The Magnitude of a Complex Number
Let's dive into the world of complex numbers. In particular, we're going to tackle the concept of magnitude, also known as modulus or absolute value, for a complex number. When you think about the magnitude of a complex number, visualize it as the distance from the origin, or point O, to a point that the complex number z=(a,b) represents on the complex plane.
This magnitude is usually symbolized as |z|.
A complex number has an algebraic representation, z = a + bi, where 'a' stands for the real part and 'b' refers to the imaginary part.
$$ z = a+bi $$
You can compute the magnitude of a complex number with Pythagoras' theorem:
$$ |z| = \sqrt{ a^2 + b^2 } $$
Put simply, the magnitude |z| of a complex number z acts as the hypotenuse of the right triangle, which is formed by the segments that represent the real and imaginary parts of the complex number z in the complex plane.
Remember, within the complex (or Gauss) plane, the complex number z=a+bi is represented by the point (a,b).
The "a" and "b" coefficients of the real and imaginary parts determine the point's coordinates.
Why do we care about the magnitude of a complex number, you may ask? The magnitude of a complex number is a cornerstone concept within the realm of complex numbers. It's an invaluable tool, both for grasping the theoretical understanding of complex numbers and for their practical applications in various domains such as electrical engineering, computer science, physics, and more.
If this snapshot overview hasn't entirely demystified things for you, fear not.
In the following section, we're going to unpack what the magnitude of a complex number is, and how it's calculated - in a straightforward, step-by-step manner.
A more detailed, step-by-step, simpler explanation
Complex numbers, despite their intimidating name, don't have to be complex. When understood, they can become a powerful tool in the realms of mathematics, physics, and engineering.
One of the key concepts when dealing with complex numbers is understanding their magnitude.
First up, what exactly is a complex number?
In simplest terms, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit with a property of i2 = -1.
$$ z = a + bi $$
In this equation, "a" signifies the real part of the complex number, while "bi" denotes the imaginary part.
$$ Re(z) = a $$
$$ Im(z) = bi $$
For example, let's consider the complex number z=4+3i.
$$ z = 4 + 3i $$
In this case, the real part of the complex number is Re(z)=4, and the imaginary part is Im(z)=3i.
$$ Re(z) = 4 $$
$$ Im(z) = 3i $$
So, in our example, the coefficients for the real and imaginary parts of the complex number are a=4 and b=3, respectively.
We represent complex numbers on a two-dimensional plane, known as the complex plane or Argand-Gauss plane.
For instance, the complex number z=4+3i is represented by the point (4,3) on this plane.
A typical complex number a + bi can be visualized as a point (a,b) on the Gauss plane, with the horizontal axis representing the real part and the vertical axis indicating the imaginary part. In essence, every point on the Gauss plane corresponds to a complex number, and vice versa.
The magnitude of a complex number, represented as |z| if our complex number is z, is the distance between the origin O and the point (4,3).
This magnitude is also the length of the vector that originates from the origin and ends at the point (a,b)=(4,3) on the complex plane, with "a" being the coefficient of the real part of the complex number and "b" being the coefficient of the imaginary part.
You'll notice that the point and the origin form a right triangle
The hypotenuse of this triangle corresponds to the magnitude of the complex number |z|.
Therefore, to calculate the magnitude of a complex number z = a + bi, you can use the formula from Pythagoras' theorem.
$$ |z| = \sqrt{ a^2 + b^2 } $$
This is why we use Pythagoras' theorem to calculate the magnitude of a complex number.
In our example, the coefficients of the real and imaginary parts of the complex number z are a=4 and b=3.
Consequently, the magnitude of the complex number is 5.
$$ |z| = \sqrt{ a^2 + b^2 } = \sqrt{ 4^2 + 3^2 } = \sqrt{ 16 + 9 } = \sqrt{ 25 } = 5 $$
This represents the length of the segment (or vector) that joins the origin O to the point (4,3) of the complex number on the Gauss plane.
So, to wrap it all up, the magnitude of the complex number z=4+3i equals |z|=5.
