Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ (1+9 i)-(1+2 i) $$

Step-by-Step Solution

Answer

The difference is \(0+7i\).

Step 1: Identify the components

To solve the expression \((1+9i)-(1+2i)\), start by identifying the real and imaginary components of each complex number. Here, \(1+9i\) has a real part of 1 and an imaginary part of 9. The complex number \(1+2i\) has a real part of 1 and an imaginary part of 2.

Step 2: Subtract the real parts

Subtract the real parts of the two complex numbers: \(1 - 1 = 0\).

Step 3: Subtract the imaginary parts

Subtract the imaginary parts of the two complex numbers: \(9 - 2 = 7\).

Step 4: Combine results

Combine the results from Steps 2 and 3 to form the new complex number: \(0+7i\). So, the result of \((1+9i)-(1+2i)\) is \(0+7i\).

Key Concepts

Addition and Subtraction of Complex Numbers

Complex numbers are a fundamental part of mathematics, combining a real and an imaginary component, typically expressed as \(a + bi\) where \(a\) is the real part and \(bi\) is the imaginary part. When performing operations such as addition or subtraction on complex numbers, you address the real and imaginary parts separately.
The process is straightforward:

• To add complex numbers, you simply add the real parts together and the imaginary parts together.
• Similarly, to subtract complex numbers, you subtract the real and imaginary parts independently.

For instance, in the expression \((1+9i)-(1+2i)\), you subtract each corresponding part separately. The real parts, 1 and 1, give you \(1 - 1 = 0\). The imaginary parts, 9i and 2i, are calculated as \(9i - 2i = 7i\). Thus, the result of the operation is the complex number \(0 + 7i\). This method ensures that all components are properly managed and combined, leading to accurate results in your calculations.

Real and Imaginary Parts

Understanding the nature of real and imaginary parts in a complex number is crucial. A complex number combines both a real part, denoted as \(a\), and an imaginary part, denoted as \(bi\). The imaginary unit \(i\) is defined as \(\sqrt{-1}\), which allows it to satisfy the property \(i^2 = -1\). This special characteristic is what distinguishes complex numbers from other numerical systems.
When dealing with complex numbers in expressions such as addition or subtraction, it's important to maintain focus on each part separately:

• Real part: The non-imaginary component, which can be added or subtracted directly.
• Imaginary part: This component includes the imaginary unit \(i\) and requires individual handling with respect to its mathematical properties.

This separation ensures clarity and precision in the computation. As shown in the example \((1+9i)-(1+2i)\), correctly distinguishing and managing these parts allowed us to arrive systematically at the final result.

Complex Number Operations

Operations involving complex numbers rely on the unique combination of both algebraic and geometric properties. Besides basic addition and subtraction, complex numbers can also be multiplied and divided, although those operations involve additional considerations like using conjugates and managing the powers of \(i\). Here, the primary focus is the proper manipulation of basic operations.
Each calculation with complex numbers smoothly integrates into broader algebraic thinking because you're treating these numbers not just as abstract symbols, but as entities with both magnitude and direction, especially when visualized on a complex plane.
Given any two complex numbers, understanding how to efficiently compute operations such as those demonstrated in the example ensures you develop a strong foundational understanding of this numerical system.