Problem 1 - 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. $$ (5-2 i)+(-3+6 i) $$

Step-by-Step Solution

Answer

The expression is \(2 + 4i\).

Step 1: Identify the Real Parts

In the expression \((5-2i) + (-3+6i)\), identify the real numbers. Here they are \(5\) from the first part and \(-3\) from the second part.

Step 2: Identify the Imaginary Parts

Look at the imaginary parts of the expression. We have \(-2i\) from \(5-2i\) and \(6i\) from \(-3+6i\).

Step 3: Combine the Real Parts

Add the real parts together: \(5 + (-3) = 2\).

Step 4: Combine the Imaginary Parts

Add the imaginary parts: \(-2i + 6i = 4i\).

Step 5: Write the Final Expression

Combine the results from Step 3 and Step 4 into the form \(a + bi\). The expression becomes \(2 + 4i\).

Key Concepts

Imaginary Numbers

Imaginary numbers play a vital role in understanding complex numbers. They introduce us to the idea of numbers that include the square root of a negative number. The fundamental imaginary unit is represented as \(i\). It is defined by the property that \(i^2 = -1\). This special property allows imaginary numbers to extend our real number system.
In the given problem, the parts of the expression featuring \(i\) are the imaginary components. For example, in the expression \(5 - 2i\) + \(-3 + 6i\), the terms \(-2i\) and \(6i\) are the imaginary parts. By learning about imaginary numbers, we can effectively handle a wider range of mathematical situations, especially in algebra and complex number calculations.
Imaginary numbers also pair with real numbers to form complex numbers, providing us with a complete framework to solve equations that would otherwise have no solution in the realm of real numbers.

Real Parts

Real parts of complex numbers resemble the ordinary numbers we use in daily life. These are the numbers without the \(i\) component. They are of the same nature as rational and irrational numbers. Understanding real parts in complex numbers helps us break them down into their core elements.
In the expression \(5 - 2i\) + \(-3 + 6i\), the real parts are \(5\) and \(-3\). These are straightforward to handle, as they only require standard arithmetic operations such as addition, subtraction, etc.
Recognizing and correctly manipulating real parts is essential when performing calculations with complex numbers. Whether you're adding or subtracting, isolating these real components allows you to simplify and resolve complex expressions accurately.

Addition of Complex Numbers

Adding complex numbers involves aligning like terms, much like simplifying algebraic expressions. We focus on pairing real numbers together and separately combining imaginary terms. This approach maintains the clarity of our calculations.
Consider the problem expression: \((5 - 2i) + (-3 + 6i)\). Here’s how we handle it:

• Real Part Addition: Add the real numbers together: \(5 + (-3) = 2\).
• Imaginary Part Addition: Add the imaginary numbers: \(-2i + 6i = 4i\).

With this method, you ensure that your expression remains logical and simplified, allowing you to write it in the standard complex number form \(a + bi\).
This systematic process of alignment helps avoid confusion and is especially useful in more complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are mathematical phrases combining numbers, variables, and operators to represent a particular relationship. When dealing with complex numbers, these expressions expand to include imaginary parts.
Complex numbers like \((5 - 2i) + (-3 + 6i)\) are considered algebraic because they involve operations on numbers containing variables—here, the variable \(i\). This combination of imaginary and real numbers presents us with a new layer of complexity within algebraic expressions.
For such expressions, the key is in correctly isolating and simplifying both real and imaginary components. By systematically handling these parts, as outlined in algebra's basic rules, one can achieve a solution that fits the form \(a + bi\), crucial for understanding and utilizing in further mathematics.
Algebraic expressions with complex numbers thus enhance our problem-solving abilities, providing powerful tools for equations beyond real number capabilities.