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1. 1. Complex NumbersComplex Numbers Complex Plane, Polar Coordinates And Euler Formula T- 1-855-694-8886 Email- info@iTutor.com By iTutor.com
2. 2. DefinitionDefinition  A complex number z is a number of the form where  x is the real part and y the imaginary part, written as x = Re z, y = Im z.  i is called the imaginary unit  If x = 0, then z = iy is a pure imaginary number.  The complex conjugate of a complex number, z = x + iy, denoted by z* , is given by z* = x – iy.  Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. 1−=i iyx + © iTutor. 2000-2013. All Rights Reserved
3. 3. Complex PlaneComplex Plane A complex number can be plotted on a plane with two perpendicular coordinate axes The horizontal x-axis, called the real axis The vertical y-axis, called the imaginary axis Represent z = x + iy geometrically as the point P(x,y) in the x-y plane, or as the vector from the origin to P(x,y).OP uuuv The complex plane x-y plane is also known as the complex plane. © iTutor. 2000-2013. All Rights Reserved
4. 4. Polar CoordinatesPolar Coordinates With siny r θ=cos ,x r θ= z takes the polar form: r is called the absolute value or modulus or magnitude of z and is denoted by |z|. *22 zzyxrz =+== 22 * ))(( yx iyxiyxzz += −+=Note that : )sin(cos θθ irz += © iTutor. 2000-2013. All Rights Reserved
5. 5. Complex plane, polar form of a complex number       = − x y1 tanθ Geometrically, |z| is the distance of the point z from the origin while θ is the directed angle from the positive x- axis to OP in the above figure. From the figure, © iTutor. 2000-2013. All Rights Reserved
6. 6. θ is called the argument of z and is denoted by arg z. Thus, For z = 0, θ is undefined. A complex number z ≠ 0 has infinitely many possible arguments, each one differing from the rest by some multiple of 2π. In fact, arg z is actually The value of θ that lies in the interval (-π, π] is called the principle argument of z (≠ 0) and is denoted by Arg z. 0tanarg 1 ≠      == − z x y zθ ,...2,1,0,2tan 1 =±      = − nn x y πθ © iTutor. 2000-2013. All Rights Reserved
7. 7. Euler Formula –Euler Formula – an alternate polar forman alternate polar form The polar form of a complex number can be rewritten as : This leads to the complex exponential function : θ θθ i re iyxirz = +=+= )sin(cos ( ) ( )θθ θθ θ θ ij ij ee j ee − − −= += 2 1 sin 2 1 cos Further leads to : ( )yiye eeee x iyxiyxz sincos += == + © iTutor. 2000-2013. All Rights Reserved
8. 8. In mathematics terms, q is referred to as the argument of z and it can be positive or negative. In engineering terms, q is generally referred to as phase of z and it can be positive or negative. It is denoted as z∠ The magnitude of z is the same both in Mathematics and engineering, although in engineering, there are also different interpretations depending on what physical system one is referring to. Magnitudes are always > 0. The application of complex numbers in engineering will be dealt with later. © iTutor. 2000-2013. All Rights Reserved
9. 9. x +θ1 x z1 z2 Im Re-θ2 r1 r2 1 11 θi erz = 2 22 θi erz − = 0,,, 2121 >θθrr © iTutor. 2000-2013. All Rights Reserved
10. 10. 4 2 4 sin 4 cos2 π ππ i eiz =      += 2)11(|| 22 =+=z rad2 4 2 1 1 tan 1       +=+      =∠ − π π π nnz A complex number, z = 1 + j , has a magnitude Example 1Example 1 and argument : Hence its principal argument is : Arg / 4z π= rad Hence in polar form : © iTutor. 2000-2013. All Rights Reserved
11. 11. A complex number, z = 1 - i , has a magnitude 2)11(|| 22 =+=z Example 2Example 2 rad2 4 2 1 1 tan 1       +−=+      − =∠ − π π π nnzand argument : Hence its principal argument is : rad Hence in polar form : 4 π −=zArg       −== − 4 sin 4 cos22 4 πππ iez i © iTutor. 2000-2013. All Rights Reserved
12. 12. What about z1=0+i, z2= 0-i, z3=2+i0, z4=-2 ? Other ExamplesOther Examples π π 5.01 1 10 5.0 1 ∠= = += i e iz π π 5.01 1 10 5.0 2 −∠= = −= −i e iz 02 2 02 0 3 ∠= = += i e iz π π −∠= = +−= − 2 2 024 i e iz © iTutor. 2000-2013. All Rights Reserved
13. 13. ● ● ● Im Re z1 = + i z2 = - i z3 = 2z4 = -2 ● π5.0 © iTutor. 2000-2013. All Rights Reserved
14. 14. Arithmetic Operations in Polar FormArithmetic Operations in Polar Form  The representation of z by its real and imaginary parts is useful for addition and subtraction.  For multiplication and division, representation by the polar form has apparent geometric meaning. © iTutor. 2000-2013. All Rights Reserved
15. 15. Suppose we have 2 complex numbers, z1 and z2 given by : 2 1 2222 1111 θ θ i i eriyxz eriyxz − =−= =+= ( ) ( ) ( ) ( )2121 221121 yyixx iyxiyxzz −++= −++=+ ( )( ) ))(( 21 2121 21 21 θθ θθ −+ − = = i ii err ererzz Easier with normal form than polar form Easier with polar form than normal form magnitudes multiply! phases add! © iTutor. 2000-2013. All Rights Reserved
16. 16. For a complex number z2 ≠ 0, )( 2 1))(( 2 1 2 1 2 1 2121 2 1 θθθθ θ θ +−− === ii i i e r r e r r er er z z 2 1 2 1 r r z z = magnitudes divide! phases subtract! 2121 )( θθθθ +=−−=∠z © iTutor. 2000-2013. All Rights Reserved
17. 17. The End Call us for more Information: www.iTutor.com 1-855-694-8886 Visit