Problem 5 - Algebra and Trigonometry

Question

Graph the complex number and find its modulus. $$ 4 i $$

Step-by-Step Solution

Answer

The modulus of the complex number \( 4i \) is 4.

Step 1: Understanding the Complex Number

The given complex number is purely imaginary since it doesn't have a real part. It can be expressed as: \( 0 + 4i \). The real part is 0, and the imaginary part is 4.

Step 2: Plotting the Complex Number

To plot the complex number on the Argand plane, locate the point at (0, 4) since the real part is 0 and the imaginary part is 4. This point is located on the imaginary axis, 4 units above the origin.

Step 3: Calculating the Modulus

The modulus of a complex number \( a + bi \) is found using the formula: \( \sqrt{a^2 + b^2} \). For our complex number \( 0 + 4i \), \( a = 0 \) and \( b = 4 \). Therefore, the modulus is: \( \sqrt{0^2 + 4^2} = 4 \).

Key Concepts

Imaginary Numbers

Imaginary numbers are quite interesting, as they extend the concept of what we typically consider a number. They are signified by the unit 'i'. This unit 'i' equates to the square root of -1. That might sound strange because, in the real number system, there is no real number whose square is negative. But in the realm of complex numbers, it makes perfect sense.
Imagine you have a number line. On it, you'd have negative, zero, and positive numbers. Imaginary numbers are like the backstage or the behind-the-scenes of this number line, making what appears impossible, possible.

• Any complex number can be composed of a real part and an imaginary part.
• The imaginary number 'i' is essential for transforming real numbers into something much larger and amazing.

If you find yourself facing a number like '4i', it means you're looking at an imaginary number because there is no real part involved in this number. It's subtly magical as it stands upright on the imaginary axis.

Argand Plane

The Argand plane is a fantastic way to visualize complex numbers. Imagine it as a regular coordinate plane, like the one used for usual Cartesian coordinates, but designed specifically for complex numbers.
On the Argand plane, the horizontal axis represents the real parts of complex numbers, and the vertical axis represents the imaginary parts.

• A point on this plane corresponds to a complex number, plotted as (real part, imaginary part).
• If you have 0 + 4i, you would plot this at (0, 4) on the Argand plane.

It's quite simple but powerful. Seeing a complex number as a point or a vector on this plane makes the abstract concept of complex numbers more concrete and also visually interpretable. This particular instance lies on the imaginary axis because the real part is zero, making it purely imaginary.

Modulus of a Complex Number

The modulus of a complex number is a measure of its 'size' or 'magnitude', quite similar to finding the length of a vector. If you have a complex number in the form of \( a + bi \), its modulus can be found using the formula:\[ \sqrt{a^2 + b^2} \]
This formula is reminiscent of the Pythagorean theorem because it involves the distance from the origin to the point representing the complex number on the Argand plane.

• For the complex number \( 0 + 4i \), the real part \( a \) is 0, and the imaginary part \( b \) is 4.
• So the modulus is \( \sqrt{0^2 + 4^2} = 4 \).

The modulus tells you how far a point is from the origin of the Argand plane. It's a simple yet deep way to understand the magnitude of a complex number, even though it might "look" like an ordinary number at first glance.