Complex Numbers: A Comprehensive Recap
Complex numbers, expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (satisfying i2 = -1), find their place in a wide array of mathematical and real-world applications. The real part, a, and the imaginary part, b, are graphically represented on the complex plane, with a and b dictating the horizontal and vertical positions, respectively.
Graphical Representation
- Real Axis: The horizontal axis, representing the real part of a complex number.
- Imaginary Axis: The vertical axis, representing the imaginary part.
The point P(a, b) on the complex plane represents the complex number z = a + bi.
Modulus of a Complex Number
Definition and Formula
The modulus of a complex number, symbolised as |z|, is essentially the Euclidean distance from the origin (0,0) to the point P representing the complex number on the complex plane. Mathematically, if z = a + bi, the modulus is calculated as: |z| = sqrt(a2 + b2)
Geometric and Algebraic Insights
- The modulus gives a scalar measure of the "magnitude" or "size" of a complex number.
- It is always non-negative and is zero if and only if the complex number is zero.
- Algebraically, it provides a means to compare the sizes of two complex numbers.
IB Maths Tutor Tip: Understanding the trigonometric form of complex numbers illuminates the geometric relationship between complex multiplication and rotation, enhancing comprehension of complex number operations and their applications in real-world scenarios.
Example 1: Calculating Modulus
Consider z = 3 + 4i. The modulus is: |z| = sqrt(32 + 42) = 5
Argument of a Complex Number
Definition and Formula
The argument of a complex number, denoted as arg(z), is the angle theta formed by the line segment OP (where O is the origin and P represents the complex number) with the positive real axis. For z = a + bi, the argument can be found using: theta = atan2(b, a)
Principal Argument
The principal argument, Arg(z), is confined to a specific interval, typically -pi < theta <= pi, ensuring a unique value for every complex number.
Geometric and Algebraic Interpretations
- The argument provides the "direction" or "orientation" of the complex number.
- It is pivotal in understanding the rotational aspect of complex numbers in operations like multiplication.
Example 2: Finding Argument
For z = 1 + sqrt(3)i, the argument is: theta = atan2(sqrt(3), 1) theta = pi/3
IB Tutor Advice: Practise converting complex numbers between rectangular and trigonometric forms to master their interrelations and simplify complex calculations, especially useful for solving problems in multiplication, division, and finding powers or roots.
Trigonometric Form of Complex Numbers
Derivation and Formula
A complex number z = a + bi can be expressed in trigonometric form as: z = r(cos(theta) + i sin(theta)) where r = |z| and theta is the argument of z.
Euler's Formula
Euler's Formula, e(itheta) = cos(theta) + i sin(theta), allows us to express the trigonometric form of z as: z = r e(itheta)
Applications and Significance
- Multiplication and Division: The trigonometric form simplifies these operations, reducing them to addition and subtraction of angles and multiplication and division of moduli.
- Powers and Roots: Finding powers and roots becomes algebraically simpler using De Moivre’s Theorem.
Example 3: Expressing in Trigonometric Form
For z = 3 + 4i, given that |z| = 5 and theta = atan2(4, 3), z = 5(cos(theta) + i sin(theta))
FAQ
Euler’s Formula, expressed as e(iθ) = cos(θ) + i sin(θ), provides a profound connection between exponential and trigonometric functions in the context of complex numbers. When a complex number is expressed in its trigonometric form, z = r(cos(θ) + i sin(θ)), Euler’s Formula allows us to rewrite it as z = r e(iθ). This form is particularly useful in simplifying operations such as multiplication, division, and exponentiation of complex numbers, as it allows for easy manipulation of the moduli and arguments, thereby providing a seamless bridge between algebraic and geometric interpretations of complex numbers.
No, a complex number cannot have a negative modulus. The modulus of a complex number, defined as the square root of the sum of the squares of its real and imaginary parts, is always non-negative. Mathematically, if z = a + bi, then |z| = sqrt(a2 + b2), which is always greater than or equal to zero. The modulus represents the distance of the point representing the complex number from the origin in the complex plane, and distance cannot be negative. Thus, the modulus of a complex number is always a non-negative real number.
The principal argument is significant in ensuring consistency and uniformity in mathematical communication and computation. When performing operations like multiplication and division of complex numbers in trigonometric form, the arguments of the numbers are added or subtracted. The principal argument ensures that the resulting argument of the operation remains within a standardised interval, typically (-π, π]. This avoids ambiguity and maintains a standard form, making mathematical work more coherent and interpretable across various contexts and applications, thereby avoiding potential misinterpretations or mathematical errors.
The argument of a complex number is restricted to a specific interval to ensure uniqueness. Without such a restriction, the argument could have infinitely many values, differing by multiples of 2π, which could lead to ambiguity. The typical interval chosen is (-π, π] because it symmetrically surrounds the real axis, providing a balanced perspective in both the positive and negative imaginary directions. This choice is somewhat arbitrary and is made for the sake of mathematical consistency and convention, ensuring that computations and communications among mathematicians and scientists are coherent and unified.
In electrical engineering, the trigonometric form of complex numbers is pivotal in analysing AC circuits. The voltage and current in an AC circuit can be represented as phasors, which are complex numbers. The modulus of the complex number represents the amplitude of the wave, while the argument represents the phase shift. When multiplying phasors (complex numbers), engineers can easily determine the resultant wave's amplitude and phase shift by multiplying the amplitudes and adding the phase shifts, respectively. This simplifies the analysis of AC circuits, particularly when dealing with impedance and calculating the voltage and current across circuit elements.
Practice Questions
The trigonometric form of a complex number z = a + bi is given by z = r(cos(theta) + i sin(theta)), where r is the modulus of z and theta is the argument of z. The modulus r is calculated as r = sqrt(a2 + b2) and the argument theta can be found using theta = atan2(b, a). For z = 4 + 3i, we find that r = sqrt(42 + 32) = 5 and theta = atan2(3, 4). Therefore, the trigonometric form of z is z = 5(cos(theta) + i sin(theta)).
To find the product of two complex numbers in trigonometric form, we multiply their moduli and add their arguments. If z1 = r1(cos(theta1) + i sin(theta1)) and z2 = r2(cos(theta2) + i sin(theta2)), then the product z1 x z2 is given by z1 x z2 = r1 x r2 (cos(theta1 + theta2) + i sin(theta1 + theta2)). For z1 = 3(cos(pi/4) + i sin(pi/4)) and z2 = 4(cos(pi/6) + i sin(pi/6)), we have r1 = 3, r2 = 4, theta1 = pi/4, and theta2 = pi/6. Therefore, z1 x z2 = 12(cos(pi/4 + pi/6) + i sin(pi/4 + pi/6)) = 12(cos(5pi/12) + i sin(5pi/12)).
Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.