Problem 27 - Precalculus Mathematics for Calculus

Question

Evaluate the expression and write the result in the form \(a+b i\) $$(3-4 i)(5-12 i)$$

Step-by-Step Solution

Answer

The expression evaluates to \(-33 - 56i\).

Step 1: Distribute the Terms

First, we apply the distributive property (also known as the FOIL method for complex numbers) to multiply the expressions. This implies: \((3 - 4i)(5 - 12i) = 3 \cdot 5 + 3 \cdot (-12i) - 4i \cdot 5 - 4i \cdot (-12i)\).

Step 2: Perform Individual Multiplications

Carry out the individual multiplications:- \(3 \cdot 5 = 15\)- \(3 \cdot (-12i) = -36i\)- \(-4i \cdot 5 = -20i\)- \(-4i \cdot (-12i) = 48i^2\)

Step 3: Simplify Using the Imaginary Unit

Recall that \(i^2 = -1\). Thus, substitute \(i^2\) with \(-1\) in the expression: \(48i^2 = 48(-1) = -48\). This results in the expression: \(15 - 36i - 20i - 48\).

Step 4: Combine Like Terms

Now, combine the real parts and imaginary parts separately:- Real parts: \(15 - 48 = -33\)- Imaginary parts: \(-36i - 20i = -56i\)Hence, the expression becomes \(-33 - 56i\).

Step 5: Write the Result in Standard Form

The final result of the multiplication in the form \(a + bi\) is:\(-33 - 56i\).

Key Concepts

Multiplication of Complex Numbers

When you multiply complex numbers, you're really playing with both real and imaginary parts. Complex numbers have the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. To multiply, you need to distribute each part of one complex number by each part of the other.In our example, with \((3 - 4i)(5 - 12i)\), you take the number \(3\) and multiply it by \(5\) and \(-12i\), and then take \(-4i\) and do the same. This ensures every term is distributed among the others:

• \( 3 \times 5 \)
• \( 3 \times -12i \)
• \(-4i \times 5 \)
• \(-4i \times -12i \)

This approach covers all interactions between the components, similar to multiplying binomials (often referred to as the FOIL method in basic algebra). It allows for the complete multiplication of each part in its correct order, leading us to the final result.

Distributive Property

The distributive property is a fundamental rule of algebra that applies to the multiplication of complex numbers as well. It states that a single term can be distributed across a sum of terms. Basically, for any numbers \(a\), \(b\), and \(c\), it shows as: \( a(b + c) = ab + ac \).In our exercise, it allows us to take each of the terms in \((3 - 4i)\) and multiply them by each of the terms in \((5 - 12i)\), enabling us to simplify complex expressions effectively. By applying this property:

• The real parts (pure numbers) get multiplied together and the imaginary terms with each other, ensuring a comprehensive pairing of terms.
• It helps to keep the process organized so we don't miss or duplicate terms.

This property helps in rearranging and solving parts of the product in a logical order, ultimately making it much easier to simplify and combine the results.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a mathematical concept used to extend the real numbers. It's defined as satisfying \(i^2 = -1\). When you square an imaginary unit, it magically transforms the product into a real number but with a negative sign.In the given example, when we calculated \(-4i \times -12i\), the \(i\) terms multiply to \(i^2\), giving us \(48i^2\). Applying \(i^2 = -1\), this becomes \(48 \times -1 = -48\). It converts parts of the complex result back into real numbers, which are crucial for combining like terms effectively.This simplification form is essential to solve and maintain the expression real, even as it contains imaginary components. It lets those working with it handle complex numbers logically and ensures calculations blend seamlessly into the standard form, \(a + bi\). Understanding this unit is key to effectively managing and interpreting complex solutions.