91Ó°ÊÓ

The concept of the imaginary unit is pivotal in understanding complex numbers. A complex number is a number that comprises a real part and an imaginary part and is often expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit is denoted by \(i\) and is defined as \(i = \sqrt{-1}\). This definition allows us to express the square roots of negative numbers in terms of \(i\).

For example, the square root of \(-4\) is \(\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i\), and similarly, \(\sqrt{-16} = 4i\). Utilizing \(i\) enables the simplification of expressions involving square roots of negative numbers, paving the way for handling complex numbers with ease.

The distributive property is a fundamental rule in algebra that helps in the expansion of expressions. It states that for any numbers \(a\), \(b\), and \(c\), the equation \(a(b+c) = ab + ac\) holds true. This property is especially useful when expanding products of binomials, such as in our example.

When expanding the expression \((2-2i)(3-4i)\), the distributive property is applied by treating each term separately:

• Multiply each term in the first binomial \((2 - 2i)\) by each term in the second binomial \((3 - 4i)\).
• This results in: \(2 \cdot 3\), \(2 \cdot (-4i)\), \(-2i \cdot 3\), and \(-2i \cdot 4i\).

By multiplying term by term and simplifying, we eventually achieve the fully expanded form, which we can then combine and simplify further to find the real and imaginary parts.

Binomial expansion in the context of complex numbers involves applying the distributive property to obtain a final expression in its simplest form. A binomial expression is one that includes two terms, as seen in \((2-2i)\) and \((3-4i)\) in our example. The process of expanding these binomials involves using the FOIL method, an acronym that stands for First, Outside, Inside, Last:

• First: Multiply the first terms in each binomial: \(2 \cdot 3\).
• Outside: Multiply the outer terms: \(2 \cdot (-4i)\).
• Inside: Multiply the inner terms: \(-2i \cdot 3\).
• Last: Multiply the last terms: \(-2i \cdot 4i\).

Each multiplication is done separately, and then these individual products are added together. You must remember that \(i^2 = -1\), which helps to simplify the last multiplication term. After simplification, combining the terms gives the result in the standard form \(a + bi\). This method clarifies how complex numbers and their expansions can be calculated systematically.