Problem 4 - Algebra and Trigonometry

Question

Find the real and imaginary parts of the complex number. $$ \frac{4+7 i}{2} $$

Step-by-Step Solution

Answer

The real part is 2 and the imaginary part is \( \frac{7}{2} \).

Step 1: Identify the Complex Number

The given complex number is \( \frac{4+7i}{2} \). This is in the form \( \frac{a+bi}{c} \) where \( a = 4 \), \( b = 7 \), and \( c = 2 \). Our goal is to find the real and imaginary parts separately by dividing both parts by 2.

Step 2: Simplify the Real Part

To find the real part, divide the numerical part (without \( i \)) by 2. The real part is \( \frac{4}{2} = 2 \).

Step 3: Simplify the Imaginary Part

For the imaginary part, divide the coefficient of \( i \) by 2. The imaginary part is \( \frac{7}{2} \). This means the imaginary part is \( \frac{7}{2}i \), so \( b = \frac{7}{2} \).

Key Concepts

Real Part of a Complex Number

When dealing with complex numbers, each number has a real part and an imaginary part. The real part is simply the actual number that is not associated with the imaginary unit \(i\). In the expression of a complex number like \(a + bi\), \(a\) represents the real part. Let’s apply this to the complex number: \(\frac{4+7i}{2}\).

To find the real part, we focus on the numerical value without the \(i\). Here, the real component is \(4\). Since we have \(\frac{4}{2}\), the real part of the expression is \(2\). It’s as easy as simplifying a fraction!

After simplifying, the solution to the exercise reveals that the real part of \(\frac{4+7i}{2}\) is a straightforward \(2\). So, whenever you encounter similar expressions in complex numbers, just focus on the part without \(i\) for the real portion.

Imaginary Part of a Complex Number

The imaginary part of a complex number involves the term associated with \(i\), which is the mathematical representation for the square root of -1. In a complex number \(a + bi\), the \(b\) stood before \(i\) is referred to as the imaginary part.

In the exercise \(\frac{4+7i}{2}\), we isolate the imaginary portion, which is \(7i\). To find just the coefficient or the imaginary part itself, without including the \(i\) symbol, you divide \(7\) by \(2\). This results in \(\frac{7}{2}\). Therefore, the simplified imaginary portion is \(\frac{7}{2}i\), meaning \(b = \frac{7}{2}\).

Understanding this simplification is crucial for working with complex expressions. It allows you to express the imaginary part in its simplest form, making calculations in further problems more manageable.

Simplifying Complex Expressions

Simplifying complex expressions involves breaking down both the real and imaginary parts separately. It’s important to handle them individually first before performing any additional operations.

Given the expression \(\frac{4+7i}{2}\), you separate it into real and imaginary components, then simplify each part individually. Specifically:

• For the real part: Divide \(4\) by \(2\), giving you \(2\).
• For the imaginary part: Divide \(7\) by \(2\), resulting in \(\frac{7}{2}i\).

Once simplified, you can reassemble them for a concise complex number: \(2 + \frac{7}{2}i\).

Simplifying complex expressions is essentially about making them cleaner and easier to interpret. It aids in solving equations involving complex numbers, as it gives clear visibility to both components, thereby easing further analytical or computational steps.