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Complex Numbers:

Complex Numbers Definition of i Addition, subtraction and equality Multiplication Complex conjugate and division Argand diagram Solving quadratic equations Modulus and argument form De Moivre’s theorem. Instructor: Dr Pradeep Malakar

Complex Numbers:

Complex Numbers x 2 must be greater than 0 for every real number, x , so the equation x 2 = -1 has no real solutions. To deal with this problem, Mathematicians of the eighteenth century introduced the “imaginary number Which they assumed had the property but otherwise could be treated as a real number.

Complex Numbers:

Complex Numbers A complex number is an ordered pair of real numbers, denoted either by ( a,b ) or a+bi, or sometimes a+bj . If b = 0 then the number is real. NB: Some text books write ‘j’ instead of ‘i’

Slide 5:

6. -16 7. -5 = (-1)(16) = (-1)(16) = 4i = (-1)(5) = (-1)(5) = i 5

Slide 6:

Cycle of i

Slide 7:

Divide the exponent by 4 and look at the remainder.

Slide 8:

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

Slide 9:

Complex Plane Imaginary Axis Real Axis (0,0) (a,b) (0,b) a+bi (a,0) Geometrically, a complex number can be viewed either as a point or a vector in an “xy” plane.

Complex Numbers:

Complex Numbers We define a complex number with the form z = a + ib where a, b are real numbers. The complex number z has a real part , a, written Re{z}. The imaginary part of z, written Im{z}, is b. Notice that the imaginary part is a real number. So we may write z as z = Re{z} + iIm{z}

Equating Complex Numbers:

Equating Complex Numbers Two complex numbers z 1 = x 1 + i y 1 z 2 = x 2 + i y 2 are equal if and only if their real parts are equal and their imaginary parts are equal. That is, z 1 = z 2 if and only if Re{z 1 } = Re{z 2 } and Im{z 1 } = Im{z 2 }

Complex Arithmetic:

Complex Arithmetic In order to add two complex numbers, separately add the real parts and imaginary parts. ( x 1 + i y 1 ) + ( x 2 + i y 2 ) = ( x 1 + x 2 ) + i ( y 1 + y 2 ) The product of two complex numbers works as expected if you remember that i 2 = -1. (1 + 2i)(2 + 3i) = 2 + 3i + 4i + 6i 2 = 2 + 7i – 6 = -4 + 7i In general, (x 1 + iy 1 )(x 2 + iy 2 ) = (x 1 x 2 - y 1 y 2 ) + i(x 1 y 2 + x 2 y 1 )

Complex Arithmetic:

Complex Arithmetic Arithmetic with imaginary works as expected: i + i = 2i 3i – 4i = -i 5 (3i) = 15 i To take the product of two imaginary numbers, remember that i 2 = -1: i • i = -1 i 3 = i • i 2 = -i i 4 = 1 2i • 7i = -14 Dividing two imaginary numbers produces a real number: 6i / 2i = 3

Slide 14:

( 3 + 2 i )( 3 – 2 i ) 13

Multiplication by a scalar::

Both components of the complex number are multiplied by the scalar k(a+bi)=(ka)+(kb)i (if k is real) k(a+bi)=(-kb)+(ka)i (if k is purely imaginary) Multiplication by a scalar: 4(2 + 5i) = 4(2) + 4(5i) = 8 + 20i 2i(5 + 7i) = 2i(5) + 2i(7i) = 10i + 14i 2 = 10i + 14(- 1) = - 14 + 10i

Similarities between complex number arithmetic and vector arithmetic:

Similarities between complex number arithmetic and vector arithmetic Sum of 2 complex numbers Difference of 2 complex numbers z 1 z 2 z 1 + z 2 z 1 z 2 z 1 - z 2

Similarities between complex number arithmetic and vector arithmetics:

Similarities between complex number arithmetic and vector arithmetics K > 0 K < 0

Slide 18:

A complex number and its conjugate. It is interesting to note that if and only if z is a real number . If z=a+bi is any complex number, then the conjugate of z, denoted by is defined by In words, is obtained by reversing the sign of the imaginary part of z . Geometrically, z is the reflection of z about the real axis Complex Conjugate

Slide 19:

If a complex number is viewed as a vector in 2-D space, then the norm or length of the vector is called the modulus (or absolute value) of z . The modulus of a complex number z= a+bi , denoted by | z |, is defined by Note that if b=0 , then z=a is a real number, and so the modulus of a real number is simply its absolute value. It is for this reason, that the modulus of z is called the absolute value of z .

Complex Conjugate:

Complex Conjugate The complex conjugate of x + iy is defined to be x – iy. To take the conjugate, replace each i with –i. Some useful properties of the conjugate are: z + z* = 2 Re{z} z - z * = 2i Im{z} zz * = Re{z} 2 + Im{z} 2 Notice that zz* is a positive real number. Its positive square root is called the modulus or magnitude of z, and is written |z|.

Prove that:

Prove that

Dividing Complex Numbers:

Dividing Complex Numbers The way to divide two complex numbers is not as obvious. But, there is a procedure to follow : 1. Multiply both numerator and denominator by the complex conjugate of the denominator. The denominator is now real; divide the real part and imaginary part of the numerator by the denominator.

Dividing Complex Numbers:

Multiply the numerator and denominator by the conjugate of the denominator. Dividing Complex Numbers

Cartestian Coordinates:

Cartestian Coordinates The representation of a complex number as a sum of a real and imaginary number z = x + iy is called its Cartesian form . The Cartesian form is also referred to as rectangular form . The name “Cartesian” suggests that we can represent a complex number by a point in the real plane, Reals 2 . We often do this, with the real part x representing the horizontal position, and the imaginary part y representing the vertical position. The set Complex is even referred to as the “complex plane”.

Complex Plane:

Complex Plane

Slide 26:

If z= a+ib is a nonzero complex number, r = z and θ indicates the angle from the positive real axis to the vector z , then The projection of the vector on the X axis is the projection of the vector on the Y axis is Such that z= a+ib can be written as or r is the amplitude (modulus) of the complex number and θ is the angle between the vector and the "x" axis, ( arg ( z ) or phase angle) Note that the angle , θ , can be determined using Polar Form

Slide 27:

However, care must be taken in the calculation of θ as it will depend on the quadrant location of the complex number

Polar Form:

Polar Form In addition to the Cartesian form, a complex number z may also be represented in polar form : z = r e i θ = r(cos θ +isin θ ) Here, r is a real number representing the magnitude of z, and θ represents the angle of z in the complex plane. Multiplication and division of complex numbers is easier in polar form: Addition and subtraction of complex numbers is easier in Cartesian form.

Slide 29:

Multiplication in polar coordinates Division in polar coordinates

Slide 30:

Express in the form re i θ , each of the following complex numbers Converting Between Forms

Different forms for a complex number:

Different forms for a complex number A complex number can be expressed in different forms: Cartesian form : z = a + ib Modulus-argument form : r is the modulus of the complex number θ is the argument of the complex number, radians. Sometimes, it is convenient to use the following short-hand notation for a complex number in modulus-argument form: z = [ r , θ]. Exponential form : re i θ r is the modulus of the complex number θ is the argument of the complex number, radians.

Complex Exponentials:

Complex Exponentials The exponential of a real number x is defined by a series: sine and cosine have similar expansions: We can use these expansions to define these functions for complex numbers.

Complex Exponentials:

Complex Exponentials Put an imaginary number iy into the exponential series formula: Look at the real and imaginary parts of e iy : This is cos(y)… This is sin(y)…

Euler’s Formula:

Euler’s Formula This gives us the famous identity known as Euler’s formula: From this, we get two more formulas: Exponential functions are often easier to work with than sinusoids, so these formulas can be useful. The following property of exponentials is still valid for complex z: Using the formulas on this page, we can prove many common trigonometric identities next week.

Euler’s Formula:

Euler’s Formula any complex numbers can be written in any of the forms… The form re i θ is known as the exponential form of a complex number Can you use the above to explain why e i π = -1

De Moivre’s Theorem:

De Moivre’s Theorem

Applications of De Moivre’s Theorem:

Applications of De Moivre’s Theorem Find

Applications of De Moivre’s Theorem:

Applications of De Moivre’s Theorem Find As it is in the wrong form, we begin by writing it in modulus-argument form

Quadratic equation:

Quadratic equation

Solve x3 + 2x +3 = 0:

Solve x 3 + 2 x +3 = 0

Solve 9x2 – 6x + 37 = 0 :

Solve 9 x 2 – 6 x + 37 = 0

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