Complex Numbers And Appsfeb

Complex Numbers and applications N Patel

A complex number can be defined as a ordered pair of real numbers, denoted either by (a,b) or a+bi where i 2 = (√-1) 2 = -1 y a + bi As a vector Real axis Imaginary axis Let Z be the complex number Z = a +bi The real number ‘a’ is called the real part of Z ‘ b’ is called the imaginary part of Z The Agrand diagram x

a + bi -a + bi a - bi -a - bi As a vector

Real Numbers Magnitude Only Sign indicates Positive or Negative value only. One Dimensional Number (magnitude) Horizontal or X-axis based.

Imaginary Numbers Y axis representation Right Angle to X-axis j or i notation

Vectors and complex numbers Real Numbers and Angles X Axis and Y Axis Two and Three Dimensional Represented in Complex Number format Real and Imaginary Number 3 + 2i ( Mathematical form) Due to engineering symbols it is standard to replace I with j hence 3+2j ( Engineering form)

Example of Complex Numbers 3 is Real Number on x axis 2 is seen on imaginary axis. Imaginary Numbers are placed perpendicular to X axis. Therefore Z = 3 +2j Using Pythagoras Theorem a 2 =3 2 + 2 2 a = 3.605 angle = tan -1 ( 2/3) 33.69 degrees.

Resultant Forces SUMMARY Any Force at an Angle θ has Vertical and Horizontal parts. Horizontal = Adjacent Vertical = Opposite. HYP = Resultant Apply Pythagorean, Sine or Cosine rules as required. θ F

Remember this Example 4x 2 +3x + 2 = 0 ax 2 +bx +c =0 This is a complex Root and cannot go any further as the square root of a negative number cannot be defined X = -0.375 ± 0.599j

Adding two vectors Z 1 + Z 2 Rule Z 1 + Z 2 = (a+bj) + (c+dj) = (a+c) + (b+d)j Add real parts (x) and add j part separately. Find Z 1 + Z 2 When Z 1 = 3 + 2j added to another vector Z 2 =1 +2j = 4 +4j

Adding two vectors Z 1 + Z 2 Rule Z 1 + Z 2 = (a+bj) + (c+dj) = (a+c) + (b+d)j Add real parts (x) and add j part separately. Find Z 1 + Z 2 When Z 1 = 4 -5jadded to another vector Z 2 =-1 +6j = 3 +j

subtracting two vectors Z 1 - Z 2 Rule Z 1 - Z 2 = (a+bj) - (c+dj) = (a-c) + (b-d)j subtract real parts (x) and subtract j part separately. Find Z 1 - Z 2 When Z 1 = 1 - 2j subtracted from vector Z 2 =4 +5j

Rule Z 1 - Z 2 = (a+bj) + (c+dj) = (a-c) + (b-d)j Find Z 1 - Z 2 When Z 1 = 4 -5j subtract from vector Z 2 =-1 +6j = 5-11j

Student Task Try following examples: 3 +5j + 2-3j 2-6j + 4-j 5+4j – (3+ 3j) 2+ j –(5-7j) 5+2j 6-7j 2+j -3+8j

J notation Multiplication (a + i b)*(c + i d) = ? To multiply just expand out the terms and group as follows: Rule: (a + i b)*(c + i d) = (a*c - b*d) + i (a*d + b*c)

J notation Multiplication Treat the real and imaginary parts separate when multiplying. Example: (3+4j) (2+6j) = (a + i b)*(c + i d) = ? Rule: (a + i b)*(c + i d) = (a*c - b*d) + i (a*d + b*c) -18 + 26j

J notation Multiplication Treat the real and imaginary parts separate when multiplying. Example: (3+2j) (4+5j) = (a + i b)*(c + i d) = ? Rule: (a + i b)*(c + i d) = (a*c - b*d) + i (a*d + b*c) 5 -14j

Try the following (2+5j)(4-2j) (-3-5j)(2-4j) Answers 18+16j -26+2j

Conjugate Complex Numbers The conjugate of a + i b is a - i b Example: A = 4 +3j Zbar = 4-3j = conjugate Therefore these numbers differ by the sign of the imaginary number only. Useful Tip to remember: (a + i b)*conj(a + i b) = a*a + b*b

Division of Complex Numbers Rule = Z 1 x Z(bar)⁄ Z 2 Z(bar) when Z 1 =(3+4j) and Z 2 =(1-2j) -1 +2j

Try the following 5+3j/2+j Answer: 2.6 +0.2j

Example of Complex Numbers converting to Polar and Rectangular Form 3 is Real Number on x axis 2 is seen on imaginary axis. Imaginary Numbers are placed perpendicular to X axis. Therefore Z = 3 +2j Using Pythagoras Theorem a 2 =3 2 + 2 2 a = 3.605 angle = tan -1 ( 2/3) 33.69 degrees. 3+2j = 3.605∟33.69°

Polar and Rectangular Form A vector can be represented in polar form by use of Pythagoras Theorem and trig: (a +bj) where r = hypotenuse. 5+2j = 5.39∟21.80° a= r cos α 5.39 cos 21.8 = check = 5 b= r sin α 5.39 sin 21.8 = check = 2 If Z=7.62∟336.8°, in rectangular form = a= r cos α 7.62 x 0.92 =7 b= r sin α 7.62 x -.39 =-3 Z= 7 -3j

Dividing complex numbers in polar form Divide the magnitudes and subtract the angles 6∟75°/3∟25° = 2∟50°

Multiplying polar numbers Multiply the magnitudes and add angles 2∟50° x 7∟30° = 14∟80°

Complex Numbers And Appsfeb

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to Complex Numbers And Appsfeb (16)

Complex Numbers And Appsfeb