Problem 15 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ \left(7-\frac{1}{2} i\right)-\left(5+\frac{3}{2} i\right) $$

Step-by-Step Solution

Answer

The result is \( 2 - 2i \).

Step 1: Identify the Complex Numbers

The expression given is \( \left(7 - \frac{1}{2} i \right) - \left(5 + \frac{3}{2} i \right) \). Identify each complex number separately as \( (a + bi) \) and \( (c + di) \).- First complex number: \( a + bi = 7 - \frac{1}{2} i \)- Second complex number: \( c + di = 5 + \frac{3}{2} i \).

Step 2: Subtract Real Parts

Subtract the real part of the second complex number from the real part of the first complex number.- Real parts are 7 and 5.- Calculation: \( 7 - 5 = 2 \).

Step 3: Subtract Imaginary Parts

Subtract the imaginary part of the second complex number from the imaginary part of the first complex number.- Imaginary parts are \( -\frac{1}{2} \) and \( \frac{3}{2} \).- Calculation: \( -\frac{1}{2} - \frac{3}{2} = -\frac{4}{2} = -2 \).

Step 4: Write the Resulting Complex Number

Combine the results from Step 2 and Step 3 to write the result in the form \( a + bi \).- Result: \( 2 - 2i \).

Key Concepts

Subtracting Complex Numbers

Subtracting complex numbers is similar to subtracting real numbers. You deal with two parts: the real part and the imaginary part. Complex numbers are usually written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part, with \(i\) representing \(\sqrt{-1}\). When you subtract one complex number from another, you need to separately subtract the corresponding real parts and imaginary parts.
For example, let's consider the complex numbers from the original exercise: \((7 - \frac{1}{2}i)\) and \((5 + \frac{3}{2}i)\). To subtract these, follow these steps:

• Subtract the real part of the second number from the real part of the first number: \(7 - 5 = 2\).
• Subtract the imaginary part of the second number from the imaginary part of the first number: \(-\frac{1}{2} - \frac{3}{2} = -2\).

This gives us the resulting complex number: \(2 - 2i\).
Each step ensures that both real and imaginary components are carefully handled, resulting in a correct subtraction outcome.

Real and Imaginary Parts

In complex numbers, the real and imaginary parts have distinct roles. Understanding these can greatly simplify the process of working with complex numbers. The general form of a complex number is \(a + bi\).
The real part is simply \(a\), a regular number that does not involve \(i\). The imaginary part is \(bi\), which includes the imaginary unit \(i\).

• For example, in the complex number \(7 - \frac{1}{2}i\), the real part is 7, and the imaginary part is \(-\frac{1}{2}i\).
• In another complex number like \(5 + \frac{3}{2}i\), the real part is 5, and the imaginary part is \(\frac{3}{2}i\).

Identifying these parts correctly is crucial for operations such as addition, subtraction, multiplication, or division of complex numbers. Separating real and imaginary components allows for systematic handling, ensuring clarity and preventing mistakes in calculations.

Complex Number Operations

Operations involving complex numbers are straightforward once you know the basics. The main operations include addition, subtraction, multiplication, and division.
Let’s focus on subtraction, the operation at hand:

• Subtract the real components of the two complex numbers.
• Subtract the imaginary components, considering the imaginary unit \(i\).

For more advanced operations like multiplication, follow these general steps:

• Distribute each part of one complex number to every part of the other complex number, usually following the FOIL method (First, Outer, Inner, Last).
• Simplify and combine like terms, converting \(i^2\) to \(-1\) when it appears.

For division, you require the conjugate of the denominator to rationalize it. This ensures the denominator becomes a real number.
Understanding these operations lets you handle any complex number scenarios with confidence and accuracy, building a strong foundation for advanced mathematical concepts.