MEM 23004 A APPLY TECHNICAL MATHEMATICS COMPLEX NUMBERS

MEM 23004 A APPLY TECHNICAL MATHEMATICS COMPLEX NUMBERS– Part 1

Complex numbers • Presentation developed by Les Smith • Mechanical Engineering • Granville College of TAFE

Complex Numbers Imaginary numbers

Complex Numbers Imaginary numbers In some text books, the imaginary number has be given the symbol “j ” instead of “i ” This is because i has been used as a symbol for some other quantity (such as current flow) In these notes, I will use i as the symbol for the imaginary number.

Complex Numbers Imaginary numbers Used in mechanical engineering in numerous areas, such as: § Vibration analysis; § Vector analysis; § Control engineering; § Fluid mechanics; § etc

Complex Numbers Imaginary numbers Consider the following equation. x 2 + 9 = 0

Complex Numbers Imaginary numbers But in the imaginary number system. x 2 + 9 = 0

Complex Numbers A complex number is a number that is made up of two parts; • A real part and; • An imaginary part. For example 4 + 3 i The real part The imaginary part

Complex Numbers

Complex Numbers Complex numbers The real part The imaginary part

Complex Numbers Adding complex numbers If Z 1 = a + b i and Then Z 1 + Z 2 = (a + bi ) + (c + di ) = (a + c) + (b + d)i Z 2 = c + d i

Complex Numbers Example of adding complex numbers If Z 1 = 4 + 5 i and Z 2 = 7 – 3 i Then Z 1 + Z 2 = (4 + 5 i ) + (7 + -3 i ) = (4 + 7) + (5 + -3)i = 11 + 4 i

Complex Numbers Subtracting complex numbers If Z 1 = a + b i and Then Z 1 – Z 2 = (a + bi ) – (c + di ) = (a – c) + (b – d)i Z 2 = c + d i

Complex Numbers Example of subtracting complex numbers If Z 1 = 4 + 5 i and Z 2 = 7 – 3 i Then Z 1 – Z 2 = (4 + 5 i ) – (7 + -3 i ) = (4 – 7) + (5 – -3)i = – 3 + 8 i

Complex Numbers Multiplying complex numbers If Z 1 = a + b i and Then Z 1 x Z 2 = (a + bi )(c + di ) Similar to algebraic expansion Z 2 = c + d i = ac + adi + bci + bdi 2 = ac + adi + bci + bd(-1) = (ac-bd) + (ad+bc)i

Complex Numbers Example of multiplying complex numbers If Z 1 = 4 + 5 i and Then Z 1 x Z 2 = (4 + 5 i )(7 + -3 i ) Z 2 = 7 – 3 i = 28 -12 i + 35 i + (5)(-3)(i 2) = (28 + 15) + (-12 + 35)i = 43 + 23 i

Complex Numbers Dividing complex numbers - 1 This is a bit more complicated than the other operations We use the conjugate of the denominator (bottom of a fraction) in the calculation The conjugate is the complex number with the sign in the middle changed. The conjugate is written with a bar over the top of the complex number.

Complex Numbers Note the plus sign Note the bar which denotes that this is the conjugate of Z Note the negative sign in the conjugate

Complex Numbers

$Complex Numbers We will multiply both the top and bottom of this fraction by$

Complex Numbers We will multiply both the top and bottom of this fraction by the conjugate of Z 2.

Complex Numbers This is the equivalent of multiplying by 1

Complex Numbers

Complex Numbers Exercises Given that Z 1=4 -5 i; Z 2=9+15 i; Z 3=-13 -12 i; Z 4=-7+9 i

Complex Numbers The Argand diagram • Used to represent a complex number pictorially. § Based on a Cartesian coordinate system but the x-axis becomes the real number axis and the y-axis is the imaginary number axis.

Complex Numbers The Argand diagram Imaginary numbers 5 4 3 2 -4 -3 -2 -11 -1 -2 -3 -4 1 2 3 4 5 Real numbers

Complex Numbers Plotting a point on the Argand diagram Imaginary numbers Example: A = 3 -2 i 5 4 3 2 -4 -3 -2 -11 The imaginary number value of the complex number -1 -2 -3 -4 The real number value of the complex number 1 2 3 4 5 A Real numbers

Complex Numbers Plotting a point on the Argand diagram Imaginary numbers Example: A = 3 -2 i B = -2+4 i 5 B 4 3 2 -4 -3 -2 -11 -1 -2 -3 -4 1 2 3 4 5 A Real numbers

-1 -2 -3 -4 θ 1 2 3 4 5 |A 2 -4 -3 -2 -11 |o r“ r” Complex Numbers Writing complex numbers using polar Imaginary numbers coordinates Example: 5 A 4 A = 3+4 i 3 Real numbers

Complex Numbers

Complex Numbers Example Representing 2 complex numbers on an Argand diagram Z 1 =Imaginary -2 + 3 numbers i and Z 2 = 3 – 3 i 4 3 2 -4 -3 -2 1 1 2 3 4 5 -1 -2 -3 -4 Real numbers

Complex Numbers

Determinants Link to the complex number review/video: • http: //betterexplained. com/articles/a-visual-intuitive-guide-to-imaginary -numbers/ • http: //betterexplained. com/? s=complex+numbers • https: //www. youtube. com/watch? v=ys. Vc. AYo 7 UPI • https: //www. youtube. com/watch? v=kpywdu 1 afas • https: //www. youtube. com/watch? v=b. Pq. B 9 a 1 uk_8 • https: //www. youtube. com/watch? v=Fwu. PXch. H 2 r. A • https: //www. youtube. com/watch? v=N 0 Y 8 ia 57 C 24