Problem 98 - Algebra and Trigonometry

Question

If \(z=2-5 i\) and \(w=4+i,\) find \(z \cdot w\).

Step-by-Step Solution

Answer

The product \(z \cdot w = 13 - 18i\)

Step 1: Write down the given complex numbers

The given complex numbers are defined as: \(z = 2 - 5i\) and \(w = 4 + i\)

Step 2: Multiply the complex numbers

To find the product of \(z\) and \(w\), use the distributive property (also known as the FOIL method for binomials in real numbers): \((2 - 5i)(4 + i)\)

Step 3: Apply the distributive property

First, distribute each term in the first complex number by each term in the second complex number: \((2)(4) + (2)(i) + (-5i)(4) + (-5i)(i)\)

Step 4: Simplify the expression

Perform the multiplications: \(8 + 2i - 20i - 5i^2\)

Step 5: Use the property \(i^2 = -1\)

Substitute \(i^2\) with \(-1\): \(8 + 2i - 20i - 5(-1)\)

Step 6: Combine like terms

Combine the real numbers and the imaginary numbers: \(8 + 5 + (2i - 20i)\) simplifies to \(13 - 18i\)

Key Concepts

complex numbers

Complex numbers are a type of number that extends the idea of one-dimensional numbers (like real numbers) to a two-dimensional plane. This plane consists of a real part and an imaginary part. The imaginary part is usually represented with the letter 'i', where \(i = \sqrt{-1}\). For example, a complex number can be represented as \(z = a + bi\), where 'a' is the real part and 'b' is the imaginary part.

FOIL method

The FOIL method is a technique used in algebra to multiply two binomials. FOIL stands for First, Outer, Inner, Last. This method helps in systematically distributing terms when multiplying expressions. For instance, when multiplying \(z = 2 - 5i\) and \(w = 4 + i\), we use:

• First terms: (2)(4) = 8.
• Outer terms: (2)(i) = 2i.
• Inner terms: (-5i)(4) = -20i.
• Last terms: (-5i)(i) = -5i^2.

Cleverly applying the FOIL method helps ensure we do not miss any term during calculation.

distributive property

The distributive property is an algebraic rule that says multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. It's written as: \(a(b + c) = ab + ac\)
When working with complex numbers, this property is useful for expanding expressions like \((2 - 5i)(4 + i)\). Expanding each term using FOIL actually relies on the distributive property. For \( (2 - 5i)(4 + i) \), we apply \(2(4 + i) + (-5i)(4 + i)\), which simplifies step by step and helps in combining like terms.

i^2 property

In complex numbers, the imaginary unit 'i' has a crucial property: \( i^2 = -1 \). This is because 'i' is defined as the square root of -1. Recognizing this property is essential when multiplying complex numbers. For example, during the process of solving \( (2 - 5i)(4 + i) \), we get \ (-5i)(i) = -5i^2 \. Knowing that \( i^2 = -1 \) allows us to replace \ -5i^2 \ with 5, because \ -5i^2 = -5(-1) = 5 \.