Question

Multiple Choice If \(z=x+y i\) is a complex number, then the magnitude of \(z\) is: (a) \(x^{2}+y^{2}\) (b) \(|x|+|y|\) (c) \(\sqrt{x^{2}+y^{2}}\) (d) \(\sqrt{|x|+|y|}\)

Short answer

(c)

Step-by-step solution

Understand the given complex number

The complex number is given as \( z = x + yi \), where \(x\) is the real part and \( y \) is the imaginary part.

Recall the formula for the magnitude of a complex number

The magnitude (or modulus) of a complex number \( z = x + yi \) is given by the formula \( |z| = \sqrt{x^2 + y^2} \).

Match the formula with the given options

Comparing the calculated result with the given options, the correct choice is (c) \( \sqrt{x^2 + y^2} \).

Key concepts

complex number

A complex number is a number that combines both a real part and an imaginary part. It is usually written in the form of \((z = x + yi)\), where \((x)\) is the real part and \((y)\) is the imaginary part, and \((i)\) is the imaginary unit satisfying the equation \((i^2 = -1)\).
Complex numbers are essential in various fields of science and engineering because they offer a way to work with two-dimensional quantities.
For example, in electrical engineering, complex numbers are used to analyze AC circuits. Their unique properties allow for representation and manipulation of rotations and oscillations, which is crucial in many applications.

magnitude formula

The magnitude formula of a complex number provides the length or size of the number on the complex plane.
For a complex number \((z = x + yi)\), the magnitude, denoted by \((|z|)\), is calculated using the formula:
\[ |z| = \sqrt{x^2 + y^2} \]
This formula arises from the Pythagorean theorem and essentially measures the distance from the origin to the point \((x, y)\) in the complex plane.
It’s important to note that the magnitude is always a non-negative real number, even if the components \((x)\) and \((y)\) are negative.
The magnitude tells us how 'big' the complex number is, irrespective of its direction.

modulus of complex number

The modulus of a complex number is another term for its magnitude.
Mathematicians and engineers often use the terms interchangeably.
The modulus \( |z| \) of a complex number provides critical information about its absolute size and is widely used in solving mathematical equations and in signal processing.
To find the modulus, follow these steps:

• Identify the real part \((x)\) and the imaginary part \((y)\) of the complex number \((z = x + yi)\).
• Square both the real and imaginary parts: \((x^2)\) and \((y^2)\).
• Add the squares: \((x^2 + y^2)\).
• Take the square root of the sum: \((|z| = \sqrt{x^2 + y^2})\).

The modulus is helpful in many theoretical and practical applications, such as determining the amplitude of a wave in physics or the magnitude of a vector in linear algebra.

complex plane

The complex plane is a two-dimensional plane used to represent complex numbers graphically.
In the complex plane, the horizontal axis (x-axis) represents the real part of the complex number, while the vertical axis (y-axis) represents the imaginary part.
Each complex number \((z = x + yi)\) is represented as a point \((x, y)\) in the complex plane.
This graphical representation allows for easy visualization of complex number operations, such as addition, subtraction, and finding the modulus or magnitude.
For instance, the magnitude of \((z)\) is the distance from the origin \((0, 0)\) to the point \((x, y)\).
The complex plane also helps in understanding more advanced concepts like Euler's formula and complex functions.
Overall, it is an indispensable tool for anyone working with complex numbers, whether in mathematics, physics, or engineering.