EngMaths - Week 1 - Complex Numbers

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Last updated 12:36 PM on 7/19/26
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20 Terms

1
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Describe what is meant by a complex number

  • z = a + bi

    • a and b are real

    • i is imaginary, i² = -1

  • Re(z) = a

  • Im(z) = b

<ul><li><p>z = a + bi</p><ul><li><p>a and b are real</p></li><li><p>i is imaginary, i² = -1</p></li></ul></li><li><p>Re(z) = a</p></li><li><p>Im(z) = b</p></li></ul><p></p>
2
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Describe how an imaginary number is described graphically

  • Im is the y-axis

  • Re is the x-axis

  • z is the modulus

    • |z| = sqrt (a² + b²)

<ul><li><p>Im is the y-axis</p></li><li><p>Re is the x-axis</p></li><li><p>z is the modulus</p><ul><li><p>|z| = sqrt (a² + b²) </p></li></ul></li></ul><p></p>
3
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Describe the addition and subtraction of complex numbers z1 (a + bi) and z2 (c + di)

  • z1 + z2 = (a + c ) + (b + d)i

  • z1 - z2 = (a - c ) + (b - d)i

<ul><li><p>z1 + z2 = (a + c ) + (b + d)i</p></li><li><p>z1 - z2 = (a - c ) + (b - d)i</p></li></ul><p></p>
4
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Describe the multiplication of complex numbers z1 (a + bi) and z2 (c + di)

  • z1 · z2 = (a + bi) + (c + di)

    • (a · c) + (a · di) + (bi · c) + (b · d · i²) [i² = -1]

    • (a · c) + (- b · d) + (a · d + b ·c)i

    • (ac−bd) + (ad+bc)i

<ul><li><p>z1 · z2 = (a + bi) + (c + di)</p><ul><li><p>(a · c) + (a · di) + (bi · c) + (b · d · i²) [i² = -1]</p></li><li><p>(a · c) + (- b · d) + (a · d + b ·c)i</p></li><li><p>(ac−bd) + (ad+bc)i</p></li></ul></li></ul><p></p>
5
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Describe the division of complex numbers z1 (a + bi) and z2 (c + di)

  • 1 / z2 = 1 / (c + di) x [ (c - di) / (c - di) ]

    • c / (c² + d²) - d / (c² + d²)i

  • z1 / z2

    • z1 · 1 / z2 = z1 · ( c / (c² + d²) - d / (c² + d²)i

    • (a + bi) · ( c / (c² + d²) - d / (c² + d²)i

    • ( ac / (c² + d²) + bd / (c² + d²) + ( ac / (c² + d²) - bd / (c² + d²)i

  • z1 / z2 = [ ac + bd / (c² + d²) ] + [ ac - bd / (c² + d²)i ]

<ul><li><p>1 / z2 = 1 / (c + di) x [ (c - di) / (c - di) ]</p><ul><li><p>c / (c² + d²) - d / (c² + d²)i</p></li></ul></li><li><p>z1 / z2</p><ul><li><p>z1 · 1 / z2 = z1 · ( c / (c² + d²) - d / (c² + d²)i</p></li><li><p>(a + bi) · ( c / (c² + d²) - d / (c² + d²)i</p></li><li><p>( ac / (c² + d²) + bd / (c² + d²) + ( ac / (c² + d²) - bd / (c² + d²)i</p></li></ul></li><li><p>z1 / z2 = [ ac + bd / (c² + d²) ] + [ ac - bd / (c² + d²)i ]</p></li></ul><p></p>
6
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State what is meant by the conjugate of z (a + bi)

bar z = a - bi

<p>bar z = a - bi</p>
7
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state what bar bar z is

bar bar z = z

<p>bar bar z = z</p>
8
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state what bar (z1 + z2) is

  • bar (z1 + z2 ) = bar z1 + bar z2

<ul><li><p>bar (z1 + z2 ) = bar z1 + bar z2</p></li></ul><p></p>
9
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state what bar (z1 · z2) is

  • bar (z1 · z2) = bar z1 · bar z2

<ul><li><p>bar (z1 · z2) = bar z1 · bar z2</p></li></ul><p></p>
10
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state what bar (z^n) is

  • bar (z^n) = (bar z) ^ n

<ul><li><p>bar (z^n) = (bar z) ^ n</p></li></ul><p></p>
11
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state what z + bar z is

  • a + bi + a - bi = 2a

<ul><li><p>a + bi + a - bi = 2a</p></li></ul><p></p>
12
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state what z - bar z is

  • z - bar z

    • a + bi - a + bi = 2bi

<ul><li><p>z - bar z</p><ul><li><p>a + bi - a + bi = 2bi</p></li></ul></li></ul><p></p>
13
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state what z · bar z is

  • z · bar z = (a + bi) · (a - bi) = a² + b²

<ul><li><p>z · bar z = (a + bi) · (a - bi) = a² + b²</p></li></ul><p></p>
14
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State what is meant by the absolute value of a complex number

  • modulus of a complex number

  • distance from the origin to that complex number

<ul><li><p>modulus of a complex number</p></li><li><p>distance from the origin to that complex number</p></li><li><p></p></li></ul><p></p>
15
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Describe what is meant by the triangle inequality

  • the magnitude of the sum of two complex numbers is always less than or equal to the sum of the magnitudes of the complex numbers

  • can be used for any number of complex numbers

<ul><li><p>the magnitude of the sum of two complex numbers is always less than or equal to the sum of the magnitudes of the complex numbers</p></li><li><p>can be used for any number of complex numbers</p></li></ul><p></p>
16
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Describe what is meant by the polar form of a complex number

  • z = r(cos x + isin x )

  • x is the angle of z to the real axis

  • r is the magnitude of z

<ul><li><p>z = r(cos x + isin x )</p></li><li><p>x is the angle of z to the real axis</p></li><li><p>r is the magnitude of z</p></li></ul><p></p>
17
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Describe how to convert a complex number into its polar form

  • r = |z| = sqrt ( a² + b² )

  • theta = arctan ( b / a )

  • a = rcos theta

  • b = rsin theta

<ul><li><p>r = |z| = sqrt ( a² + b² )</p></li><li><p>theta = arctan ( b / a ) </p></li><li><p>a = rcos theta</p></li><li><p>b = rsin theta</p></li></ul><p></p>
18
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Describe how to calculate the principal argument

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19
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Describe multiplication and division in polar form

  • z1 * z2

    • arg(z1 * z2) = arg(z1) + arg(z2)

    • |z1 * z2| = r1r2

  • z1 / z2

    • arg ( z1 / z2) = arg(z1) - arg(z2)

    • |z1 / z2| = r1 / r2

<ul><li><p>z1 * z2 </p><ul><li><p>arg(z1 * z2) = arg(z1) + arg(z2)</p></li><li><p>|z1 * z2| = r1r2</p></li></ul></li><li><p>z1 / z2</p><ul><li><p>arg ( z1 / z2) = arg(z1) - arg(z2)</p></li><li><p>|z1 / z2| = r1 / r2</p></li></ul></li></ul><p></p>
20
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Describe de Moivre’s formula

  • z = r( cos x+ isin x )

  • z^n = [r(cos x + isin x) ]^n

    • r^n (cos ( nx) + isin (nx) )

  • |z|^n = |z^n|

  • arg(z^n) = n arg (z)

<ul><li><p>z = r( cos x+ isin x )</p></li><li><p>z^n = [r(cos x + isin x) ]^n</p><ul><li><p>r^n (cos ( nx) + isin (nx) )</p></li></ul></li><li><p>|z|^n = |z^n|</p></li><li><p>arg(z^n) = n arg (z)</p></li></ul><p></p>