The imaginary unit \( i \) is defined by \( i^2 = -1 \). A complex number in Cartesian form is \( z = a + bi, \) where \( a = \) Re(\( z) \) and \( b = \) Im(\( z) \).
Division: To divide by a complex number, multiply top and bottom by the conjugate of the denominator: (\( a+bi)/(c+di) \times (c-di)/(c-di). \)
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Common error: arctan(\( b/a) \) only gives the correct angle for Q1 and Q4. For \( z = -1 + i, \) arctan(\( -1) = -45 \)° but the actual argument is 135° (Q2). Always plot the point first.
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Argand Diagram
The Argand diagram represents complex numbers as points in a plane: horizontal axis = Real, vertical axis = Imaginary.
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Key interpretations: |\( z_1 - z_2| = \) distance between two points. |\( z - w| = r \) describes a circle centre \( w, \) radius \( r \). arg(\( z - w) = \theta \) describes a half-line from \( w \).
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Polar (Modulus-Argument) Form
Polar form
\( z = r cis \theta = r(\cos \theta + i \sin \theta ) \)
where \( r = \) |\( z| \) and \( \theta = \) arg(\( z) \)
Converting between forms
\( Cartesian \to Polar: r = \sqrt{a^2+b^2}, \theta = arctan(b/a) \) \( Polar \to Cartesian: a = r \cos \theta , b = r \sin \theta \)
Worked Example
Write \( z = -1 + \sqrt{3} i \) in polar form.
\( r = \sqrt{1 + 3} = 2 \)
\( \theta = \) arctan(\( \sqrt{3}/-1) \) — point is in Q2, so \( \theta = \pi - \pi /3 = 2\pi /3 \)
Geometric meaning: Multiplying by \( r \) cis \( \theta \) scales by \( r \) and rotates by \( \theta . \) Multiplying by cis(\( \pi /2) = \) rotation of 90° anticlockwise.
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De Moivre’s Theorem
De Moivre’s theorem (given in formula booklet)
\( (r cis \theta )^n = r^n cis(n\theta ) \)
Equivalently: [\( r(\cos \theta + i \) sin \( \theta )] \)\( n \) = \( r^{n} \)(cos \( n\theta + i \) sin \( n\theta ) \)
Worked Example
Find (1 + \( i)^8 \).
Convert to polar: |1+\( i| = \sqrt{2}, \) arg = \( \pi /4. \) So 1+\( i = \sqrt{2} \) cis(\( \pi /4) \)
Common error: Applying De Moivre’s theorem to Cartesian form directly. You MUST convert to polar form first: (\( a+bi)^{n} \) \( \neq \) \( a^{n} \) + (\( bi)^{n} \).
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Adding Sinusoidal Functions
Complex numbers can be used to combine sinusoidal functions of the same frequency into a single sinusoidal function.
Combining sinusoids
\( A_1 \sin(\omega t + \phi _1) + A_2 \sin(\omega t + \phi _2) = R \sin(\omega t + \alpha ) \)
Represent each term as a complex number \( A \) cis \( \phi , \) add them, then read off \( R \) and \( \alpha \)
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Method: Write \( z_1 = A_1 \) cis \( \phi _1 \) and \( z_2 = A_2 \) cis \( \phi _2. \) Then \( z_1 + z_2 = R \) cis \( \alpha \) where \( R = \) |\( z_1+z_2| \) and \( \alpha = \) arg(\( z_1+z_2) \).
GDC: Complex Numbers
TI-84 Plus CE
[MODE] → set to a+b\( i \) or re^(\( \theta i) \)
Enter \( i \) with [2ND][.]
abs( for modulus, angle( for argument
Powers: (1+\( i)^8 \) [ENTER]
TI-Nspire CX II
Enter \( i \) from keyboard or [\( \pi ] \) menu abs(z) for modulus, angle(z) for argument conj(z) for conjugate real(z), imag(z) for parts
Polar: type in r\( \cdot e^(i\cdot \theta ) \) or use ▸Polar conversion
Casio fx-CG50
[MENU] → Run-Matrix [SHIFT][0] for \( i \) [OPTN] → COMPLEX → Abs, Arg, Conj
Toggle form: [SHIFT][MENU] → Complex Mode → a+bi or r∠\( \theta \)
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Exam Traps & Key Reminders
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Powers of \( i \): \( i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, \) then repeats. For \( i^{n} \), find \( n \) mod 4.
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Argument in radians. The IB expects arguments in radians (usually in terms of \( \pi ), \) not degrees, unless otherwise stated.
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Formula booklet: De Moivre’s theorem, modulus, and polar/Cartesian conversion are given. The multiplication/division rules for polar form follow directly from De Moivre’s theorem.
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