Problem 10 - Advanced Engineering Mathematics

Question

$$\frac{3}{4}+\frac{2}{3} i$$

Step-by-Step Solution

Answer

This complex number is \( \frac{3}{4} + \frac{2}{3}i \). It's in its simplest form.

Step 1: Identify the Complex Number

The given complex number is \( \frac{3}{4} + \frac{2}{3} i \). It is composed of a real part and an imaginary part.

Step 2: Recognize the Parts of the Complex Number

A complex number is generally expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Here, \( a = \frac{3}{4} \) and \( b = \frac{2}{3} \).

Step 3: Simplification (if necessary)

In this problem, the expression \( \frac{3}{4} + \frac{2}{3} i \) is already in its simplest form, since both the real and imaginary parts are reduced fractions and cannot be simplified further.

Step 4: Conclusion

Since the complex number is already in standard form, we conclude that no further simplification or manipulation is needed.

Key Concepts

Real Part of Complex Numbers

Complex numbers might seem a bit tricky at first, but once you get the hang of them, they're quite straightforward. Let's break it down, starting with the 'real part' of a complex number. In any complex number expressed in the form \( a + bi \), the 'real part' is represented by \( a \). It is called 'real' because it does not involve the imaginary unit \( i \), which is a mathematical representation of the square root of -1.

For example, given the complex number \( \frac{3}{4} + \frac{2}{3} i \), the real part is \( \frac{3}{4} \). This part is simple and contains no imaginary components. Just remember that you're simply taking the coefficient that stands alone without the \( i \).

Understanding the real part is vital because it corresponds to the point on the horizontal axis in the complex plane. It tells us exactly how far from the origin the real value of the number extends.

Imaginary Part of Complex Numbers

Now, let's take a look at what we call the 'imaginary part' of a complex number. In the standard form \( a + bi \), the imaginary part is denoted by \( b \). But why is it 'imaginary'? It's all due to the presence of \( i \), the imaginary unit. This part represents numbers that are multiples of \( i \), which are necessary when solving equations like \( x^2 = -1 \).

In our example, \( \frac{3}{4} + \frac{2}{3} i \), \( \frac{2}{3} \) is the imaginary part. Don't let the term 'imaginary' throw you off; it's very real in terms of use and application in mathematics and engineering.

The imaginary part is plotted on the vertical axis of the complex plane, giving the complex number its vertical position. It is important in describing oscillations, waves, and many phenomena in the physical world.

Standard Form of Complex Numbers

Complex numbers are almost like coordinates in a two-dimensional space. The way we usually write them, \( a + bi \), is known as the 'standard form of complex numbers'. Let's discuss why this is the standard.

This form makes it clear at a glance what the real and imaginary parts of the number are, which is very convenient for both calculations and graphical representation. In standard form, you always have \( a \) (the real part) and \( bi \) (the imaginary part) clearly separated.

Take our example \( \frac{3}{4} + \frac{2}{3} i \). Written in standard form as it already is, we can instantly recognize \( \frac{3}{4} \) as the real part and \( \frac{2}{3} i \) as the imaginary part. No surprises here!

• It simplifies operations such as addition, subtraction, and multiplication of complex numbers.
• It also makes it possible to easily draw complex numbers on the complex plane, giving them a visual representation which can be extremely helpful.

Understanding the standard form is crucial as it allows for an easier grasp of more complex concepts, like polar form and Euler's formula, which build on this foundation.