Problem 8 - 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. $$ (8+2 i)(7-3 i) $$

Step-by-Step Solution

Answer

The expression is \(62 - 10i\).

Step 1: Apply the distributive property

To multiply two complex numbers, apply the distributive property (also known as the FOIL method for binomials). So, calculate: \((8+2i)(7-3i)=(8)(7)+(8)(-3i)+(2i)(7)+(2i)(-3i)\). This breaks down into four separate multiplications.

Step 2: Compute each part of the expansion

- Calculate \(8 \times 7 = 56\).- Calculate \(8 \times -3i = -24i\).- Calculate \(2i \times 7 = 14i\).- Calculate \(2i \times -3i = -6i^2\). Remember that \(i^2 = -1\), so \(-6i^2 = 6\).

Step 3: Combine real parts and imaginary parts separately

Combine the real parts: \(56 + 6 = 62\). Combine the imaginary parts: \(-24i + 14i = -10i\).

Step 4: Write in the form \(a + bi\)

The expression \((8+2i)(7-3i)\) simplifies to \(62 - 10i\). This matches the form \(a + bi\), where \(a = 62\) and \(b = -10\).

Key Concepts

Distributive Property

Imagine the distributive property as a technique that helps you to simplify expressions, especially when we're dealing with complex numbers. It allows us to multiply a pair of binomials by distributing each part of the first binomial across each part of the second. In the case of complex numbers, like our exercise

• (8 + 2i)(7 - 3i), we distribute numbers and letters in a sequence of steps.

This is often called the FOIL method, an acronym for First, Outside, Inside, and Last elements in each binomial:

• First: Multiply the first terms in the binomials, 8 and 7, to get 56.
• Outside: Multiply the outer terms, 8 and -3i, resulting in -24i.
• Inside: Multiply the inside terms, 2i and 7, producing 14i.
• Last: Multiply the last terms, 2i and -3i, which gives -6i².

Understanding the distributive property is like having a toolbox. You can take apart complex expressions, compute each part separately, and then piece them back together in a simplified format like a + bi.

Real Numbers

Real numbers consist of all the numbers we encounter in everyday life: integers, fractions, and decimals. In complex numbers, the real part is the 'a' in the expression 'a + bi'.
In our example, after applying the distributive property to (8 + 2i)(7 - 3i), we identify the real numbers that result from the multiplications:

• 8 multiplied by 7, giving us 56, is clearly a real number.
• From the multiplication of imaginary parts, -6i² becomes 6, since i² equals -1 and hence, -6i² becomes 6.

Then, we combine these real parts, 56 and 6, to get 62.
Real numbers anchor complex number expressions, helping them fit within familiar numeric systems we use regularly.

Imaginary Numbers

Imaginary numbers are part of the foundation for understanding complex numbers. Represented by 'bi' in a complex number setup, 'b' is the coefficient for the imaginary unit 'i'.
The imaginary unit, 'i', is defined by the property i² = -1. This property allows us to handle expressions involving square roots of negative numbers.
For our expression (8 + 2i)(7 - 3i):

• The multiplication 8 times -3i gives us -24i, and 2i times 7 gives 14i.
• Combining these imaginary components involves simple arithmetic: -24i plus 14i results in -10i.

This -10i is the 'b' in our a + bi expression, showcasing why imaginary numbers are vital to fully describing complex numbers.