Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (-3+8 i)-(2+3 i) $$

Step-by-Step Solution

Answer

-5 + 5i

Step 1: Identify the Problem

The exercise asks us to simplify the expression (-3 + 8i) - (2 + 3i). The goal is to express it in the form of a+bi by performing subtraction on complex numbers.

Step 2: Separate Real and Imaginary Parts

In complex subtraction, each component (real and imaginary) of one number is subtracted by each respective component of the other number. In the expression (-3 + 8i) - (2 + 3i), separate the real parts (-3 and 2) and the imaginary parts (8i and 3i).

Step 3: Subtract Real Parts

Subtract the real parts: -3 - 2 = -5

Step 4: Subtract Imaginary Parts

Subtract the imaginary parts: 8i - 3i = 5i

Step 5: Compose the Result

Combine the results from the subtraction of real and imaginary parts to form the complex number: -5 + 5i. This expression is now in the form of a + bi, where a = -5 and b = 5.

Key Concepts

Complex Number Subtraction

Subtracting complex numbers might seem tricky at first, but it's actually pretty straightforward. Imagine you have two complex numbers: one is represented as \(-3 + 8i\) and the other as \(2 + 3i\). When subtracting these numbers, you need to handle the real parts and the imaginary parts separately. First, subtract the real numbers \(-3\) and \(2\). Then, subtract the imaginary numbers, \(8i\) and \(3i\). This method ensures that both components – real and imaginary – are treated correctly to give you the right answer.

Real and Imaginary Parts

Understanding the difference between real and imaginary parts is vital for working with complex numbers. A complex number such as \(-3 + 8i\) consists of two parts: the real part \(-3\), and the imaginary part \(8i\). The real part is just like any regular number you deal with, whereas the imaginary part is multiplied by \(i\), where \(i\) is the square root of \(-1\). When dealing with complex number operations, always keep these two components in your mind. They function independently but conjointly represent the complex number as a whole.

Complex Number Form a+bi

The goal of many complex number problems is to express your answer in the form \(a + bi\). This form uses 'a' to represent the real component and 'b' for the imaginary component. So, if you have a complex number like \(-5 + 5i\), \(-5\) is 'a', and \(5\) is 'b'. This standard form makes it easy to compare and manipulate complex numbers, as it clearly shows both components. The given problem, \((-3 + 8i) - (2 + 3i)\), simplifies to this form: the answer \(-5 + 5i\) shows us directly that \(a = -5\) and \(b = 5\). Being comfortable with this notation allows you to easily identify and work with real and imaginary parts within any complex number.