Important Facts Topic: Complex Plane Learn Math for Free. Visit www.ganitgurooz.com
  • Fact # 6 Class - XI MB1105: Complex Number and Quadratic Equations Important Facts Topic: Operations on Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com 1 2 The addition of complex numbers satisfy the following properties: (a) The sum of two complex numbers is a complex number i.e., is a complex num berz zThe closure law : 1 2 1 2 1 2 2 1 1 2 3 1 2 3 1 2 3 for all complex numbers and . (b) For any two complex numbers and , . (c) For any three complex numbers , , , . z z z z z z z z z z z z z z z z z The com m utative law : The associative law : (d) There exists the complex numbers 0 0 denoted as 0 , called the additive identity or the zero iThe existence of additive identity : complex number, such that, for every complex numbers , 0 . (e) For every complex number , z z z z a ibThe existence of additive inverse : we have the complex number (denoted as ), called the additive inverse or negativea i b z of , such that 0.z z z
  • Fact # 7 Class - XI MB1105: Complex Number and Quadratic Equations Important Facts Topic: Operations on Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com 1 2 (a) The product of two complex numbers is a complex number, the product is a complex numberz z The m ultiplication of com plex num bers satisfy the follow ing properties : The closure law : 1 2 1 2 1 2 2 1 1 2 3 1 for all complex numbers and . (b) For any two complex numbers and , (c) For any three complex numbers , , , z z z z z z z z z z z z z The com m utative law : The associative law : 2 3 1 2 3 . (d) There exists the complex number 1 0 (denoted as 1), called the m z z z z iThe existence of m ultiplicative identity : ultiplicative identity such that .1 , for every complex number . (e) For every non-zero complex number , we have the complex number z z z z a ibThe existence of m ultiplicative inverse : 1 2 2 2 2 1 denoted by or called the multiplicative inverse of a b z z a b a b z 1 2 3 1 2 3 1 2 1 3 1 2 3 1 3 2 3 1 such that . 1 (here, 1 is the multiplicative identity). (f) For any three complex numbers , , , (a) (b) . z z z z z z z z z z z z z z z z z z z The distributive law :
  • Fact # 8 While reducing a complex number to polar form, we always take the principle value of the argument. Class - XI MB1105: Complex Number and Quadratic Equations Important Facts Topic: Polar Representation of Complex Number Learn Math for Free. Visit www.ganitgurooz.com
  • Fact # 9 The fundamental theorem of algebra states that every non- constant single-variable polynomials with complex coefficients has at least one complex root. Class - XI MB1105: Complex Number and Quadratic Equations Important Facts Topic: Quadratic Equations Learn Math for Free. Visit www.ganitgurooz.com
  • Fact # 10 Every polynomial equation of degree n has n roots. Class - XI MB1105: Complex Number and Quadratic Equations Important Facts Topic: Quadratic Equations Learn Math for Free. Visit www.ganitgurooz.com
  • Fact # 11 Class - XI MB1105: Complex Number and Quadratic Equations Important Facts Topic: Quadratic Equations Learn Math for Free. Visit www.ganitgurooz.com 2 If in the quadratic equation , , , and is one root of the quadratic, then the other root must be the conjugate and vice-versa , , 0 . ax bx c a b c R p iq p iq p q R q
  • Class - XI MB1105: Complex Number and Quadratic Equations Important Formulae Topic: Complex Numbers 1. iIn te g ra l p o w e rs o f : 2 3 2 4 2 2 2 2 2 1 2 W e have 1, 1. T herefore, 1 , 1 1 1. W e note that for any 1, w hen is even (a) 1 . 1, w hen is odd , w hen is even (b) 1 , w hen i n nn nn n i i i i i i i i i i n N n i i n i n i i i i i n . s odd 1 Also, for any , the value of is found out by w riting this as and solving . T hus, any integral pow er of can be expressed in term s of 1 or . n n n n N i i i i i Learn Math for Free. Visit www.ganitgurooz.com
  • Class - XI MB1105: Complex Number and Quadratic Equations Important Formulae Topic: Operations on Complex Numbers 2. Identities for com plex num bers : 1 2 2 2 2 1 2 1 2 1 2 2 2 2 1 2 1 1 2 2 3 3 2 2 3 1 2 1 1 2 1 2 2 3 3 2 2 3 1 2 1 1 2 1 2 2 2 2 1 2 1 2 1 2 For all com plex num bers and w e get the follow ing: (a) 2 . (b) 2 . (c) 3 3 . (d) 3 3 . (e) . z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z Learn Math for Free. Visit www.ganitgurooz.com
  • Class - XI MB1105: Complex Number and Quadratic Equations Important Formulae Topic: Conjugate of Complex Numbers 3. P roperties of conjugate : 2 2 (a) ; (b) 2 R e ; (c) 2 Im ; (d) R e Im ; (e) if and only of is purely real; z z z z z z z i z zz z z z z z 1 2 1 2 1 2 1 2 1 1 2 2 2 (f) if and only if is purely im aginary; (g) ; (h) ; (i) , w here 0. z z z z z z z z z z z z z z z z Learn Math for Free. Visit www.ganitgurooz.com
  • Class - XI MB1105: Complex Number and Quadratic Equations Important Formulae Topic: Modulus of Complex Numbers 4. P roperties of m odulus of a com plex num ber : 1 2 1 2 1 2 1 1 2 2 2 For any tw o com plex num ber and , w e hav e (a) (b) provided 0. z z z z z z z z z z z Learn Math for Free. Visit www.ganitgurooz.com
  • Method to find the multiplicative inverse of a non zero complex number x + iy: Class – XI,CBSE Important Procedures Topic: Conjugate of Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com MB1105: Complex Number and Quadratic Equations
  • Method to find the multiplicative inverse of a non zero complex number x + iy: Class – XI,CBSE Important Procedures Topic: Conjugate of Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com 2 2 2 2 2 2 2 2 2 1 1 M ultiplicative inverse of . x iy x iy x iy x iy x iy x iy x iy x i y x y x y i x y x MB1105: Complex Number and Quadratic Equations
  • Method to write a complex number in the form A + iB: Class – XI,CBSE Important Procedures Topic: Conjugate of Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com MB1105: Complex Number and Quadratic Equations a ib c id
  • Method to write a complex number in the form A + iB: Class – XI,CBSE Important Procedures Topic: Conjugate of Complex Numbers Learn Math for Free. Visit www.ganitgurooz.com 2 2 2 2 2 2 2 2 2 2 W e have , where and . a ib c ida ib c id c id c id ac bd i bc ad ac bd bc ad i c d c d c d ac bd bc ad A iB A B c d c d MB1105: Complex Number and Quadratic Equations a ib c id