By the end of this chapter, you will be able to:
Mastering these skills will help you solve real-world engineering problems that involve complex numbers with confidence and precision.
Complex numbers are fundamental in Mechanical Production Technician work, especially in analyzing oscillations, vibrations, and signal processing in machinery and control systems. Understanding the real and imaginary parts of complex numbers enables technicians to interpret system behaviors and solve engineering problems involving alternating currents, mechanical resonances, and dynamic responses. This chapter focuses on the properties of the real and imaginary parts of complex numbers, emphasizing their application in mechanical production contexts in Kenya.
A complex number \(z\) is expressed as \(z = x + jy\), where \(x\) and \(y\) are real numbers, and \(j\) is the imaginary unit with the property \(j^2 = -1\). The real part of \(z\) is the component \(x\), which lies on the horizontal axis of the complex plane. It represents the magnitude of the number along the real axis and is used to quantify measurable physical quantities such as displacement or voltage in mechanical systems.
$$\text{Real part of } z = \operatorname{Re}(z) = x$$
Example 1: A mechanical vibration is represented by the complex number \(z = 5 + j3\). Find the real part.
Given: \(z = 5 + j3\)
$$\operatorname{Re}(z) = x = 5$$
Answer: 5
Example 2: In signal analysis for a conveyor motor, the complex representation of the voltage is \(z = -7 + j4\). Determine the real part.
Given: \(z = -7 + j4\)
$$\operatorname{Re}(z) = x = -7$$
Answer: -7
Example 3: A control system’s response is modelled as \(z = 12.5 - j8.2\). Calculate the real part.
Given: \(z = 12.5 - j8.2\)
$$\operatorname{Re}(z) = x = 12.5$$
Answer: 12.5
Example 4: The impedance of a mechanical system is \(z = 3.2 + j0\). Find the real part and interpret its meaning.
Given: \(z = 3.2 + j0\)
$$\operatorname{Re}(z) = x = 3.2$$
Answer: 3.2
Interpretation: The impedance is purely real, indicating no reactive component (no energy stored in the system).
Example 5: A complex number representing displacement is \(z = -4.7 - j6.1\). Find the real part.
Given: \(z = -4.7 - j6.1\)
$$\operatorname{Re}(z) = x = -4.7$$
Answer: -4.7
When adding two complex numbers \(z_1 = x_1 + jy_1\) and \(z_2 = x_2 + jy_2\), the real part of the sum is the sum of the real parts:
$$\operatorname{Re}(z_1 + z_2) = x_1 + x_2$$
This operation is crucial when combining signals or forces in mechanical systems.
Example 1: Two forces represented by \(z_1 = 8 + j5\) N and \(z_2 = 3 - j2\) N act on a machine part. Find the real part of the resultant force.
Given: \(z_1 = 8 + j5\), \(z_2 = 3 - j2\)
$$\operatorname{Re}(z_1 + z_2) = 8 + 3 = 11$$
Answer: 11 N
Example 2: Voltages in a motor control circuit are \(z_1 = -6 + j7\) V and \(z_2 = 4 + j3\) V. Compute the real part of their sum.
Given: \(z_1 = -6 + j7\), \(z_2 = 4 + j3\)
$$\operatorname{Re}(z_1 + z_2) = -6 + 4 = -2$$
Answer: -2 V
Example 3: Displacements represented as \(z_1 = 2.5 - j1.5\) mm and \(z_2 = -3.7 + j0.9\) mm are combined. Find the real part of the total displacement.
Given: \(z_1 = 2.5 - j1.5\), \(z_2 = -3.7 + j0.9\)
$$\operatorname{Re}(z_1 + z_2) = 2.5 + (-3.7) = -1.2$$
Answer: -1.2 mm
Example 4: Currents in a mechanical sensor circuit are \(z_1 = 0 + j4\) A and \(z_2 = 5 + j0\) A. Find the real part of the total current.
Given: \(z_1 = 0 + j4\), \(z_2 = 5 + j0\)
$$\operatorname{Re}(z_1 + z_2) = 0 + 5 = 5$$
Answer: 5 A
Example 5: Two complex impedances \(z_1 = 10 - j5\) Ω and \(z_2 = -4 + j3\) Ω are connected in series. Calculate the real part of the total impedance.
Given: \(z_1 = 10 - j5\), \(z_2 = -4 + j3\)
$$\operatorname{Re}(z_1 + z_2) = 10 + (-4) = 6$$
Answer: 6 Ω
When multiplying two complex numbers, the real part of the product is given by:
$$(x_1 + jy_1)(x_2 + jy_2) = (x_1x_2 - y_1y_2) + j(x_1y_2 + y_1x_2)$$
The real part of the product is:
$$\operatorname{Re}(z_1 z_2) = x_1 x_2 - y_1 y_2$$
This is important in calculating power and energy in mechanical systems.
Example 1: Multiply \(z_1 = 3 + j2\) and \(z_2 = 1 + j4\). Find the real part of the product.
Given: \(z_1 = 3 + j2\), \(z_2 = 1 + j4\)
$$\operatorname{Re}(z_1 z_2) = 3 \times 1 - 2 \times 4 = 3 - 8 = -5$$
Answer: -5
Example 2: \(z_1 = -2 + j3\), \(z_2 = 4 - j1\). Compute the real part of the product.
Given: \(z_1 = -2 + j3\), \(z_2 = 4 - j1\)
$$\operatorname{Re}(z_1 z_2) = (-2)(4) - 3(-1) = -8 + 3 = -5$$
Answer: -5
Example 3: \(z_1 = 5 + j0\), \(z_2 = 0 + j7\). Find the real part of the product.
Given: \(z_1 = 5 + j0\), \(z_2 = 0 + j7\)
$$\operatorname{Re}(z_1 z_2) = 5 \times 0 - 0 \times 7 = 0 - 0 = 0$$
Answer: 0
Example 4: \(z_1 = 1.5 - j2\), \(z_2 = -3 + j4\). Calculate the real part of their product.
Given: \(z_1 = 1.5 - j2\), \(z_2 = -3 + j4\)
$$\operatorname{Re}(z_1 z_2) = 1.5 \times (-3) - (-2) \times 4 = -4.5 + 8 = 3.5$$
Answer: 3.5
Example 5: \(z_1 = -1 + j5\), \(z_2 = -2 - j3\). Find the real part of the product.
Given: \(z_1 = -1 + j5\), \(z_2 = -2 - j3\)
$$\operatorname{Re}(z_1 z_2) = (-1)(-2) - 5(-3) = 2 + 15 = 17$$
Answer: 17
The imaginary part of a complex number \(z = x + jy\) is the coefficient of \(j\), which is \(y\). It lies on the vertical axis of the complex plane and represents the magnitude along the imaginary axis. In mechanical production, this part is used to model phase shifts, oscillations, and reactive components in dynamic systems.
$$\operatorname{Im}(z) = y$$
Example 1: For \(z = 6 + j9\), find the imaginary part.
Given: \(z = 6 + j9\)
$$\operatorname{Im}(z) = 9$$
Answer: 9
Example 2: The complex signal \(z = -4 - j7\) represents a mechanical vibration. Calculate its imaginary part.
Given: \(z = -4 - j7\)
$$\operatorname{Im}(z) = -7$$
Answer: -7
Example 3: An electrical current is represented by \(z = 0 + j15\) A. Find the imaginary part.
Given: \(z = 0 + j15\)
$$\operatorname{Im}(z) = 15$$
Answer: 15 A
Example 4: The impedance in a sensor circuit is \(z = 10 - j5\) Ω. Determine the imaginary part.
Given: \(z = 10 - j5\)
$$\operatorname{Im}(z) = -5$$
Answer: -5 Ω
Example 5: A displacement vector is \(z = -3.7 + j2.1\) mm. Find the imaginary part.
Given: \(z = -3.7 + j2.1\)
$$\operatorname{Im}(z) = 2.1$$
Answer: 2.1 mm
When adding two complex numbers, the imaginary part of the sum is the sum of their imaginary parts:
$$\operatorname{Im}(z_1 + z_2) = y_1 + y_2$$
This is useful when combining phase-shifted signals or oscillations.
Example 1: \(z_1 = 7 + j3\), \(z_2 = 2 + j6\). Find the imaginary part of the sum.
Given: \(z_1 = 7 + j3\), \(z_2 = 2 + j6\)
$$\operatorname{Im}(z_1 + z_2) = 3 + 6 = 9$$
Answer: 9
Example 2: Voltages \(z_1 = -5 + j4\), \(z_2 = 3 - j7\). Calculate the imaginary part of the sum.
Given: \(z_1 = -5 + j4\), \(z_2 = 3 - j7\)
$$\operatorname{Im}(z_1 + z_2) = 4 + (-7) = -3$$
Answer: -3
Example 3: Displacements \(z_1 = 4.1 - j2.3\), \(z_2 = -1.5 + j3.8\). Find the imaginary part of the total displacement.
Given: \(z_1 = 4.1 - j2.3\), \(z_2 = -1.5 + j3.8\)
$$\operatorname{Im}(z_1 + z_2) = -2.3 + 3.8 = 1.5$$
Answer: 1.5
Example 4: Currents \(z_1 = 0 + j9\), \(z_2 = 7 + j0\). Find the imaginary part of the sum.
Given: \(z_1 = 0 + j9\), \(z_2 = 7 + j0\)
$$\operatorname{Im}(z_1 + z_2) = 9 + 0 = 9$$
Answer: 9
Example 5: Impedances \(z_1 = 6 - j4\), \(z_2 = -2 + j7\). Calculate the imaginary part of the combined impedance.
Given: \(z_1 = 6 - j4\), \(z_2 = -2 + j7\)
$$\operatorname{Im}(z_1 + z_2) = -4 + 7 = 3$$
Answer: 3
For two complex numbers \(z_1 = x_1 + jy_1\) and \(z_2 = x_2 + jy_2\), the imaginary part of their product is:
$$\operatorname{Im}(z_1 z_2) = x_1 y_2 + y_1 x_2$$
This calculation is critical in determining phase relationships and power factors in mechanical-electrical systems.
Example 1: Multiply \(z_1 = 2 + j3\) and \(z_2 = 4 + j5\). Find the imaginary part of their product.
Given: \(z_1 = 2 + j3\), \(z_2 = 4 + j5\)
$$\operatorname{Im}(z_1 z_2) = 2 \times 5 + 3 \times 4 = 10 + 12 = 22$$
Answer: 22
Example 2: \(z_1 = -1 + j4\), \(z_2 = 3 - j2\). Compute the imaginary part of the product.
Given: \(z_1 = -1 + j4\), \(z_2 = 3 - j2\)
$$\operatorname{Im}(z_1 z_2) = (-1) \times (-2) + 4 \times 3 = 2 + 12 = 14$$
Answer: 14
Example 3: \(z_1 = 0 + j7\), \(z_2 = 5 + j0\). Find the imaginary part of the product.
Given: \(z_1 = 0 + j7\), \(z_2 = 5 + j0\)
$$\operatorname{Im}(z_1 z_2) = 0 \times 0 + 7 \times 5 = 0 + 35 = 35$$
Answer: 35
Example 4: \(z_1 = 1.5 - j3\), \(z_2 = -2 + j1\). Calculate the imaginary part of their product.
Given: \(z_1 = 1.5 - j3\), \(z_2 = -2 + j1\)
$$\operatorname{Im}(z_1 z_2) = 1.5 \times 1 + (-3) \times (-2) = 1.5 + 6 = 7.5$$
Answer: 7.5
Example 5: \(z_1 = -3 + j2\), \(z_2 = -1 - j4\). Find the imaginary part of the product.
Given: \(z_1 = -3 + j2\), \(z_2 = -1 - j4\)
$$\operatorname{Im}(z_1 z_2) = (-3) \times (-4) + 2 \times (-1) = 12 - 2 = 10$$
Answer: 10
The Argand diagram is a graphical representation of complex numbers on a two-dimensional plane, essential for mechanical production technicians when analysing vibrations or alternating currents. In Kenya’s manufacturing and mechanical workshops, visualising complex numbers aids in understanding oscillations and signal processing in equipment diagnostics and control systems. The horizontal axis represents the real component, while the vertical axis represents the imaginary component of the complex number.
The complex number \( z = x + jy \) is plotted as the point \((x, y)\) on the Argand plane, where \(x\) is the real part and \(y\) is the imaginary part. The Argand diagram helps to visualise operations such as addition, subtraction, and multiplication of complex numbers geometrically.
The position of \(z\) in the Argand diagram is given by:
$$ z = x + jy $$
where:
Example 1:
A mechanical technician measures a vibration signal represented by the complex number \(z = 3 + j4\). Plot this point on the Argand diagram.
Given:
\(x = 3\), \(y = 4\)
Plot the point at coordinates \((3, 4)\) on the Argand plane.
Answer: The point is located 3 units along the real axis and 4 units along the imaginary axis.
Example 2:
Plot and add the complex numbers \(z_1 = 2 + j3\) and \(z_2 = -1 + j5\) on the Argand diagram, and find the resulting point.
Given:
\(z_1 = 2 + j3\), \(z_2 = -1 + j5\)
Addition of complex numbers:$$ z = z_1 + z_2 = (2 - 1) + j(3 + 5) $$$$ z = 1 + j8 $$
Plot points \((2,3)\), \((-1,5)\), and the result \((1,8)\).
Answer: The sum corresponds to the point \((1, 8)\) on the Argand diagram.
Example 3:
A mechanical system has a complex impedance represented by \(z = 4 - j3\). Locate this point on the Argand diagram.
Given:
\(x = 4\), \(y = -3\)
Plot at \((4, -3)\).
Answer: The point lies 4 units on the real axis and 3 units below the real axis on the imaginary axis.
Example 4:
Determine the geometric representation of the complex number \(z = -3 + j2\) on the Argand diagram used for signal analysis in a manufacturing plant.
Given:
\(x = -3\), \(y = 2\)
Plot at \((-3, 2)\).
Answer: The point is located 3 units left of the origin and 2 units up on the imaginary axis.
Example 5:
Find the resulting point after subtracting \(z_2 = 1 + j2\) from \(z_1 = 5 + j6\) and plot all three points.
Given:
\(z_1 = 5 + j6\), \(z_2 = 1 + j2\)
Subtraction:$$ z = z_1 - z_2 = (5 - 1) + j(6 - 2) $$$$ z = 4 + j4 $$
Plot \((5,6)\), \((1,2)\), and \((4,4)\).
Answer: The difference is at \((4, 4)\) on the Argand diagram.
The modulus or magnitude of a complex number is its distance from the origin in the Argand diagram, a critical measure in mechanical production for assessing amplitude of oscillations or signal strength. In Kenyan engineering workshops, accurately determining the magnitude of complex numbers is vital for analysing alternating current circuits, vibrations, and dynamic systems. The modulus of \( z = x + jy \) is denoted by \(|z|\) and calculated using the Pythagorean theorem.
The modulus formula is:
$$ |z| = \sqrt{x^2 + y^2} $$
where \(x\) is the real part and \(y\) is the imaginary part of the complex number.
Example 1:
Calculate the modulus of the complex number \(z = 3 + j4\), representing a vibration amplitude.
Given:
\(x = 3\), \(y = 4\)
Formula:$$ |z| = \sqrt{x^2 + y^2} $$
Substitution:$$ |z| = \sqrt{3^2 + 4^2} $$$$ |z| = \sqrt{9 + 16} $$$$ |z| = \sqrt{25} $$
Answer: \(\boxed{5}\) units
Example 2:
Find the modulus of \(z = -5 + j12\), a complex impedance in an electrical motor control system.
Given:
\(x = -5\), \(y = 12\)
Formula:$$ |z| = \sqrt{x^2 + y^2} $$
Substitution:$$ |z| = \sqrt{(-5)^2 + 12^2} $$$$ |z| = \sqrt{25 + 144} $$$$ |z| = \sqrt{169} $$
Answer: \(\boxed{13}\) units
Example 3:
Determine the magnitude of \(z = 7 - j24\), used in analyzing signal strength in production machinery.
Given:
\(x = 7\), \(y = -24\)
Formula:$$ |z| = \sqrt{x^2 + y^2} $$
Substitution:$$ |z| = \sqrt{7^2 + (-24)^2} $$$$ |z| = \sqrt{49 + 576} $$$$ |z| = \sqrt{625} $$
Answer: \(\boxed{25}\) units
Example 4:
Calculate the modulus of \(z = -8 - j15\) representing a complex load in a mechanical system.
Given:
\(x = -8\), \(y = -15\)
Formula:$$ |z| = \sqrt{x^2 + y^2} $$
Substitution:$$ |z| = \sqrt{(-8)^2 + (-15)^2} $$$$ |z| = \sqrt{64 + 225} $$$$ |z| = \sqrt{289} $$
Answer: \(\boxed{17}\) units
Example 5:
A signal in a factory control panel is represented by \(z = 0 + j9\). Find its magnitude.
Given:
\(x = 0\), \(y = 9\)
Formula:$$ |z| = \sqrt{x^2 + y^2} $$
Substitution:$$ |z| = \sqrt{0^2 + 9^2} $$$$ |z| = \sqrt{81} $$
Answer: \(\boxed{9}\) units
The argument or angle of a complex number is the angle formed between the positive real axis and the line representing the complex number in the complex plane (Argand diagram). For Mechanical Production Technicians in Kenya, understanding the argument is crucial in analyzing oscillations, vibrations, and alternating currents in manufacturing machines and control systems. The angle helps convert between Cartesian form and polar form, enabling easier multiplication, division, and interpretation of complex numbers in practical engineering problems.
The argument \( \theta \) of a complex number \( z = x + jy \) is calculated by
$$ \theta = \tan^{-1} \frac{y}{x} $$
where \( x \) is the real part and \( y \) is the imaginary part. The angle \( \theta \) is usually expressed in radians or degrees and must be adjusted according to the quadrant of the point \( (x, y) \).
Example 1: Find the argument of the complex number \( z = 3 + j4 \) representing a voltage in an electrical motor control circuit.
Given: \( x = 3 \), \( y = 4 \)
$$ \theta = \tan^{-1} \frac{y}{x} $$
$$ \theta = \tan^{-1} \frac{4}{3} $$
$$ \theta = \tan^{-1} 1.3333 $$
$$ \theta = 53.13^\circ $$
Answer: \( \theta = 53.13^\circ \)
Example 2: Calculate the argument of \( z = -5 + j5 \) describing a force vector in a mechanical linkage.
Given: \( x = -5 \), \( y = 5 \)
$$ \theta = \tan^{-1} \frac{y}{x} $$
$$ \theta = \tan^{-1} \frac{5}{-5} = \tan^{-1} (-1) $$
$$ \theta = -45^\circ $$
Since \( x < 0 \) and \( y > 0 \), \( z \) lies in the second quadrant, so
$$ \theta = 180^\circ - 45^\circ = 135^\circ $$
Answer: \( \theta = 135^\circ \)
Example 3: Determine the argument of \( z = -3 - j3\sqrt{3} \) representing a complex impedance in a production line circuit.
Given: \( x = -3 \), \( y = -3\sqrt{3} \approx -5.196 \)
$$ \theta = \tan^{-1} \frac{y}{x} $$
$$ \theta = \tan^{-1} \frac{-5.196}{-3} = \tan^{-1} 1.732 $$
$$ \theta = 60^\circ $$
Since \( x < 0 \), \( y < 0 \), \( z \) lies in the third quadrant, so
$$ \theta = 180^\circ + 60^\circ = 240^\circ $$
Answer: \( \theta = 240^\circ \)
Example 4: A mechanical sensor outputs a complex signal \( z = 0 + j7 \). Find its argument.
Given: \( x = 0 \), \( y = 7 \)
Since \( x = 0 \), the argument is
Here, \( y = 7 > 0 \)
Answer: \( \theta = 90^\circ \)
Example 5: Calculate the argument of \( z = 8 - j6 \) representing a velocity vector in a manufacturing robot arm.
Given: \( x = 8 \), \( y = -6 \)
$$ \theta = \tan^{-1} \frac{y}{x} $$
$$ \theta = \tan^{-1} \frac{-6}{8} = \tan^{-1} (-0.75) $$
$$ \theta = -36.87^\circ $$
Since \( x > 0 \), \( y < 0 \), \( z \) lies in the fourth quadrant, so angle can be expressed as
$$ \theta = 360^\circ - 36.87^\circ = 323.13^\circ $$
Answer: \( \theta = 323.13^\circ \)
The complex conjugate of a complex number \( z = x + jy \) is \( \overline{z} = x, jy \). This operation reflects the point representing \( z \) across the real axis in the Argand diagram. Mechanical Production Technicians use conjugates in simplifying division of complex numbers, calculating magnitudes, and solving equations related to oscillations and alternating currents in systems such as CNC machines and signal processing in quality control.
The conjugate operation follows:
$$ \overline{z} = x, jy $$
with the property that
$$ z \times \overline{z} = x^2 + y^2 $$
which is a real number equal to the square of the magnitude of \( z \).
Example 1: Find the conjugate of \( z = 7 + j2 \) representing a signal amplitude in a sensor.
Given: \( z = 7 + j2 \)
$$ \overline{z} = 7 - j2 $$
Answer: \( \overline{z} = 7 - j2 \)
Example 2: Calculate the product \( z \times \overline{z} \) for \( z = 4 - j3 \) representing a complex velocity.
Given: \( z = 4 - j3 \)
$$ \overline{z} = 4 + j3 $$
$$ z \times \overline{z} = (4 - j3)(4 + j3) $$
$$ = 4 \times 4 + 4 \times j3 - j3 \times 4 - j3 \times j3 $$
$$ = 16 + j12 - j12 - j^2 9 $$
Since \( j^2 = -1 \)
$$ = 16 + 0 + 9 = 25 $$
Answer: \( z \times \overline{z} = 25 \)
Example 3: Divide \( z_1 = 3 + j4 \) by \( z_2 = 1 - j2 \) using conjugates, representing force ratios in a mechanical system.
Given: \( z_1 = 3 + j4 \), \( z_2 = 1 - j2 \)
Step 1: Find conjugate of denominator
$$ \overline{z_2} = 1 + j2 $$
Step 2: Multiply numerator and denominator by \( \overline{z_2} \)
$$ \frac{z_1}{z_2} = \frac{(3 + j4)(1 + j2)}{(1 - j2)(1 + j2)} $$
Step 3: Calculate numerator
$$ (3)(1) + (3)(j2) + (j4)(1) + (j4)(j2) = 3 + j6 + j4 + j^2 8 $$
$$ = 3 + j10 - 8 = -5 + j10 $$
Step 4: Calculate denominator
$$ 1 - j^2 4 = 1 + 4 = 5 $$
Step 5: Divide real and imaginary parts by 5
$$ \frac{-5}{5} + j \frac{10}{5} = -1 + j2 $$
Answer: \( \frac{z_1}{z_2} = -1 + j2 \)
Example 4: Find the magnitude of \( z = -6 + j8 \) using conjugate multiplication, representing displacement in a machine arm.
Given: \( z = -6 + j8 \)
$$ \overline{z} = -6 - j8 $$
$$ z \times \overline{z} = (-6 + j8)(-6 - j8) $$
$$ = 36 + 48j - 48j, j^2 64 = 36 + 0 + 64 = 100 $$
Magnitude \( r = \sqrt{z \times \overline{z}} = \sqrt{100} = 10 \)
Answer: Magnitude \( r = 10 \)
Example 5: Simplify \( \frac{5 - j7}{2 + j3} \) using conjugate operations, representing current ratios in an industrial robot.
Given: \( z_1 = 5 - j7 \), \( z_2 = 2 + j3 \)
$$ \overline{z_2} = 2 - j3 $$
Multiply numerator and denominator by \( \overline{z_2} \):
$$ \frac{5 - j7}{2 + j3} = \frac{(5 - j7)(2 - j3)}{(2 + j3)(2 - j3)} $$
Calculate numerator:
$$ (5)(2) - 5 j3 - j7 \times 2 + j7 \times j3 = 10 - j15 - j14 + j^2 21 $$
$$ = 10 - j29 - 21 = -11 - j29 $$
Calculate denominator:
$$ 4 - j^2 9 = 4 + 9 = 13 $$
Divide:
$$ \frac{-11}{13} - j \frac{29}{13} $$
Answer: \( \frac{z_1}{z_2} = -\frac{11}{13} - j \frac{29}{13} \approx -0.846 - j2.231 \)
Find the argument of the complex number \( z = 6 - j8 \) representing a force vector in a mechanical press. (4 marks)
Calculate the argument of \( z = -4 - j4 \) representing an alternating current phasor in a motor control system. (4 marks)
Determine the conjugate of \( z = 9 + j12 \) and find \( z \times \overline{z} \) for a vibration signal. (5 marks)
Using conjugate operations, divide \( z_1 = 7 + j5 \) by \( z_2 = 3 - j4 \) representing torque ratios in a lathe machine. (6 marks)
Find the magnitude and argument of \( z = -8 + j15 \) using conjugates and argument calculation. (6 marks)
Addition of complex numbers is a fundamental operation that mechanical production technicians encounter when dealing with oscillations, vibrations, and alternating current circuits in Kenyan manufacturing environments. Complex numbers represent quantities having both magnitude and direction, such as impedance in electrical circuits or displacement in mechanical vibrations. Adding complex numbers involves combining their real parts and imaginary parts separately.
A complex number is generally expressed as \(z = a + jb\), where \(a\) is the real part and \(b\) is the imaginary part. The sum of two complex numbers \(z_1 = a_1 + jb_1\) and \(z_2 = a_2 + jb_2\) is given by:
$$ z_1 + z_2 = (a_1 + a_2) + j(b_1 + b_2) $$
Example 1: A machine vibration displacement is represented by \(z_1 = 3 + j4\) mm and another displacement by \(z_2 = 1 + j2\) mm. Find the resultant displacement.
Given: \(z_1 = 3 + j4\) mm, \(z_2 = 1 + j2\) mm
$$ z_1 + z_2 = (3 + 1) + j(4 + 2) $$
$$ = 4 + j6 $$
Answer: \(4 + j6\) mm
Example 2: In an AC circuit, the impedance of two components are \(z_1 = 5 + j12\) ohms and \(z_2 = 7 + j3\) ohms. Calculate the total impedance when connected in series.
Given: \(z_1 = 5 + j12\) Ω, \(z_2 = 7 + j3\) Ω
$$ z_1 + z_2 = (5 + 7) + j(12 + 3) $$
$$ = 12 + j15 $$
Answer: \(12 + j15\) ohms
Example 3: A signal in a control system is \(z_1 = -2 + j7\) volts and another signal is \(z_2 = 4 - j5\) volts. Find the combined signal voltage.
Given: \(z_1 = -2 + j7\) V, \(z_2 = 4 - j5\) V
$$ z_1 + z_2 = (-2 + 4) + j(7 - 5) $$
$$ = 2 + j2 $$
Answer: \(2 + j2\) volts
Example 4: In a mechanical system, two forces are represented as complex numbers \(z_1 = 10 + j6\) kN and \(z_2 = -3 + j8\) kN. Find the resultant force.
Given: \(z_1 = 10 + j6\) kN, \(z_2 = -3 + j8\) kN
$$ z_1 + z_2 = (10 - 3) + j(6 + 8) $$
$$ = 7 + j14 $$
Answer: \(7 + j14\) kN
Example 5: Two alternating voltage sources have voltages \(z_1 = 15 - j9\) V and \(z_2 = -7 + j11\) V. Find their resultant voltage.
Given: \(z_1 = 15 - j9\) V, \(z_2 = -7 + j11\) V
$$ z_1 + z_2 = (15 - 7) + j(-9 + 11) $$
$$ = 8 + j2 $$
Answer: \(8 + j2\) volts
Subtraction of complex numbers is essential in mechanical production, particularly when calculating net displacements, differences in electrical voltages, or force components in machinery maintenance and design. The process involves subtracting the real parts and imaginary parts separately, which is useful for analyzing phase differences or error signals in control systems.
For two complex numbers \(z_1 = a_1 + jb_1\) and \(z_2 = a_2 + jb_2\), their difference is:
$$ z_1 - z_2 = (a_1 - a_2) + j(b_1 - b_2) $$
Example 1: A machine displacement is \(z_1 = 6 + j8\) mm and the previous displacement was \(z_2 = 3 + j5\) mm. Find the change in displacement.
Given: \(z_1 = 6 + j8\) mm, \(z_2 = 3 + j5\) mm
$$ z_1 - z_2 = (6 - 3) + j(8 - 5) $$
$$ = 3 + j3 $$
Answer: \(3 + j3\) mm
Example 2: In an AC circuit, two voltage sources have voltages \(z_1 = 12 + j10\) V and \(z_2 = 7 + j4\) V. Calculate the voltage difference.
Given: \(z_1 = 12 + j10\) V, \(z_2 = 7 + j4\) V
$$ z_1 - z_2 = (12 - 7) + j(10 - 4) $$
$$ = 5 + j6 $$
Answer: \(5 + j6\) volts
Example 3: A force vector in a mechanical system is \(z_1 = 9 + j15\) kN and another force is \(z_2 = 4 + j7\) kN. Find the resultant force difference.
Given: \(z_1 = 9 + j15\) kN, \(z_2 = 4 + j7\) kN
$$ z_1 - z_2 = (9 - 4) + j(15 - 7) $$
$$ = 5 + j8 $$
Answer: \(5 + j8\) kN
Example 4: A signal voltage changes from \(z_1 = 18 - j9\) V to \(z_2 = 5 - j3\) V. Find the change in voltage.
Given: \(z_1 = 18 - j9\) V, \(z_2 = 5 - j3\) V
$$ z_1 - z_2 = (18 - 5) + j(-9 + 3) $$
$$ = 13 - j6 $$
Answer: \(13 - j6\) volts
Example 5: In a control system, two signals are \(z_1 = -6 + j11\) V and \(z_2 = 3 + j4\) V. Find the difference signal voltage.
Given: \(z_1 = -6 + j11\) V, \(z_2 = 3 + j4\) V
$$ z_1 - z_2 = (-6 - 3) + j(11 - 4) $$
$$ = -9 + j7 $$
Answer: \(-9 + j7\) volts
Multiplication of complex numbers is fundamental in mechanical production technology, especially when dealing with oscillations, vibrations, and signal processing in control systems. Mechanical production technicians often encounter problems involving rotating machinery where complex numbers simplify calculations of phase shifts and amplitudes. Multiplying complex numbers combines their magnitudes and adds their angles when expressed in polar form, streamlining analysis of periodic mechanical processes.
The multiplication of two complex numbers \(z_1 = a + jb\) and \(z_2 = c + jd\) in Cartesian form is given by:
$$ z_1 \times z_2 = (ac, bd) + j(ad + bc) $$
Alternatively, if the numbers are in polar form \(z_1 = r_1 \angle \theta_1\) and \(z_2 = r_2 \angle \theta_2\), their product is:
$$ z_1 \times z_2 = r_1 r_2 \angle (\theta_1 + \theta_2) $$
Multiplying complex numbers in Cartesian form involves multiplying and combining real and imaginary parts using the distributive property.
Example 1: Find the product of \(z_1 = 3 + j2\) and \(z_2 = 1 + j4\).
Given: \(z_1 = 3 + j2\), \(z_2 = 1 + j4\)
$$ z_1 \times z_2 = (3)(1) - (2)(4) + j[(3)(4) + (2)(1)] $$
$$ = 3 - 8 + j(12 + 2) $$
$$ = -5 + j14 $$
Answer: \(-5 + j14\)
Example 2: Calculate the product of \(z_1 = -2 + j5\) and \(z_2 = 4 - j3\).
Given: \(z_1 = -2 + j5\), \(z_2 = 4 - j3\)
$$ z_1 \times z_2 = (-2)(4) - (5)(-3) + j[(-2)(-3) + (5)(4)] $$
$$ = -8 + 15 + j(6 + 20) $$
$$ = 7 + j26 $$
Answer: \(7 + j26\)
Example 3: A mechanical system has two oscillating components represented by \(z_1 = 6 + j7\) and \(z_2 = -3 + j2\). Find their product.
Given: \(z_1 = 6 + j7\), \(z_2 = -3 + j2\)
$$ z_1 \times z_2 = (6)(-3) - (7)(2) + j[(6)(2) + (7)(-3)] $$
$$ = -18 - 14 + j(12 - 21) $$
$$ = -32 - j9 $$
Answer: \(-32 - j9\)
Multiplying complex numbers in polar form simplifies to multiplying magnitudes and adding angles, useful for phase angle calculations in rotating machinery.
Given \(z_1 = r_1 \angle \theta_1\) and \(z_2 = r_2 \angle \theta_2\), the product is:
$$ z_1 \times z_2 = r_1 r_2 \angle (\theta_1 + \theta_2) $$
Example 1: Multiply \(z_1 = 5 \angle 30^\circ\) and \(z_2 = 3 \angle 45^\circ\).
Given: \(r_1 = 5\), \(\theta_1 = 30^\circ\), \(r_2 = 3\), \(\theta_2 = 45^\circ\)
$$ r = 5 \times 3 = 15 $$
$$ \theta = 30^\circ + 45^\circ = 75^\circ $$
Answer: \(15 \angle 75^\circ\)
Example 2: Find the product of \(z_1 = 8 \angle 120^\circ\) and \(z_2 = 2 \angle -60^\circ\).
Given: \(r_1 = 8\), \(\theta_1 = 120^\circ\), \(r_2 = 2\), \(\theta_2 = -60^\circ\)
$$ r = 8 \times 2 = 16 $$
$$ \theta = 120^\circ + (-60^\circ) = 60^\circ $$
Answer: \(16 \angle 60^\circ\)
Example 3: A rotating shaft vibration is represented by \(z_1 = 10 \angle 150^\circ\) and an applied force by \(z_2 = 4 \angle 75^\circ\). Find the product.
Given: \(r_1 = 10\), \(\theta_1 = 150^\circ\), \(r_2 = 4\), \(\theta_2 = 75^\circ\)
$$ r = 10 \times 4 = 40 $$
$$ \theta = 150^\circ + 75^\circ = 225^\circ $$
Answer: \(40 \angle 225^\circ\)
Often, complex numbers are given in Cartesian form but multiplication is easier in polar form. Conversion is essential.
Conversion formulas:
$$ r = \sqrt{a^2 + b^2} $$
$$ \theta = \tan^{-1}(\frac{b}{a}) $$
Multiplication steps:
Example 1: Multiply \(z_1 = 1 + j1\) and \(z_2 = \sqrt{3} + j1\).
Given: \(z_1 = 1 + j1\), \(z_2 = \sqrt{3} + j1\)
Convert \(z_1\):
$$ r_1 = \sqrt{1^2 + 1^2} = \sqrt{2} $$
$$ \theta_1 = \tan^{-1}(\frac{1}{1}) = 45^\circ $$
Convert \(z_2\):
$$ r_2 = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2 $$
$$ \theta_2 = \tan^{-1}(\frac{1}{\sqrt{3}}) = 30^\circ $$
Multiply:
$$ r = \sqrt{2} \times 2 = 2\sqrt{2} $$
$$ \theta = 45^\circ + 30^\circ = 75^\circ $$
Convert back to Cartesian:
$$ a = r \cos \theta = 2\sqrt{2} \times \cos 75^\circ = 2\sqrt{2} \times 0.2588 = 0.732 $$
$$ b = r \sin \theta = 2\sqrt{2} \times \sin 75^\circ = 2\sqrt{2} \times 0.9659 = 2.732 $$
Answer: Approximately \(0.732 + j2.732\)
Example 2: Multiply \(z_1 = 2 - j2\) and \(z_2 = 1 + j\).
Convert \(z_1\):
$$ r_1 = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} $$
$$ \theta_1 = \tan^{-1}(\frac{-2}{2}) = \tan^{-1}(-1) = -45^\circ $$
Convert \(z_2\):
$$ r_2 = \sqrt{1^2 + 1^2} = \sqrt{2} $$
$$ \theta_2 = \tan^{-1}(\frac{1}{1}) = 45^\circ $$
Multiply:
$$ r = 2\sqrt{2} \times \sqrt{2} = 2\sqrt{2} \times \sqrt{2} = 2 \times 2 = 4 $$
$$ \theta = -45^\circ + 45^\circ = 0^\circ $$
Convert back:
$$ a = 4 \times \cos 0^\circ = 4 \times 1 = 4 $$
$$ b = 4 \times \sin 0^\circ = 0 $$
Answer: \(4 + j0 = 4\)
Example 3: Multiply \(z_1 = -3 + j4\) and \(z_2 = 5 - j12\).
Convert \(z_1\):
$$ r_1 = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$
$$ \theta_1 = \tan^{-1}(\frac{4}{-3}) = \tan^{-1}(-1.333) = 126.87^\circ \quad \text{(2nd quadrant)} $$
Convert \(z_2\):
$$ r_2 = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 $$
$$ \theta_2 = \tan^{-1}(\frac{-12}{5}) = \tan^{-1}(-2.4) = -67.38^\circ $$
Multiply:
$$ r = 5 \times 13 = 65 $$
$$ \theta = 126.87^\circ + (-67.38^\circ) = 59.49^\circ $$
Convert back:
$$ a = 65 \times \cos 59.49^\circ = 65 \times 0.5095 = 33.12 $$
$$ b = 65 \times \sin 59.49^\circ = 65 \times 0.8604 = 55.93 $$
Answer: Approximately \(33.12 + j55.93\)
Division of complex numbers is critical in mechanical production technology, especially in signal processing, impedance calculations, and system stability analysis. Dividing complex quantities allows technicians to determine ratios of amplitudes and phase differences accurately. Division is often performed by multiplying numerator and denominator by the conjugate of the denominator to remove imaginary parts in the denominator.
Given two complex numbers \(z_1 = a + jb\) and \(z_2 = c + jd\), division in Cartesian form is:
$$ \frac{z_1}{z_2} = \frac{(a + jb)(c, jd)}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + j \frac{bc, ad}{c^2 + d^2} $$
In polar form, division is simpler:
$$ \frac{z_1}{z_2} = \frac{r_1}{r_2} \angle (\theta_1 - \theta_2) $$
Division in Cartesian form requires multiplying numerator and denominator by the complex conjugate of the denominator to rationalize it.
Example 1: Divide \(z_1 = 4 + j3\) by \(z_2 = 1 - j2\).
Given: \(z_1 = 4 + j3\), \(z_2 = 1 - j2\)
Multiply numerator and denominator by conjugate of denominator \(1 + j2\):
$$ \frac{4 + j3}{1 - j2} \times \frac{1 + j2}{1 + j2} = \frac{(4 + j3)(1 + j2)}{(1)^2 + (2)^2} $$
Calculate numerator:
$$ (4)(1) + (4)(j2) + (j3)(1) + (j3)(j2) = 4 + j8 + j3 + j^2 6 $$
Recall \(j^2 = -1\):
$$ 4 + j11 - 6 = (4 - 6) + j11 = -2 + j11 $$
Calculate denominator:
$$ 1 + 4 = 5 $$
Divide:
$$ \frac{-2}{5} + j \frac{11}{5} = -0.4 + j2.2 $$
Answer: \(-0.4 + j2.2\)
Example 2: Find \(\frac{z_1}{z_2}\) where \(z_1 = 5 - j6\) and \(z_2 = 2 + j3\).
Multiply numerator and denominator by conjugate of denominator \(2 - j3\):
$$ \frac{5 - j6}{2 + j3} \times \frac{2 - j3}{2 - j3} = \frac{(5 - j6)(2 - j3)}{2^2 + 3^2} $$
Calculate numerator:
$$ (5)(2) - (5)(j3) - (j6)(2) + (j6)(j3) = 10 - j15 - j12 + j^2 18 $$
$$ 10 - j27 - 18 = (10 - 18) - j27 = -8 - j27 $$
Denominator:
$$ 4 + 9 = 13 $$
Divide:
$$ \frac{-8}{13} - j \frac{27}{13} = -0.615 - j2.077 $$
Answer: \(-0.615 - j2.077\)
Example 3: A mechanical system parameter is \(z_1 = -3 + j7\), divided by \(z_2 = 1 + j1\). Find the quotient.
Multiply numerator and denominator by conjugate of denominator \(1 - j1\):
$$ \frac{-3 + j7}{1 + j1} \times \frac{1 - j1}{1 - j1} = \frac{(-3 + j7)(1 - j1)}{1^2 + 1^2} $$
Calculate numerator:
$$ (-3)(1) + (-3)(-j1) + (j7)(1) + (j7)(-j1) = -3 + j3 + j7 - j^2 7 $$
$$ -3 + j10 + 7 = 4 + j10 $$
Denominator:
$$ 1 + 1 = 2 $$
Divide:
$$ \frac{4}{2} + j \frac{10}{2} = 2 + j5 $$
Answer: \(2 + j5\)
Division in polar form is more straightforward, involving division of magnitudes and subtraction of angles, useful in impedance and phase calculations.
Given \(z_1 = r_1 \angle \theta_1\) and \(z_2 = r_2 \angle \theta_2\):
$$ \frac{z_1}{z_2} = \frac{r_1}{r_2} \angle (\theta_1 - \theta_2) $$
Example 1: Divide \(z_1 = 10 \angle 60^\circ\) by \(z_2 = 2 \angle 30^\circ\).
Given: \(r_1 = 10\), \(\theta_1 = 60^\circ\), \(r_2 = 2\), \(\theta_2 = 30^\circ\)
$$ r = \frac{10}{2} = 5 $$
$$ \theta = 60^\circ - 30^\circ = 30^\circ $$
Answer: \(5 \angle 30^\circ\)
Example 2: Calculate \(\frac{z_1}{z_2}\) where \(z_1 = 8 \angle 150^\circ\) and \(z_2 = 4 \angle 45^\circ\).
Given: \(r_1 = 8\), \(\theta_1 = 150^\circ\), \(r_2 = 4\), \(\theta_2 = 45^\circ\)
$$ r = \frac{8}{4} = 2 $$
$$ \theta = 150^\circ - 45^\circ = 105^\circ $$
Answer: \(2 \angle 105^\circ\)
Example 3: A sensor output is \(z_1 = 15 \angle 210^\circ\), and the input is \(z_2 = 3 \angle 45^\circ\). Find the ratio \(z_1 / z_2\).
Given: \(r_1 = 15\), \(\theta_1 = 210^\circ\), \(r_2 = 3\), \(\theta_2 = 45^\circ\)
$$ r = \frac{15}{3} = 5 $$
$$ \theta = 210^\circ - 45^\circ = 165^\circ $$
Answer: \(5 \angle 165^\circ\)
When complex numbers are in Cartesian form, division is easier by converting to polar form, performing division, then converting back.
Steps:
Example 1: Divide \(z_1 = 3 + j4\) by \(z_2 = 1 - j1\).
Convert \(z_1\):
$$ r_1 = \sqrt{3^2 + 4^2} = 5 $$
$$ \theta_1 = \tan^{-1}(\frac{4}{3}) = 53.13^\circ $$
Convert \(z_2\):
$$ r_2 = \sqrt{1^2 + (-1)^2} = \sqrt{2} = 1.414 $$
$$ \theta_2 = \tan^{-1}(\frac{-1}{1}) = -45^\circ $$
Divide:
$$ r = \frac{5}{1.414} = 3.535 $$
$$ \theta = 53.13^\circ - (-45^\circ) = 98.13^\circ $$
Convert back:
$$ a = 3.535 \times \cos 98.13^\circ = 3.535 \times (-0.139) = -0.491 $$
$$ b = 3.535 \times \sin 98.13^\circ = 3.535 \times 0.990 = 3.500 $$
Answer: Approximately \(-0.491 + j3.500\)
Example 2: Divide \(z_1 = -4 + j3\) by \(z_2 = 2 + j2\).
Convert \(z_1\):
$$ r_1 = \sqrt{(-4)^2 + 3^2} = 5 $$
$$ \theta_1 = \tan^{-1}(\frac{3}{-4}) = 143.13^\circ $$
Convert \(z_2\):
$$ r_2 = \sqrt{2^2 + 2^2} = \sqrt{8} = 2.828 $$
$$ \theta_2 = \tan^{-1}(\frac{2}{2}) = 45^\circ $$
Divide:
$$ r = \frac{5}{2.828} = 1.768 $$
$$ \theta = 143.13^\circ - 45^\circ = 98.13^\circ $$
Convert back:
$$ a = 1.768 \times \cos 98.13^\circ = 1.768 \times (-0.139) = -0.246 $$
$$ b = 1.768 \times \sin 98.13^\circ = 1.768 \times 0.990 = 1.750 $$
Answer: Approximately \(-0.246 + j1.750\)
Example 3: Divide \(z_1 = 7 - j24\) by \(z_2 = 1 + j2\).
Convert \(z_1\):
$$ r_1 = \sqrt{7^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 $$
$$ \theta_1 = \tan^{-1}(\frac{-24}{7}) = -73.74^\circ $$
Convert \(z_2\):
$$ r_2 = \sqrt{1^2 + 2^2} = \sqrt{5} = 2.236 $$
$$ \theta_2 = \tan^{-1}(\frac{2}{1}) = 63.43^\circ $$
Divide:
$$ r = \frac{25}{2.236} = 11.18 $$
$$ \theta = -73.74^\circ - 63.43^\circ = -137.17^\circ $$
Convert back:
$$ a = 11.18 \times \cos(-137.17^\circ) = 11.18 \times (-0.731) = -8.17 $$
$$ b = 11.18 \times \sin(-137.17^\circ) = 11.18 \times (-0.682) = -7.62 $$
Answer: Approximately \(-8.17 - j7.62\)
Complex numbers are widely used in mechanical production engineering, especially in signal processing, vibration analysis, and control systems. Being able to convert between different forms of complex numbers is essential for simplifying calculations and interpreting results in practical engineering problems. The two main forms are the rectangular (Cartesian) form and the polar form. This section focuses on converting from polar form to rectangular form.
Complex numbers can be represented in polar form as \( z = r(\cos \theta + j \sin \theta) \), where \( r \) is the magnitude (modulus) and \( \theta \) is the argument (angle) of the complex number. Converting from polar to rectangular form means expressing the complex number as \( z = x + jy \), where \( x \) is the real part and \( y \) is the imaginary part.
The conversion formula is:
$$ x = r \cos \theta $$
$$ y = r \sin \theta $$
$$ z = x + jy = r \cos \theta + j r \sin \theta $$
where
Example 1: A rotating shaft in a manufacturing machine has a vibration represented by the complex number in polar form as \( 5 \angle 30^\circ \). Find the rectangular form.
Given: \( r = 5 \), \( \theta = 30^\circ \)
$$ x = r \cos \theta $$
$$ x = 5 \cos 30^\circ $$
$$ x = 5 \times 0.866 $$
$$ x = 4.33 $$
$$ y = r \sin \theta $$
$$ y = 5 \sin 30^\circ $$
$$ y = 5 \times 0.5 $$
$$ y = 2.5 $$
Answer:
$$ z = 4.33 + j 2.5 $$
Example 2: An electrical signal in a motor control circuit is described by the complex number \( 10 \angle 135^\circ \). Convert this to rectangular form.
Given: \( r = 10 \), \( \theta = 135^\circ \)
$$ x = 10 \cos 135^\circ $$
$$ x = 10 \times (-0.7071) $$
$$ x = -7.071 $$
$$ y = 10 \sin 135^\circ $$
$$ y = 10 \times 0.7071 $$
$$ y = 7.071 $$
Answer:
$$ z = -7.071 + j 7.071 $$
Example 3: A sensor measures a force vector in polar form as \( 8 \angle 210^\circ \). Find the rectangular form components.
Given: \( r = 8 \), \( \theta = 210^\circ \)
$$ x = 8 \cos 210^\circ $$
$$ x = 8 \times (-0.866) $$
$$ x = -6.928 $$
$$ y = 8 \sin 210^\circ $$
$$ y = 8 \times (-0.5) $$
$$ y = -4.0 $$
Answer:
$$ z = -6.928 - j 4.0 $$
Example 4: The impedance of a circuit in a mechanical production line is given as \( 15 \angle 300^\circ \). Express the impedance in rectangular form.
Given: \( r = 15 \), \( \theta = 300^\circ \)
$$ x = 15 \cos 300^\circ $$
$$ x = 15 \times 0.5 $$
$$ x = 7.5 $$
$$ y = 15 \sin 300^\circ $$
$$ y = 15 \times (-0.866) $$
$$ y = -12.99 $$
Answer:
$$ z = 7.5 - j 12.99 $$
Example 5: A mechanical vibration system has a complex amplitude \( 20 \angle 120^\circ \). Convert this to rectangular form for further analysis.
Given: \( r = 20 \), \( \theta = 120^\circ \)
$$ x = 20 \cos 120^\circ $$
$$ x = 20 \times (-0.5) $$
$$ x = -10 $$
$$ y = 20 \sin 120^\circ $$
$$ y = 20 \times 0.866 $$
$$ y = 17.32 $$
Answer:
$$ z = -10 + j 17.32 $$
Rectangular form of a complex number is expressed as \( z = x + jy \), where \( x \) is the real part and \( y \) is the imaginary part. To convert this to polar form, the magnitude \( r \) and the angle \( \theta \) must be found. This conversion is fundamental in mechanical production for analyzing oscillations, rotating machinery, and control systems where phase and magnitude are critical.
The formulas are:
$$ r = \sqrt{x^2 + y^2} $$
$$ \theta = \tan^{-1} ( \frac{y}{x} ) $$
The angle \( \theta \) is usually expressed in degrees or radians, and care must be taken to determine the correct quadrant based on the signs of \( x \) and \( y \).
Example 1: A force vector is given as \( 3 + j4 \) N. Find its magnitude and angle in polar form.
Given: \( x = 3 \), \( y = 4 \)
$$ r = \sqrt{3^2 + 4^2} $$
$$ r = \sqrt{9 + 16} $$
$$ r = \sqrt{25} $$
$$ r = 5 $$
$$ \theta = \tan^{-1} ( \frac{4}{3} ) $$
$$ \theta = \tan^{-1} (1.333) $$
$$ \theta = 53.13^\circ $$
Answer:
$$ z = 5 \angle 53.13^\circ $$
Example 2: An electrical signal in a machine is \( -5 + j5 \) volts. Convert to polar form.
Given: \( x = -5 \), \( y = 5 \)
$$ r = \sqrt{(-5)^2 + 5^2} $$
$$ r = \sqrt{25 + 25} $$
$$ r = \sqrt{50} $$
$$ r = 7.071 $$
$$ \theta = \tan^{-1} ( \frac{5}{-5} ) $$
$$ \theta = \tan^{-1} (-1) $$
$$ \theta = -45^\circ $$
Since \( x < 0 \) and \( y > 0 \), the angle is in the second quadrant:
$$ \theta = 180^\circ - 45^\circ = 135^\circ $$
Answer:
$$ z = 7.071 \angle 135^\circ $$
Example 3: A vibration amplitude is \( -7 - j24 \). Find magnitude and phase angle.
Given: \( x = -7 \), \( y = -24 \)
$$ r = \sqrt{(-7)^2 + (-24)^2} $$
$$ r = \sqrt{49 + 576} $$
$$ r = \sqrt{625} $$
$$ r = 25 $$
$$ \theta = \tan^{-1} ( \frac{-24}{-7} ) $$
$$ \theta = \tan^{-1} (3.429) $$
$$ \theta = 73.74^\circ $$
Since \( x < 0 \) and \( y < 0 \), the point lies in the third quadrant:
$$ \theta = 180^\circ + 73.74^\circ = 253.74^\circ $$
Answer:
$$ z = 25 \angle 253.74^\circ $$
Example 4: A current in a motor circuit is \( 12 - j16 \) A. Express in polar form.
Given: \( x = 12 \), \( y = -16 \)
$$ r = \sqrt{12^2 + (-16)^2} $$
$$ r = \sqrt{144 + 256} $$
$$ r = \sqrt{400} $$
$$ r = 20 $$
$$ \theta = \tan^{-1} ( \frac{-16}{12} ) $$
$$ \theta = \tan^{-1} (-1.333) $$
$$ \theta = -53.13^\circ $$
Since \( x > 0 \) and \( y < 0 \), the angle is in the fourth quadrant, so:
$$ \theta = -53.13^\circ \text{ or } 360^\circ - 53.13^\circ = 306.87^\circ $$
Answer:
$$ z = 20 \angle -53.13^\circ $$
or
$$ z = 20 \angle 306.87^\circ $$
Example 5: A mechanical sensor output is \( -9 + j12 \) units. Convert to polar form.
Given: \( x = -9 \), \( y = 12 \)
$$ r = \sqrt{(-9)^2 + 12^2} $$
$$ r = \sqrt{81 + 144} $$
$$ r = \sqrt{225} $$
$$ r = 15 $$
$$ \theta = \tan^{-1} ( \frac{12}{-9} ) $$
$$ \theta = \tan^{-1} (-1.333) $$
$$ \theta = -53.13^\circ $$
Since \( x < 0 \) and \( y > 0 \), the point lies in the second quadrant:
$$ \theta = 180^\circ - 53.13^\circ = 126.87^\circ $$
Answer:
$$ z = 15 \angle 126.87^\circ $$
De Moivre's theorem provides a powerful method to raise complex numbers to integer powers or extract roots using their polar form. This is frequently used in mechanical production for analyzing harmonic motions, vibrations, and rotating systems, where powers of complex numbers simplify the computation of repeated oscillations.
De Moivre's theorem states:
$$ (r \angle \theta)^n = r^n \angle n \theta $$
Or equivalently in trigonometric form:
$$ [\cos \theta + j \sin \theta]^n = \cos (n \theta) + j \sin (n \theta) $$
This allows easy calculation of powers of complex numbers by raising the magnitude to the power and multiplying the angle by the power.
Example 1: Find \( (2 \angle 45^\circ)^3 \).
Given: \( r = 2 \), \( \theta = 45^\circ \), \( n = 3 \)
$$ r^n = 2^3 = 8 $$
$$ n \theta = 3 \times 45^\circ = 135^\circ $$
Answer:
$$ (2 \angle 45^\circ)^3 = 8 \angle 135^\circ $$
Example 2: Calculate \( (5 \angle 60^\circ)^4 \).
Given: \( r = 5 \), \( \theta = 60^\circ \), \( n = 4 \)
$$ r^n = 5^4 = 625 $$
$$ n \theta = 4 \times 60^\circ = 240^\circ $$
Answer:
$$ (5 \angle 60^\circ)^4 = 625 \angle 240^\circ $$
Example 3: Evaluate \( (3 \angle 90^\circ)^2 \).
Given: \( r = 3 \), \( \theta = 90^\circ \), \( n = 2 \)
$$ r^n = 3^2 = 9 $$
$$ n \theta = 2 \times 90^\circ = 180^\circ $$
Answer:
$$ (3 \angle 90^\circ)^2 = 9 \angle 180^\circ $$
Example 4: A rotating component has a complex amplitude \( 4 \angle 150^\circ \). Find the value of the amplitude raised to the 5th power.
Given: \( r = 4 \), \( \theta = 150^\circ \), \( n = 5 \)
$$ r^n = 4^5 = 1024 $$
$$ n \theta = 5 \times 150^\circ = 750^\circ $$
Because angles greater than 360° can be reduced by subtracting multiples of 360°,
$$ 750^\circ - 2 \times 360^\circ = 750^\circ - 720^\circ = 30^\circ $$
Answer:
$$ (4 \angle 150^\circ)^5 = 1024 \angle 30^\circ $$
Example 5: Calculate \( (6 \angle 210^\circ)^3 \) and express the result in rectangular form.
Given: \( r = 6 \), \( \theta = 210^\circ \), \( n = 3 \)
$$ r^n = 6^3 = 216 $$
$$ n \theta = 3 \times 210^\circ = 630^\circ $$
Reduce angle:
$$ 630^\circ - 360^\circ = 270^\circ $$
Convert to rectangular:
$$ x = r^n \cos (n \theta) = 216 \cos 270^\circ = 216 \times 0 = 0 $$
$$ y = r^n \sin (n \theta) = 216 \sin 270^\circ = 216 \times (-1) = -216 $$
Answer:
$$ (6 \angle 210^\circ)^3 = 0 - j 216 $$
Convert the complex number \( 7 \angle 120^\circ \) into rectangular form. (5 marks)
Express the complex number \( -4 + j7 \) in polar form. (5 marks)
Use De Moivre's theorem to calculate \( (3 \angle 45^\circ)^4 \). (6 marks)
A sensor output is \( 9 \angle 330^\circ \). Convert this to rectangular form. (5 marks)
Find the polar form of \( -8 - j15 \). (5 marks)
Calculate \( (5 \angle 150^\circ)^3 \) and express the result in rectangular form. (7 marks)
Converting complex numbers from rectangular form \(z = x + jy\) to polar form \(z = r(\cos \theta + j \sin \theta)\) is essential in mechanical production tasks involving oscillations and vibrations, such as shaft balancing or signal analysis in control systems. De Moivre's theorem simplifies raising complex numbers to powers, crucial in stress analysis and dynamic system modelling. This section covers the conversion process and applies De Moivre's theorem to compute powers of complex numbers efficiently.
A complex number \(z = x + jy\) can be expressed in polar form as \(z = r(\cos \theta + j \sin \theta)\), where \(r\) is the magnitude and \(\theta\) the argument.
The magnitude \(r\) is given by
$$r = \sqrt{x^{2} + y^{2}}$$
The argument \(\theta\) (angle) is
$$\theta = \tan^{-1}(\frac{y}{x})$$
with \(\theta\) expressed in radians or degrees.
Example 1: A rotating shaft displacement in a machine is represented by \(z = 3 + j4\). Find the magnitude and argument.
Given: \(x = 3\), \(y = 4\)
$$r = \sqrt{3^{2} + 4^{2}}$$
$$r = \sqrt{9 + 16}$$
$$r = \sqrt{25}$$
$$r = 5$$
$$\theta = \tan^{-1}(\frac{4}{3})$$
$$\theta = \tan^{-1}(1.3333)$$
$$\theta = 53.13^{\circ}$$
Answer: \(z = 5 (\cos 53.13^{\circ} + j \sin 53.13^{\circ})\)
Example 2: An electrical signal in a motor control system is \(z = -2 + j2\). Find its polar form.
Given: \(x = -2\), \(y = 2\)
$$r = \sqrt{(-2)^{2} + 2^{2}}$$
$$r = \sqrt{4 + 4}$$
$$r = \sqrt{8}$$
$$r = 2.828$$
$$\theta = \tan^{-1}(\frac{2}{-2})$$
$$\theta = \tan^{-1}(-1)$$
Since \(x < 0\) and \(y > 0\), \(\theta = 180^{\circ} - 45^{\circ} = 135^{\circ}\)
Answer: \(z = 2.828(\cos 135^{\circ} + j \sin 135^{\circ})\)
Example 3: A signal in a vibration sensor is \(z = -3 - j3\). Convert to polar form.
Given: \(x = -3\), \(y = -3\)
$$r = \sqrt{(-3)^{2} + (-3)^{2}}$$
$$r = \sqrt{9 + 9}$$
$$r = \sqrt{18}$$
$$r = 4.2426$$
$$\theta = \tan^{-1}(\frac{-3}{-3})$$
$$\theta = \tan^{-1}(1)$$
Since \(x < 0\) and \(y < 0\), \(\theta = 180^{\circ} + 45^{\circ} = 225^{\circ}\)
Answer: \(z = 4.2426(\cos 225^{\circ} + j \sin 225^{\circ})\)
Example 4: A gear vibration is modelled by \(z = 0 + j5\). Find the polar form.
Given: \(x = 0\), \(y = 5\)
$$r = \sqrt{0^{2} + 5^{2}}$$
$$r = \sqrt{25}$$
$$r = 5$$
Since \(x=0\), \(\theta = 90^{\circ}\)
Answer: \(z = 5(\cos 90^{\circ} + j \sin 90^{\circ})\)
Example 5: An oscillating beam displacement is \(z = 5 - j12\). Convert to polar form.
Given: \(x = 5\), \(y = -12\)
$$r = \sqrt{5^{2} + (-12)^{2}}$$
$$r = \sqrt{25 + 144}$$
$$r = \sqrt{169}$$
$$r = 13$$
$$\theta = \tan^{-1}(\frac{-12}{5})$$
$$\theta = \tan^{-1}(-2.4)$$
Since \(x > 0\), \(\theta = -67.38^{\circ}\) or \(292.62^{\circ}\)
Answer: \(z = 13(\cos (-67.38^{\circ}) + j \sin (-67.38^{\circ}))\)
De Moivre's theorem states that for a complex number \(z = r(\cos \theta + j \sin \theta)\), its \(n\)-th power is
$$z^{n} = r^{n} (\cos n\theta + j \sin n\theta)$$
This theorem is used in mechanical production for analysing harmonics and vibrations where powers of complex numbers represent repeated oscillations.
Example 1: Calculate \((2(\cos 30^{\circ} + j \sin 30^{\circ}))^{3}\).
Given: \(r=2\), \(\theta=30^{\circ}\), \(n=3\)
$$z^{3} = 2^{3} (\cos 3 \times 30^{\circ} + j \sin 3 \times 30^{\circ})$$
$$z^{3} = 8 (\cos 90^{\circ} + j \sin 90^{\circ})$$
$$z^{3} = 8 (0 + j \times 1)$$
$$z^{3} = 0 + j8$$
Answer: \(z^{3} = j8\)
Example 2: Find \((5(\cos 45^{\circ} + j \sin 45^{\circ}))^{4}\).
Given: \(r=5\), \(\theta=45^{\circ}\), \(n=4\)
$$z^{4} = 5^{4} (\cos 4 \times 45^{\circ} + j \sin 4 \times 45^{\circ})$$
$$z^{4} = 625 (\cos 180^{\circ} + j \sin 180^{\circ})$$
$$z^{4} = 625 (-1 + j \times 0)$$
$$z^{4} = -625 + j0$$
Answer: \(z^{4} = -625\)
Example 3: Calculate \((3(\cos 60^{\circ} + j \sin 60^{\circ}))^{2}\).
Given: \(r=3\), \(\theta=60^{\circ}\), \(n=2\)
$$z^{2} = 3^{2} (\cos 2 \times 60^{\circ} + j \sin 2 \times 60^{\circ})$$
$$z^{2} = 9 (\cos 120^{\circ} + j \sin 120^{\circ})$$
$$z^{2} = 9 (-0.5 + j \times 0.8660)$$
$$z^{2} = -4.5 + j7.794$$
Answer: \(z^{2} = -4.5 + j7.794\)
Example 4: Find \((1(\cos 150^{\circ} + j \sin 150^{\circ}))^{5}\).
Given: \(r=1\), \(\theta=150^{\circ}\), \(n=5\)
$$z^{5} = 1^{5} (\cos 5 \times 150^{\circ} + j \sin 5 \times 150^{\circ})$$
$$z^{5} = 1 (\cos 750^{\circ} + j \sin 750^{\circ})$$
Since \(750^{\circ} = 750 - 2 \times 360 = 30^{\circ}\),
$$z^{5} = \cos 30^{\circ} + j \sin 30^{\circ}$$
$$z^{5} = 0.8660 + j \times 0.5$$
Answer: \(z^{5} = 0.8660 + j0.5\)
Example 5: Calculate \((4(\cos 0^{\circ} + j \sin 0^{\circ}))^{6}\).
Given: \(r=4\), \(\theta=0^{\circ}\), \(n=6\)
$$z^{6} = 4^{6} (\cos 6 \times 0^{\circ} + j \sin 6 \times 0^{\circ})$$
$$z^{6} = 4096 (1 + j \times 0)$$
$$z^{6} = 4096 + j0$$
Answer: \(z^{6} = 4096\)
De Moivre's theorem is used to model periodic phenomena such as rotating shafts and oscillating beams by raising complex numbers representing the motion to powers. This enables calculation of amplitudes and phase shifts after multiple cycles, aiding design and fault diagnosis.
Example 1: A vibration signal is \(z = 3 + j4\). Find \(z^{3}\) using De Moivre's theorem.
Given: \(z = 3 + j4\)
Step 1: Convert to polar form
$$r = \sqrt{3^{2} + 4^{2}} = 5$$
$$\theta = \tan^{-1}(\frac{4}{3}) = 53.13^{\circ}$$
Step 2: Apply De Moivre's theorem
$$z^{3} = 5^{3} (\cos 3 \times 53.13^{\circ} + j \sin 3 \times 53.13^{\circ})$$
$$z^{3} = 125 (\cos 159.39^{\circ} + j \sin 159.39^{\circ})$$
Step 3: Calculate cosine and sine
$$\cos 159.39^{\circ} = -0.940$$
$$\sin 159.39^{\circ} = 0.342$$
Step 4: Multiply
$$z^{3} = 125 (-0.940 + j \times 0.342)$$
$$z^{3} = -117.5 + j42.75$$
Answer: \(z^{3} = -117.5 + j42.75\)
Example 2: A signal \(z = 1 - j1\) represents a current in a motor. Find \(z^{4}\).
Given: \(x=1\), \(y=-1\)
Step 1: Convert to polar
$$r = \sqrt{1^{2} + (-1)^{2}} = \sqrt{2} = 1.414$$
$$\theta = \tan^{-1}(\frac{-1}{1}) = -45^{\circ}$$
Step 2: Apply De Moivre's theorem
$$z^{4} = 1.414^{4} (\cos 4 \times -45^{\circ} + j \sin 4 \times -45^{\circ})$$
$$z^{4} = (1.414^{4}) (\cos -180^{\circ} + j \sin -180^{\circ})$$
$$1.414^{4} = (1.414^{2})^{2} = (2)^{2} = 4$$
$$\cos -180^{\circ} = -1$$
$$\sin -180^{\circ} = 0$$
Step 3: Calculate
$$z^{4} = 4(-1 + j \times 0) = -4 + j0$$
Answer: \(z^{4} = -4\)
Example 3: Calculate \((2 + j2)^{5}\) using De Moivre's theorem.
Given: \(x=2\), \(y=2\)
Step 1: Convert to polar
$$r = \sqrt{2^{2} + 2^{2}} = \sqrt{8} = 2.828$$
$$\theta = \tan^{-1}(\frac{2}{2}) = 45^{\circ}$$
Step 2: Apply De Moivre's theorem
$$z^{5} = 2.828^{5} (\cos 5 \times 45^{\circ} + j \sin 5 \times 45^{\circ})$$
Calculate \(r^{5}\):
$$2.828^{2} = 8$$
$$2.828^{4} = 8^{2} = 64$$
$$2.828^{5} = 64 \times 2.828 = 181$$
Calculate angle:
$$5 \times 45^{\circ} = 225^{\circ}$$
Calculate cosine and sine:
$$\cos 225^{\circ} = -0.7071$$
$$\sin 225^{\circ} = -0.7071$$
Step 3: Calculate
$$z^{5} = 181 (-0.7071 - j \times 0.7071)$$
$$z^{5} = -128.0 - j128.0$$
Answer: \(z^{5} = -128.0 - j128.0\)
Example 4: An oscillation is represented by \(z = 4(\cos 120^{\circ} + j \sin 120^{\circ})\). Find \(z^{3}\).
Given: \(r=4\), \(\theta=120^{\circ}\), \(n=3\)
$$z^{3} = 4^{3} (\cos 360^{\circ} + j \sin 360^{\circ})$$
$$z^{3} = 64 (1 + j \times 0)$$
$$z^{3} = 64$$
Answer: \(z^{3} = 64\)
After applying De Moivre's theorem, converting back to rectangular form \(x + jy\) is often needed for practical mechanical calculations.
The rectangular form is given by
$$x = r \cos \theta$$
$$y = r \sin \theta$$
where \(r\) and \(\theta\) are the magnitude and argument after raising to the power.
Example 1: Find the rectangular form of \(z^{3} = 8(\cos 90^{\circ} + j \sin 90^{\circ})\).
Given: \(r=8\), \(\theta=90^{\circ}\)
$$x = 8 \cos 90^{\circ} = 8 \times 0 = 0$$
$$y = 8 \sin 90^{\circ} = 8 \times 1 = 8$$
Answer: \(z^{3} = 0 + j8\)
Example 2: Convert \(z^{4} = 625(\cos 180^{\circ} + j \sin 180^{\circ})\) to rectangular form.
Given: \(r=625\), \(\theta=180^{\circ}\)
$$x = 625 \cos 180^{\circ} = 625 \times (-1) = -625$$
$$y = 625 \sin 180^{\circ} = 625 \times 0 = 0$$
Answer: \(z^{4} = -625 + j0\)
Example 3: Convert \(z^{2} = 9(\cos 120^{\circ} + j \sin 120^{\circ})\) to rectangular form.
Given: \(r=9\), \(\theta=120^{\circ}\)
$$x = 9 \cos 120^{\circ} = 9 \times (-0.5) = -4.5$$
$$y = 9 \sin 120^{\circ} = 9 \times 0.8660 = 7.794$$
Answer: \(z^{2} = -4.5 + j7.794\)
Example 4: Convert \(z^{5} = 1(\cos 30^{\circ} + j \sin 30^{\circ})\) to rectangular form.
Given: \(r=1\), \(\theta=30^{\circ}\)
$$x = 1 \times \cos 30^{\circ} = 0.8660$$
$$y = 1 \times \sin 30^{\circ} = 0.5$$
Answer: \(z^{5} = 0.8660 + j0.5\)
Example 5: Convert \(z^{6} = 4096(\cos 0^{\circ} + j \sin 0^{\circ})\) to rectangular form.
Given: \(r=4096\), \(\theta=0^{\circ}\)
$$x = 4096 \times 1 = 4096$$
$$y = 4096 \times 0 = 0$$
Answer: \(z^{6} = 4096 + j0\)
Expanding powers of complex numbers in rectangular form is common in mechanical production calculations involving polynomial stress analysis, signal modulation, and dynamic system behaviour. Instead of converting to polar form, expansions use algebraic techniques to express powers of \(z = x + jy\).
The binomial theorem expands \((x + jy)^{n}\) as a sum of terms:
$$(x + jy)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} (jy)^{k}$$
where \(\binom{n}{k}\) is the binomial coefficient.
Example 1: Expand \((1 + j2)^{2}\).
Given: \(x=1\), \(y=2\), \(n=2\)
$$(1 + j2)^{2} = \binom{2}{0} 1^{2} (j2)^{0} + \binom{2}{1} 1^{1} (j2)^{1} + \binom{2}{2} 1^{0} (j2)^{2}$$
Calculate binomial coefficients:
\(\binom{2}{0} = 1\), \(\binom{2}{1} = 2\), \(\binom{2}{2} = 1\)
Calculate powers:
\((j2)^{0} = 1\)
\((j2)^{1} = j2\)
\((j2)^{2} = j^{2} 2^{2} = -1 \times 4 = -4\)
Substitute:
$$= 1 \times 1 \times 1 + 2 \times 1 \times j2 + 1 \times 1 \times (-4)$$
$$= 1 + j4 - 4$$
Combine real parts:
$$= (1 - 4) + j4 = -3 + j4$$
Answer: \((-3 + j4)\)
Example 2: Expand \((2 + j3)^{3}\).
Given: \(x=2\), \(y=3\), \(n=3\)
$$(2 + j3)^{3} = \sum_{k=0}^{3} \binom{3}{k} 2^{3-k} (j3)^{k}$$
Calculate terms:
\(k=0\): \(\binom{3}{0} 2^{3} (j3)^{0} = 1 \times 8 \times 1 = 8\)
\(k=1\): \(\binom{3}{1} 2^{2} (j3)^{1} = 3 \times 4 \times j3 = 12 j3 = j36\)
\(k=2\): \(\binom{3}{2} 2^{1} (j3)^{2} = 3 \times 2 \times j^{2} 9 = 6 \times (-1) \times 9 = -54\)
\(k=3\): \(\binom{3}{3} 2^{0} (j3)^{3} = 1 \times 1 \times j^{3} 27 = j^{3} 27\)
Recall \(j^{3} = j^{2} \times j = -1 \times j = -j\)
So:
$$= 8 + j36 - 54 - j27$$
Group real and imaginary:
Real: \(8 - 54 = -46\)
Imaginary: \(j36 - j27 = j9\)
Answer: \((-46 + j9)\)
Example 3: Expand \((1 - j1)^{4}\).
Given: \(x=1\), \(y=-1\), \(n=4\)
$$(1 - j1)^{4} = \sum_{k=0}^{4} \binom{4}{k} 1^{4-k} (-j1)^{k}$$
Calculate powers:
\(k=0\): \(1^{4} \times 1 = 1\)
\(k=1\): \(\binom{4}{1} 1^{3} (-j1) = 4 \times (-j) = -4j\)
\(k=2\): \(\binom{4}{2} 1^{2} (-j1)^{2} = 6 \times (j^{2} \times 1) = 6 \times (-1) = -6\)
\(k=3\): \(\binom{4}{3} 1^{1} (-j1)^{3} = 4 \times (-j)^{3}\)
Calculate \((-j)^{3} = (-j)(-j)(-j) = (-j)^{2} \times (-j) = (-1) \times (-j) = j\)
So:
$$4 \times j = 4j$$
Since \(j^{4} = 1\)
Sum all terms:
$$1 - 4j - 6 + 4j + 1$$
Combine real:
\(1 - 6 + 1 = -4\)
Combine imaginary:
\(-4j + 4j = 0\)
Answer: \(-4\)
Example 4: Expand \((3 + j4)^{2}\).
Given: \(x=3\), \(y=4\), \(n=2\)
$$(3 + j4)^{2} = \binom{2}{0}3^{2}(j4)^{0} + \binom{2}{1}3^{1}(j4)^{1} + \binom{2}{2}3^{0}(j4)^{2}$$
Calculate:
\(3^{2} = 9\)
\(j4 = j4\)
\((j4)^{2} = j^{2} 16 = -16\)
Substitute:
$$= 1 \times 9 \times 1 + 2 \times 3 \times j4 + 1 \times 1 \times (-16)$$
$$= 9 + j24 - 16$$
Combine real:
\(9 - 16 = -7\)
Answer: \(-7 + j24\)
Example 5: Expand \((1 + j1)^{3}\).
Given: \(x=1\), \(y=1\), \(n=3\)
$$(1 + j1)^{3} = \sum_{k=0}^{3} \binom{3}{k} 1^{3-k} (j1)^{k}$$
Calculate terms:
\(k=0\): \(1 \times 1 \times 1 = 1\)
\(k=1\): \(3 \times 1 \times j = 3j\)
\(k=2\): \(3 \times 1 \times j^{2} = 3 \times -1 = -3\)
\(k=3\): \(1 \times 1 \times j^{3} = j^{3} = -j\)
Sum:
$$1 + 3j - 3 - j = (1 - 3) + (3j, j) = -2 + 2j$$
Answer: \(-2 + 2j\)
Expansion of \( (x + jy)^{-n} \) involves reciprocal powers and can be done by expressing the number in polar form or using binomial expansion for negative exponents.
Example 1: Expand \((1 + j1)^{-1}\).
Given: \(z = 1 + j1\)
Step 1: Find magnitude and argument
$$r = \sqrt{1^{2} + 1^{2}} = \sqrt{2} = 1.414$$
Step 2: Find reciprocal magnitude
$$r^{-1} = \frac{1}{1.414} = 0.707$$
Step 3: Find argument
$$\theta = \tan^{-1}(\frac{1}{1}) = 45^{\circ}$$
Step 4: Use polar form for reciprocal
$$(1 + j1)^{-1} = 0.707 (\cos (-45^{\circ}) + j \sin (-45^{\circ}))$$
Step 5: Convert to rectangular form
$$x = 0.707 \cos (-45^{\circ}) = 0.707 \times 0.707 = 0.5$$
$$y = 0.707 \sin (-45^{\circ}) = 0.707 \times (-0.707) = -0.5$$
Answer: \(0.5 - j0.5\)
Example 2: Expand \((2 + j3)^{-1}\).
Given: \(z = 2 + j3\)
Step 1: Calculate magnitude
$$r = \sqrt{2^{2} + 3^{2}} = \sqrt{4 + 9} = \sqrt{13} = 3.606$$
Step 2: Reciprocal magnitude
$$r^{-1} = \frac{1}{3.606} = 0.277$$
Step 3: Argument
$$\theta = \tan^{-1}(\frac{3}{2}) = 56.31^{\circ}$$
Step 4: Reciprocal argument
$$-\theta = -56.31^{\circ}$$
Step 5: Polar form
$$(2 + j3)^{-1} = 0.277 (\cos (-56.31^{\circ}) + j \sin (-56.31^{\circ}))$$
Step 6: Rectangular form
$$x = 0.277 \times \cos (-56.31^{\circ}) = 0.277 \times 0.5547 = 0.154$$
$$y = 0.277 \times \sin (-56.31^{\circ}) = 0.277 \times (-0.8321) = -0.231$$
Answer: \(0.154 - j0.231\)
Example 3: Expand \((1 - j1)^{-2}\).
Given: \(z = 1 - j1\), \(n=2\)
Step 1: Find magnitude
$$r = \sqrt{1^{2} + (-1)^{2}} = \sqrt{2} = 1.414$$
Step 2: Reciprocal magnitude
$$r^{-2} = \frac{1}{1.414^{2}} = \frac{1}{2} = 0.5$$
Step 3: Argument
$$\theta = \tan^{-1}(\frac{-1}{1}) = -45^{\circ}$$
Step 4: Reciprocal argument
$$-2 \times \theta = -2 \times (-45^{\circ}) = 90^{\circ}$$
Step 5: Polar form
$$(1 - j1)^{-2} = 0.5 (\cos 90^{\circ} + j \sin 90^{\circ})$$
Step 6: Rectangular form
$$x = 0.5 \times \cos 90^{\circ} = 0.5 \times 0 = 0$$
$$y = 0.5 \times \sin 90^{\circ} = 0.5 \times 1 = 0.5$$
Answer: \(0 + j0.5\)
Pascal's triangle provides coefficients for the binomial expansion, useful in mechanical engineering calculations where powers of complex numbers arise.
Example 1: Using Pascal's triangle coefficients, expand \((1 + j1)^{3}\).
Coefficients for \(n=3\): 1, 3, 3, 1
Calculate terms:
$$1^{3} \times (j1)^{0} = 1$$
$$3 \times 1^{2} \times j1 = 3j$$
$$3 \times 1 \times (j1)^{2} = 3 \times j^{2} = -3$$
$$1 \times (j1)^{3} = j^{3} = -j$$
Sum:
$$1 + 3j - 3 - j = -2 + 2j$$
Answer: \(-2 + 2j\)
Example 2: Expand \((2 + j3)^{2}\) using Pascal's triangle coefficients (1, 2, 1).
Calculate terms:
$$2^{2} \times (j3)^{0} = 4$$
$$2 \times 2 \times j3 = 12j$$
$$1 \times (j3)^{2} = j^{2} \times 9 = -9$$
Sum:
$$4 + 12j - 9 = -5 + 12j$$
Answer: \(-5 + 12j\)
Example 3: Expand \((1 - j2)^{3}\) using Pascal's triangle (1, 3, 3, 1).
Calculate terms:
$$1^{3} \times (-j2)^{0} = 1$$
$$3 \times 1^{2} \times (-j2) = -6j$$
$$3 \times 1 \times (-j2)^{2} = 3 \times j^{2} \times 4 = -12$$
$$1 \times (-j2)^{3} = -j^{3} \times 8 = j \times 8 = 8j$$
Sum:
$$1 - 6j - 12 + 8j = (1 - 12) + (-6j + 8j) = -11 + 2j$$
Answer: \(-11 + 2j\)
Example 4: Expand \((3 + j1)^{3}\).
Coefficients: 1, 3, 3, 1
Calculate terms:
$$3^{3} \times (j1)^{0} = 27$$
$$3 \times 3^{2} \times j1 = 3 \times 9 \times j = 27j$$
$$3 \times 3 \times (j1)^{2} = 9 \times j^{2} = -9$$
$$1 \times (j1)^{3} = j^{3} = -j$$
Sum:
$$27 + 27j - 9 - j = (27 - 9) + (27j, j) = 18 + 26j$$
Answer: \(18 + 26j\)
Example 5: Expand \((1 + j3)^{4}\) using Pascal's triangle (1, 4, 6, 4, 1).
Calculate terms:
\(k=0\): \(1^{4} \times (j3)^{0} = 1\)
\(k=1\): \(4 \times 1^{3} \times j3 = 12j\)
\(k=2\): \(6 \times 1^{2} \times (j3)^{2} = 6 \times j^{2} \times 9 = -54\)
\(k=3\): \(4 \times 1 \times (j3)^{3} = 4 \times j^{3} \times 27 = 4 \times (-j) \times 27 = -108j\)
\(k=4\): \(1 \times (j3)^{4} = j^{4} \times 81 = 81\)
Sum:
$$1 + 12j - 54 - 108j + 81$$
Group real:
\(1 - 54 + 81 = 28\)
Group imaginary:
\(12j - 108j = -96j\)
Answer: \(28 - 96j\)
Finding the roots of complex numbers is essential in mechanical production technology, especially when dealing with vibrations, signal processing, and rotating machinery analysis. Roots allow us to determine multiple solutions to equations involving complex numbers, important for understanding system behaviors such as resonance frequencies or electrical circuit responses. The nth roots of a complex number can be found by expressing the number in polar form and applying De Moivre's theorem.
The general formula for the nth roots of a complex number \( z = r(\cos \theta + j \sin \theta) \) is:
$$ z_k = r^{\frac{1}{n}} ( \cos \frac{\theta + 2k\pi}{n} + j \sin \frac{\theta + 2k\pi}{n} ), \quad k = 0, 1, 2, \ldots, n-1 $$
where \( r \) is the modulus and \( \theta \) the argument of the complex number.
Example 1: Find the square roots of the complex number \( z = 4(\cos 60^\circ + j \sin 60^\circ) \).
Given: \( r = 4 \), \( \theta = 60^\circ \), \( n = 2 \).
$$ z_k = 4^{\frac{1}{2}} ( \cos \frac{60^\circ + 2k \times 180^\circ}{2} + j \sin \frac{60^\circ + 2k \times 180^\circ}{2} ) $$
For \( k=0 \):
$$ z_0 = 2 ( \cos 30^\circ + j \sin 30^\circ ) $$
$$ z_0 = 2 (0.866 + j \times 0.5) $$
$$ z_0 = 1.732 + j1.0 $$
For \( k=1 \):
$$ z_1 = 2 ( \cos 210^\circ + j \sin 210^\circ ) $$
$$ z_1 = 2 (-0.866 + j \times -0.5) $$
$$ z_1 = -1.732 - j1.0 $$
Answer: The square roots are \( 1.732 + j1.0 \) and \( -1.732 - j1.0 \).
Example 2: Determine the cube roots of \( z = 8(\cos 120^\circ + j \sin 120^\circ) \).
Given: \( r = 8 \), \( \theta = 120^\circ \), \( n = 3 \).
$$ z_k = 8^{\frac{1}{3}} ( \cos \frac{120^\circ + 2k \times 360^\circ}{3} + j \sin \frac{120^\circ + 2k \times 360^\circ}{3} ) $$
Calculate \( 8^{1/3} = 2 \).
For \( k=0 \):
$$ z_0 = 2 ( \cos 40^\circ + j \sin 40^\circ ) $$
$$ z_0 = 2 (0.766 + j \times 0.643) $$
$$ z_0 = 1.532 + j1.286 $$
For \( k=1 \):
$$ z_1 = 2 ( \cos 160^\circ + j \sin 160^\circ ) $$
$$ z_1 = 2 (-0.940 + j \times 0.342) $$
$$ z_1 = -1.880 + j0.684 $$
For \( k=2 \):
$$ z_2 = 2 ( \cos 280^\circ + j \sin 280^\circ ) $$
$$ z_2 = 2 (0.174 + j \times -0.985) $$
$$ z_2 = 0.348 - j1.970 $$
Answer: The cube roots are \( 1.532 + j1.286 \), \( -1.880 + j0.684 \), and \( 0.348 - j1.970 \).
Example 3: Find the fourth roots of \( z = 16(\cos 90^\circ + j \sin 90^\circ) \).
Given: \( r = 16 \), \( \theta = 90^\circ \), \( n = 4 \).
$$ z_k = 16^{\frac{1}{4}} ( \cos \frac{90^\circ + 2k \times 360^\circ}{4} + j \sin \frac{90^\circ + 2k \times 360^\circ}{4} ) $$
Calculate \( 16^{1/4} = 2 \).
For \( k=0 \):
$$ z_0 = 2 ( \cos 22.5^\circ + j \sin 22.5^\circ ) $$
$$ z_0 = 2 (0.924 + j \times 0.383) $$
$$ z_0 = 1.848 + j0.766 $$
For \( k=1 \):
$$ z_1 = 2 ( \cos 112.5^\circ + j \sin 112.5^\circ ) $$
$$ z_1 = 2 (-0.383 + j \times 0.924) $$
$$ z_1 = -0.766 + j1.848 $$
For \( k=2 \):
$$ z_2 = 2 ( \cos 202.5^\circ + j \sin 202.5^\circ ) $$
$$ z_2 = 2 (-0.924 + j \times -0.383) $$
$$ z_2 = -1.848 - j0.766 $$
For \( k=3 \):
$$ z_3 = 2 ( \cos 292.5^\circ + j \sin 292.5^\circ ) $$
$$ z_3 = 2 (0.383 + j \times -0.924) $$
$$ z_3 = 0.766 - j1.848 $$
Answer: The fourth roots are \( 1.848 + j0.766 \), \( -0.766 + j1.848 \), \( -1.848 - j0.766 \), and \( 0.766 - j1.848 \).
Example 4: A vibrating shaft in a milling machine has a complex amplitude represented by \( z = 25(\cos 150^\circ + j \sin 150^\circ) \). Find its square roots, which represent possible vibration modes.
Given: \( r=25 \), \( \theta=150^\circ \), \( n=2 \).
$$ z_k = 25^{\frac{1}{2}} ( \cos \frac{150^\circ + 2k \times 180^\circ}{2} + j \sin \frac{150^\circ + 2k \times 180^\circ}{2} ) $$
Calculate \( 25^{1/2} = 5 \).
For \( k=0 \):
$$ z_0 = 5 ( \cos 75^\circ + j \sin 75^\circ ) $$
$$ z_0 = 5 (0.259 + j \times 0.966) $$
$$ z_0 = 1.294 + j4.830 $$
For \( k=1 \):
$$ z_1 = 5 ( \cos 255^\circ + j \sin 255^\circ ) $$
$$ z_1 = 5 (-0.259 + j \times -0.966) $$
$$ z_1 = -1.294 - j4.830 $$
Answer: The square roots are \( 1.294 + j4.830 \) and \( -1.294 - j4.830 \).
Example 5: Find the cube roots of \( z = 27(\cos 0^\circ + j \sin 0^\circ) \), which could represent an electrical impedance in an AC motor.
Given: \( r=27 \), \( \theta=0^\circ \), \( n=3 \).
$$ z_k = 27^{\frac{1}{3}} ( \cos \frac{0^\circ + 2k \times 360^\circ}{3} + j \sin \frac{0^\circ + 2k \times 360^\circ}{3} ) $$
Calculate \( 27^{1/3} = 3 \).
For \( k=0 \):
$$ z_0 = 3 ( \cos 0^\circ + j \sin 0^\circ ) $$
$$ z_0 = 3 (1 + j \times 0) $$
$$ z_0 = 3 + j0 $$
For \( k=1 \):
$$ z_1 = 3 ( \cos 120^\circ + j \sin 120^\circ ) $$
$$ z_1 = 3 (-0.5 + j \times 0.866) $$
$$ z_1 = -1.5 + j2.598 $$
For \( k=2 \):
$$ z_2 = 3 ( \cos 240^\circ + j \sin 240^\circ ) $$
$$ z_2 = 3 (-0.5 + j \times -0.866) $$
$$ z_2 = -1.5 - j2.598 $$
Answer: The cube roots are \( 3 + j0 \), \( -1.5 + j2.598 \), and \( -1.5 - j2.598 \).
Trigonometric identities are fundamental in mechanical production, particularly in analyzing periodic motions and signal modulations. Complex numbers provide a powerful tool to derive and verify these identities efficiently by representing trigonometric functions in exponential or polar form. This approach simplifies manipulation of sines and cosines, which are common in mechanical vibrations, rotating machinery, and control systems.
Euler's formula relates complex exponentials to trigonometric functions:
$$ e^{j\theta} = \cos \theta + j \sin \theta $$
Using this, many trigonometric identities can be derived by algebraic manipulation of complex exponentials.
Euler's formula allows expressing sine and cosine in terms of complex exponentials:
$$ \cos \theta = \frac{e^{j\theta} + e^{-j\theta}}{2} $$
$$ \sin \theta = \frac{e^{j\theta} - e^{-j\theta}}{2j} $$
Example 1: Verify the identity \(\cos^2 \theta + \sin^2 \theta = 1\) using complex exponentials.
Given: Use Euler's formula expressions for sine and cosine.
Calculate:
$$ \cos^2 \theta + \sin^2 \theta = ( \frac{e^{j\theta} + e^{-j\theta}}{2} )^2 + ( \frac{e^{j\theta} - e^{-j\theta}}{2j} )^2 $$
Expand:
$$ = \frac{(e^{j\theta} + e^{-j\theta})^2}{4} + \frac{(e^{j\theta} - e^{-j\theta})^2}{-4} $$
Calculate numerator terms:
$$ (e^{j\theta} + e^{-j\theta})^2 = e^{2j\theta} + 2 + e^{-2j\theta} $$
$$ (e^{j\theta} - e^{-j\theta})^2 = e^{2j\theta} - 2 + e^{-2j\theta} $$
Substitute:
$$ = \frac{e^{2j\theta} + 2 + e^{-2j\theta}}{4} - \frac{e^{2j\theta} - 2 + e^{-2j\theta}}{4} $$
Simplify numerator:
$$ = \frac{e^{2j\theta} + 2 + e^{-2j\theta} - e^{2j\theta} + 2 - e^{-2j\theta}}{4} $$
$$ = \frac{4}{4} = 1 $$
Answer: Verified that \( \cos^2 \theta + \sin^2 \theta = 1 \).
Example 2: Derive the formula for \(\cos (a + b)\) using complex exponentials.
Given: Use the expression \( e^{j(a+b)} = e^{ja} e^{jb} \).
Write:
$$ e^{j(a+b)} = \cos (a+b) + j \sin (a+b) $$
Also,
$$ e^{j(a+b)} = e^{ja} e^{jb} = (\cos a + j \sin a)(\cos b + j \sin b) $$
Multiply:
$$ = \cos a \cos b + j \cos a \sin b + j \sin a \cos b + j^2 \sin a \sin b $$
Recall \( j^2 = -1 \), so:
$$ = \cos a \cos b + j (\cos a \sin b + \sin a \cos b) - \sin a \sin b $$
Equate real parts:
$$ \cos (a + b) = \cos a \cos b - \sin a \sin b $$
Answer: Derived \( \cos (a + b) = \cos a \cos b - \sin a \sin b \).
Example 3: Derive the formula for \(\sin (a + b)\) using complex exponentials.
Given: Using the imaginary parts of the same product as above.
From
$$ e^{j(a+b)} = (\cos a + j \sin a)(\cos b + j \sin b) $$
Imaginary part is:
$$ \sin (a + b) = \cos a \sin b + \sin a \cos b $$
Answer: Derived \( \sin (a + b) = \cos a \sin b + \sin a \cos b \).
Example 4: Use complex numbers to prove the double-angle formula for cosine: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).
Given: \( e^{j2\theta} = (e^{j\theta})^2 = (\cos \theta + j \sin \theta)^2 \).
Expand:
$$ = \cos^2 \theta + 2j \cos \theta \sin \theta + j^2 \sin^2 \theta $$
Recall \( j^2 = -1 \):
$$ = \cos^2 \theta + 2j \cos \theta \sin \theta - \sin^2 \theta $$
Real part of \( e^{j2\theta} \) is \( \cos 2\theta \):
$$ \cos 2\theta = \cos^2 \theta - \sin^2 \theta $$
Answer: Verified \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).
Example 5: Prove the double-angle formula for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \).
Given: Imaginary part of \( e^{j2\theta} = (\cos \theta + j \sin \theta)^2 \) is:
$$ 2 \cos \theta \sin \theta $$
$$ \sin 2\theta = 2 \sin \theta \cos \theta $$
Answer: Verified \( \sin 2\theta = 2 \sin \theta \cos \theta \).
Product-to-sum formulas transform products of sine and cosine into sums for easier integration and signal analysis. Complex exponentials simplify these derivations.
Recall:
$$ \cos a \cos b = \frac{1}{2} [\cos (a+b) + \cos (a-b)] $$
$$ \sin a \sin b = \frac{1}{2} [\cos (a-b) - \cos (a+b)] $$
$$ \sin a \cos b = \frac{1}{2} [\sin (a+b) + \sin (a-b)] $$
Example 1: Derive \( \cos a \cos b = \frac{1}{2} [\cos (a+b) + \cos (a-b)] \).
Given: Express cosines as complex exponentials:
$$ \cos a = \frac{e^{ja} + e^{-ja}}{2}, \quad \cos b = \frac{e^{jb} + e^{-jb}}{2} $$
Multiply:
$$ \cos a \cos b = \frac{(e^{ja} + e^{-ja})(e^{jb} + e^{-jb})}{4} $$
Expand:
$$ = \frac{e^{j(a+b)} + e^{j(a-b)} + e^{-j(a-b)} + e^{-j(a+b)}}{4} $$
Group terms:
$$ = \frac{1}{4} ( e^{j(a+b)} + e^{-j(a+b)} + e^{j(a-b)} + e^{-j(a-b)} ) $$
Recall \( e^{j\alpha} + e^{-j\alpha} = 2 \cos \alpha \):
$$ = \frac{1}{4} ( 2 \cos (a+b) + 2 \cos (a-b) ) $$
$$ = \frac{1}{2} [\cos (a+b) + \cos (a-b)] $$
Answer: Derived \( \cos a \cos b = \frac{1}{2} [\cos (a+b) + \cos (a-b)] \).
Example 2: Derive \( \sin a \sin b = \frac{1}{2} [\cos (a-b) - \cos (a+b)] \).
Given: Express sines as complex exponentials:
$$ \sin a = \frac{e^{ja} - e^{-ja}}{2j}, \quad \sin b = \frac{e^{jb} - e^{-jb}}{2j} $$
Multiply:
$$ \sin a \sin b = \frac{(e^{ja} - e^{-ja})(e^{jb} - e^{-jb})}{4(-1)} $$
Since \( j \times j = -1 \), denominator is 4(-1) = -4.
Expand numerator:
$$ = \frac{e^{j(a+b)} - e^{j(a-b)} - e^{-j(a-b)} + e^{-j(a+b)}}{-4} $$
Group:
$$ = \frac{ - (e^{j(a+b)} + e^{-j(a+b)}) + (e^{j(a-b)} + e^{-j(a-b)}) }{4} $$
Using \( e^{j\alpha} + e^{-j\alpha} = 2 \cos \alpha \):
$$ = \frac{-2 \cos (a+b) + 2 \cos (a-b)}{4} $$
Simplify:
$$ = \frac{1}{2} [\cos (a-b) - \cos (a+b)] $$
Answer: Derived \( \sin a \sin b = \frac{1}{2} [\cos (a-b) - \cos (a+b)] \).
Example 3: Derive \( \sin a \cos b = \frac{1}{2} [\sin (a+b) + \sin (a-b)] \).
Given:
$$ \sin a = \frac{e^{ja} - e^{-ja}}{2j}, \quad \cos b = \frac{e^{jb} + e^{-jb}}{2} $$
Multiply:
$$ \sin a \cos b = \frac{(e^{ja} - e^{-ja})(e^{jb} + e^{-jb})}{4j} $$
Expand numerator:
$$ = \frac{e^{j(a+b)} + e^{j(a-b)} - e^{-j(a-b)} - e^{-j(a+b)}}{4j} $$
Group:
$$ = \frac{(e^{j(a+b)} - e^{-j(a+b)}) + (e^{j(a-b)} - e^{-j(a-b)})}{4j} $$
Recall \( e^{j\alpha} - e^{-j\alpha} = 2j \sin \alpha \):
$$ = \frac{2j \sin (a+b) + 2j \sin (a-b)}{4j} $$
Simplify:
$$ = \frac{2j [\sin (a+b) + \sin (a-b)]}{4j} = \frac{1}{2} [\sin (a+b) + \sin (a-b)] $$
Answer: Derived \( \sin a \cos b = \frac{1}{2} [\sin (a+b) + \sin (a-b)] \).
Example 4: Calculate \( \cos 75^\circ \cos 15^\circ \) using the product-to-sum formula.
Given:
$$ \cos 75^\circ \cos 15^\circ = \frac{1}{2} [\cos (75^\circ + 15^\circ) + \cos (75^\circ - 15^\circ)] $$
Calculate angles:
$$ = \frac{1}{2} [\cos 90^\circ + \cos 60^\circ] $$
Use trigonometric values:
$$ = \frac{1}{2} [0 + 0.5] = \frac{1}{2} \times 0.5 = 0.25 $$
Answer: \( \cos 75^\circ \cos 15^\circ = 0.25 \).
Example 5: Calculate \( \sin 40^\circ \sin 20^\circ \) using the product-to-sum formula.
Given:
$$ \sin 40^\circ \sin 20^\circ = \frac{1}{2} [\cos (40^\circ - 20^\circ) - \cos (40^\circ + 20^\circ)] $$
Calculate angles:
$$ = \frac{1}{2} [\cos 20^\circ - \cos 60^\circ] $$
Use trigonometric values:
$$ = \frac{1}{2} [0.940 - 0.5] = \frac{1}{2} \times 0.44 = 0.22 $$
Answer: \( \sin 40^\circ \sin 20^\circ = 0.22 \).
This chapter began by exploring the fundamental components of complex numbers, including the real part and the imaginary part, and their representation on the Argand diagram. It introduced the modulus or magnitude of a complex number, which measures its distance from the origin, and the argument or angle, describing its direction relative to the positive real axis. The concept of conjugate operations was also covered, showing how to find the conjugate of a complex number and its significance. The chapter then detailed arithmetic operations with complex numbers, including addition, subtraction, multiplication, and division, providing methods to combine and manipulate these numbers effectively. It explained how to convert complex numbers between rectangular and polar forms, emphasizing the importance of each for different applications, and introduced De Moivre's theorem as a powerful tool for raising complex numbers to powers. Finally, the chapter examined the expansion of complex numbers, finding roots of complex numbers, and applying trigonometric identities using complex numbers, thus linking algebraic and trigonometric perspectives in engineering mathematics.
Find the real part and imaginary part of the complex number \( z = 7 + 4i \). (2 marks)
Plot the complex number \( z = -3 + 5i \) on an Argand diagram and determine its modulus. (3 marks)
Calculate the argument (angle) of the complex number \( z = 1 - \sqrt{3}i \). Express your answer in degrees. (3 marks)
Find the conjugate of the complex number \( z = 6 - 2i \) and verify that \( z \times \overline{z} \) equals the square of the modulus of \( z \). (4 marks)
Perform the addition and subtraction of the complex numbers \( z_1 = 3 + 2i \) and \( z_2 = -1 + 4i \). (4 marks)
Multiply the complex numbers \( z_1 = 2 + 3i \) and \( z_2 = 4 - i \). Express your answer in rectangular form. (4 marks)
Divide \( z_1 = 5 + 2i \) by \( z_2 = 1 - 3i \). Express the result in rectangular form. (5 marks)
Convert the complex number \( z = 6 (\cos 60^\circ + i \sin 60^\circ) \) to rectangular form. (3 marks)
Convert the complex number \( z = -3 + 3\sqrt{3}i \) to polar form, giving modulus and argument in degrees. (4 marks)
Using De Moivre's theorem, calculate \( (1 + i)^4 \) and express the result in rectangular form. (5 marks)
1. Real and Imaginary parts of \( z = 7 + 4i \)
Given: \( z = 7 + 4i \)
Answer: Real part = 7, Imaginary part = 4
2. Plot \( z = -3 + 5i \) and find modulus
Given: \( z = -3 + 5i \)
Modulus formula:$$ |z| = \sqrt{(-3)^2 + 5^2} $$$$ |z| = \sqrt{9 + 25} $$$$ |z| = \sqrt{34} $$$$ |z| \approx 5.83 $$
Answer: Modulus \(\approx 5.83\)
3. Argument of \( z = 1 - \sqrt{3}i \)
Given: \( z = 1 - \sqrt{3}i \)
Calculate argument \( \theta = \tan^{-1} (\frac{\operatorname{Im}(z)}{\operatorname{Re}(z)}) \)$$ \theta = \tan^{-1} (\frac{-\sqrt{3}}{1}) $$$$ \theta = \tan^{-1}(-\sqrt{3}) $$$$ \theta = -60^\circ $$
Since \( z \) lies in the 4th quadrant, argument = \( 360^\circ - 60^\circ = 300^\circ \)
Answer: Argument \( = 300^\circ \)
4. Conjugate and verification
Given: \( z = 6 - 2i \)
Conjugate:$$ \overline{z} = 6 + 2i $$
Calculate:$$ z \times \overline{z} = (6 - 2i)(6 + 2i) $$$$ = 6^2 + 6 \times 2i - 2i \times 6 - (2i)^2 $$$$ = 36 + 12i - 12i - 4i^2 $$$$ = 36 - 4(-1) $$$$ = 36 + 4 = 40 $$
Calculate modulus squared:$$ |z|^2 = 6^2 + (-2)^2 = 36 + 4 = 40 $$
Answer: Conjugate \(= 6 + 2i\), and \( z \times \overline{z} = |z|^2 = 40 \)
5. Addition and subtraction
Given:
\( z_1 = 3 + 2i \)
\( z_2 = -1 + 4i \)
Addition:$$ z_1 + z_2 = (3 - 1) + (2 + 4)i = 2 + 6i $$
Subtraction:$$ z_1 - z_2 = (3 - (-1)) + (2 - 4)i = 4 - 2i $$
Answer:
Addition = \( 2 + 6i \)
Subtraction = \( 4 - 2i \)
6. Multiplication
Given:
\( z_1 = 2 + 3i \)
\( z_2 = 4 - i \)
Multiply:$$ z_1 \times z_2 = (2)(4) + (2)(-i) + (3i)(4) + (3i)(-i) $$$$ = 8 - 2i + 12i - 3i^2 $$$$ = 8 + 10i - 3(-1) $$$$ = 8 + 10i + 3 = 11 + 10i $$
Answer: \( 11 + 10i \)
7. Division
Given:
\( z_1 = 5 + 2i \)
\( z_2 = 1 - 3i \)
Divide:$$ \frac{z_1}{z_2} = \frac{5 + 2i}{1 - 3i} \times \frac{1 + 3i}{1 + 3i} = \frac{(5 + 2i)(1 + 3i)}{1^2 + 3^2} $$
Calculate numerator:$$ (5)(1) + (5)(3i) + (2i)(1) + (2i)(3i) = 5 + 15i + 2i + 6i^2 $$$$ = 5 + 17i + 6(-1) = 5 + 17i - 6 = -1 + 17i $$
Denominator:$$ 1 + 9 = 10 $$
Final result:$$ \frac{-1 + 17i}{10} = -0.1 + 1.7i $$
Answer: \( -0.1 + 1.7i \)
8. Convert polar to rectangular
Given:$$ z = 6(\cos 60^\circ + i \sin 60^\circ) $$
Calculate:$$ x = 6 \cos 60^\circ = 6 \times 0.5 = 3 $$$$ y = 6 \sin 60^\circ = 6 \times 0.866 = 5.196 $$
Answer: \( z = 3 + 5.196i \)
9. Convert rectangular to polar
Given:
\( z = -3 + 3\sqrt{3}i \)
Calculate modulus:$$ |z| = \sqrt{(-3)^2 + (3\sqrt{3})^2} = \sqrt{9 + 27} = \sqrt{36} = 6 $$
Calculate argument:$$ \theta = \tan^{-1} (\frac{3\sqrt{3}}{-3}) = \tan^{-1}(-\sqrt{3}) = -60^\circ $$
Since \( x < 0 \), \( y > 0 \), \( z \) is in the second quadrant, so$$ \theta = 180^\circ - 60^\circ = 120^\circ $$
Answer:
Modulus = 6, Argument = \( 120^\circ \)
10. De Moivre's theorem: \( (1 + i)^4 \)
Convert \( 1 + i \) to polar form:
Modulus:$$ r = \sqrt{1^2 + 1^2} = \sqrt{2} $$
Argument:$$ \theta = \tan^{-1} (\frac{1}{1}) = 45^\circ = \frac{\pi}{4} \text{ radians} $$
Apply De Moivre's theorem:$$ (r(\cos \theta + i \sin \theta))^4 = r^4 (\cos 4\theta + i \sin 4\theta ) $$$$ = (\sqrt{2})^4 (\cos (4 \times 45^\circ) + i \sin (4 \times 45^\circ) ) $$$$ = (2^2) (\cos 180^\circ + i \sin 180^\circ ) = 4(-1 + 0i) = -4 $$
Answer: \( -4 + 0i \)
Question 11 (Compulsory - 20 marks)
A mechanical production technician at Kenyatta National Hospital is analysing vibration data represented by complex numbers to diagnose equipment faults. The vibration vector is given as \(z = 7 (\cos 45^\circ + i \sin 45^\circ)\).
a) Convert \(z\) from polar form to rectangular form. (10 marks)
b) Using De Moivre’s theorem, find \(z^4\) and express your answer in rectangular form. (10 marks)
Question 12 (20 marks)
Given the complex number \(z = -1 + i\) used in control system feedback at a SACCO’s automated teller machine, find all the cube roots of \(z\). Show your steps using trigonometric form and express the roots in rectangular form.
Question 13 (20 marks)
A production line motor’s impedance is modelled by the complex number \(Z = 4 - 3i\).
a) Find the modulus and argument of \(Z\). (10 marks)
b) Calculate the conjugate of \(Z\) and explain its importance in power factor correction. (10 marks)
Question 14 (20 marks)
Using complex numbers, prove the trigonometric identity:
\[
\cos 3\theta = 4\cos^3 \theta - 3\cos \theta
\]
by expanding \((\cos \theta + i \sin \theta)^3\) and equating real parts. Show all steps clearly.
Question 11
a) Given \(z = 7(\cos 45^\circ + i \sin 45^\circ)\),
\[
\cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}
\]
Rectangular form:
\[
z = 7 \times \frac{\sqrt{2}}{2} + i \times 7 \times \frac{\sqrt{2}}{2} = \frac{7\sqrt{2}}{2} + i \frac{7\sqrt{2}}{2}
\]
Numerically,
\[
\frac{7 \times 1.414}{2} = 4.95
\]
\[
z = 4.95 + 4.95i
\]
b) Using De Moivre’s theorem:
\[
z^4 = 7^4 (\cos 4 \times 45^\circ + i \sin 4 \times 45^\circ ) = 2401 (\cos 180^\circ + i \sin 180^\circ)
\]
\[
\cos 180^\circ = -1, \quad \sin 180^\circ = 0
\]
\[
z^4 = 2401 \times (-1 + 0i) = -2401 + 0i
\]
Question 12
Given \(z = -1 + i\). First convert to polar form:
\[
r = \sqrt{(-1)^2 + 1^2} = \sqrt{1 +1} = \sqrt{2}
\]
\[
\theta = \tan^{-1}(\frac{1}{-1}) = \tan^{-1}(-1)
\]
Since real part negative and imaginary positive, \(\theta = 135^\circ\) or \(\frac{3\pi}{4}\) radians.
Cube roots:
\[
z^{1/3} = r^{1/3} (\cos \frac{\theta + 2k\pi}{3} + i \sin \frac{\theta + 2k\pi}{3}), \quad k=0,1,2
\]
Calculate:
\[
r^{1/3} = (\sqrt{2})^{1/3} = 2^{1/6}
\]
For each root:
Express roots in rectangular form by evaluating cos and sin at each angle and multiply by \(2^{1/6}\approx 1.122\).
Question 13
a) Modulus:
\[
|Z| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
Argument:
\[
\theta = \tan^{-1}(\frac{-3}{4}) = -36.87^\circ
\]
b) Conjugate of \(Z\) is \(4 + 3i\). The conjugate is used in power factor correction to cancel out the reactive component (imaginary) of impedance, improving efficiency.
Question 14
Expand:
\[
(\cos \theta + i \sin \theta)^3 = \cos 3\theta + i \sin 3\theta
\]
Using binomial expansion:
\[
(\cos \theta)^3 + 3(\cos \theta)^2 (i \sin \theta) + 3(\cos \theta)(i \sin \theta)^2 + (i \sin \theta)^3
\]
Simplify powers of \(i\):
\[
= \cos^3 \theta + 3i \cos^2 \theta \sin \theta - 3 \cos \theta \sin^2 \theta, i \sin^3 \theta
\]
Group real and imaginary parts:
Real part:
\[
\cos^3 \theta - 3 \cos \theta \sin^2 \theta
\]
Imaginary part:
\[
3 \cos^2 \theta \sin \theta - \sin^3 \theta
\]
Equate real parts:
\[
\cos 3\theta = \cos^3 \theta - 3 \cos \theta \sin^2 \theta
\]
Use \(\sin^2 \theta = 1 - \cos^2 \theta\):
\[
\cos 3\theta = \cos^3 \theta - 3 \cos \theta (1 - \cos^2 \theta) = \cos^3 \theta - 3 \cos \theta + 3 \cos^3 \theta = 4 \cos^3 \theta - 3 \cos \theta
\]
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