Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Complex Numbers Summary Academic Skills Advice What does a complex number mean? A complex number has a βrealβ part and an βimaginaryβ part (the imaginary part involves the square root of a negative number). e.g. π = π + ππ We use Z to denote a complex number: Real Example: Imaginary You might see the π before or after itβs number - it doesnβt matter which. Z = 4 + 3i Re(Z) = 4 Im(Z) = 3 Sometimes (especially in engineering) a j is used instead of π β they mean the same thing. Powers of i π stands for ββ1 so: 2 π 2 = (ββ1) = -1 π 4 = (π 2)2 = (-1)2 = 1 Summary: π = ββπ ππ = βπ ππ = π For any power of π substitute as many π 2βs as possible and they will all end up as ±π or ±1. Examples: π 20 = (π 2 )10 = (β1)10 = 1 π 33 = (π 2 )16 π = (β1)16 π = π Adding & Subtracting This is easy β just add or subtract the real part and add or subtract the imaginary parts: Examples: (4 + 3π ) + (2 + 6π ) = (6 + 9π ) (3 + 7π ) β (1 β 3π ) = (2 + 10π ) Multiplying Multiply out the 2 brackets (and remember that π 2 = β1). Example: (3 + 5π )(4 β 2π ) 2 = 12 β 6π + 20π β 10π = 12 + 14π β 10 (β1) = 22 + 14π H Jackson 2010 / 2015 / 2016 / Academic Skills 1 Except where otherwise noted, this work is licensed under http://creativecommons.org/licenses/by-nc-sa/3.0/ Complex Conjugate The conjugate is exactly the same as the complex number but with the opposite sign in the middle. When multiplied together they always produce a real number because the middle terms disappear (like the difference of 2 squares with quadratics). Example: 2 (4 + 6π )(4 β 6π ) = 16β 24π + 24πβ 36π = 16β 36(β1) = 16 + 36 = 52 Complex conjugate Complex number Dividing Dividing by a real number: divide the real part and divide the imaginary part. Dividing by a complex number: Multiply top and bottom of the fraction by the complex conjugate of the denominator so that it becomes real, then continue as above. Examples: 3+4π 2 4β5π 3+2π 3 = 2 4 + π = 1.5 + 2π 2 4β5π = 3+2π × 3β2π 3β2π = 12β8πβ15π+10π 2 9β6π+6πβ4π 2 = 12β23π+10(β1) 9β4(β1) = 2β23π 13 = π ππ β ππ ππ π Graphical Representation A complex number can be represented on an Argand diagram by plotting the real part on the π₯-axis and the imaginary part on the y-axis. * When finding π either: Imaginary P (π§ = π₯ + π¦π) y NB tan(ΞΈ) = π¦ so π₯ π π½ = πππβπ ( ) π 0 π π₯ ο· ο· do a sketch to see where the angle is or remember: if π < π (i.e. negative) add 180 (ππ π ) to your answer. Real Modulus: is written as |π| or π and is the length of OP, therefore |π| = π = βππ + ππ Argument: is the angle ΞΈ that is made with the horizontal axis (denoted by β ). Example: P (π§ = 4 + 5π) 5 |π| = βππ + ππ = βππ = π. π π π½ = πππβπ ( ) = π. π radians π 0 π 4 H Jackson 2010 / 2015 / 2016 / Academic Skills Real 2 Polar & Exponential Form As well as the cartesian form (π§ = π₯ + ππ¦) there are 2 alternative ways of writing a complex number: π = π(ππππ½ + πππππ½) Polar: Where: π is the modulus (length of the line) |π§| π is the argument (angle it makes with the π₯-axis) Exponential: π = ππππ½ (this should be in radians for the exponential form). Remember: To find the modulus (length), π: use Pythagoras π To find the argument (angle), π½: use πππβπ (π) * Converting between the different forms: Cartesian Need to find π and π½ π = βπ₯ 2 + π¦ 2 Polar or Exponential π¦ π = π‘ππβ1 ( ) π₯ Polar or Exponential Example: 0 Need to find π and π π₯ = ππππ π π¦ = ππ πππ Cartesian Express π = π + ππ in polar and exponential form imaginary π * 4 π 3 Modulus: π = β32 + 42 = β16 + 9 = β25 = 5 Argument: π = π‘ππβ1 (3) = 53.1o Polar form: π§ = 5(cos(53.1) + ππ ππ(53.1)) Exp form: π§ = 5π 0.927π 4 (in radians π = 0.927π ) real NB It is advisable to do a quick sketch of the complex number and check that the angle you calculated matches the diagram. If itβs in a different quadrant adjust the angle as necessary. Remember that angles are positive when measured anticlockwise ( ) Example: Express π π = ππππ in cartesian form π π₯ = ππππ π β΄ π₯ = 7 cos (3 ) = 3.5 π¦ = ππ πππ β΄ π¦ = 7 sin ( 3 ) = 6.1 Cartesian form: π π§ = 3.5 + 6.1π H Jackson 2010 / 2015 / 2016 / Academic Skills 3 A reminder of the 3 forms: Cartesian π = π + ππ Polar π = π(ππππ½ + πππππ½) π = βπ₯ 2 + π¦ 2 π₯ = ππππ π π¦ = ππ πππ Conversions: Exponential π = ππππ½ π¦ π = π‘ππβ1 ( ) * π₯ Multiplying with Polar or Exponential form Let This means, when multiplying 2 complex numbers: π§1 = π§2 π§3 Multiply the πβ²π Add the angles (πβ²π ) Then |π§1 | = |π§2 | × |π§3 | And β π§1 = β π§2 + β π§3 π π Example: If ππ = πππ π and ππ = πππ π find ππ ππ New modulus: New angle: 5 × 3 = 15 π π 5π + = 2 3 6 5π β΄ π§1 π§2 = 15π 6 π Dividing with Polar or Exponential form Let π§1 = Then |π§1 | = And π§2 This means, when dividing 2 complex numbers: π§3 Divide the πβ²π Subtract the angles (πβ²π ) |π§2 | |π§3 | β π§1 = β π§2 β β π§3 π π π π π Example: if ππ = π (πππ (π ) + ππππ ( π )) and ππ = π (πππ ( π ) + ππππ ( π )) find ππ New modulus: 5 ÷ 3 = New angle: π π 2 β΄ π β3 = π§1 π§2 = H Jackson 2010 / 2015 / 2016 / Academic Skills 5 3 5 3 π 6 π π (πππ (6 ) + ππ ππ (6 )) 4 De Moivres Theorem: Think: Raise π to the power of π and multiply the angle by π. Is used for raising a complex number to a power. ππ = ππ (πππ(ππ½) + ππππ(ππ½)) e.g π π If π = π (πππ ( π ) + ππππ ( π )) then π§ 5 = 35 (πππ 5π 3 + ππ ππ 5π 3 ) If the power is a fraction (root) there are π solutions: Remember that roots can be written as fractions: Just has a 2ππ extra π π π π π = π (πππ ( π½+πππ π ) + ππππ ( π½+πππ π )) 1 π βπ§ = π§ π Where k = 0,1,2β¦β¦.n-1 If you struggle to remember the theorem for when the power is a fraction β just use the original (the 1st equation) exactly the same but remember that there will be π solutions, all with the same modulus but with different arguments. In this case, to find the arguments you need to keep adding e.g. ππ π to your previous answer. Find the fifth roots of π = π(πππ(π ) + ππππ(π )) Raise π to the power of π and multiply the angle by π. Using the original equation: 1st answer: 1 1 π π π§ 5 = 25 (πππ (5 ) + ππ ππ ( 5 )) (Remember there are 5 answers β so keep adding 2π 5 to find the other 4) 1 1 3π 3π 1 1 5π 5π 1 1 7π 7π 1 1 9π 9π 2nd answer: π§ 5 = 25 (πππ ( 5 ) + ππ ππ ( 5 )) 3rd answer: π§ 5 = 25 (πππ ( 5 ) + ππ ππ ( 5 )) 4th answer: π§ 5 = 25 (πππ ( 5 ) + ππ ππ ( 5 )) 5th answer: π§ 5 = 25 (πππ ( 5 ) + ππ ππ ( 5 )) H Jackson 2010 / 2015 / 2016 / Academic Skills 5