Problem 53 - Algebra A Combined Function

Question

Multiply. Write your answers in the form \(a+b i\). $$ (1-i)^{2} $$

Step-by-Step Solution

Answer

The expression \((1-i)^{2}\) simplifies to \(-2i\).

Step 1: Expand the Expression

To solve the expression \((1-i)^2\), let's first expand it. Use the algebraic identity \((a-b)^2 = a^2 - 2ab + b^2\). Here, your \(a = 1\) and \(b = i\).

Step 2: Calculate Each Term

Plug \(a = 1\) and \(b = i\) into the identity: \[ (1-i)^2 = 1^2 - 2(1)(i) + i^2 \] Now, simplify each term separately: - \(1^2 = 1\).- \(-2(1)(i) = -2i\).- \(i^2 = -1\) because \(i\) is the imaginary unit.

Step 3: Combine the Terms

Substitute the simplified terms back into the expanded expression:\[ 1 - 2i + (-1) \] Combine like terms:\[ 1 - 1 - 2i = 0 - 2i \]

Step 4: Write in the Form \(a + b i\)

The expression \(0 - 2i\) can be written in the form \(a + bi\) where \(a = 0\) and \(b = -2\). Hence: \[ (1-i)^2 = 0 - 2i \] or equivalently \[ -2i \].

Key Concepts

Imaginary Unit

In the world of complex numbers, the imaginary unit is a crucial element. It is represented by the letter \(i\). The defining property of \(i\) is that \(i^2 = -1\). This means that the square of the imaginary unit results in a negative one, a concept not found in traditional, real numbers.
This fundamental rule allows complex numbers to be expressed in the form of \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(bi\) represents the imaginary part of the complex number.

• Understanding \(i^2 = -1\) is key to simplifying expressions involving complex numbers.
• It allows us to find real and imaginary components when performing algebraic operations.

For example, in the solution step using \((1 - i)^2\), we use \(i^2 = -1\) to simplify \(i^2\) terms into real numbers to further solve the expression.

Algebraic Identities

Algebraic identities are powerful tools used to simplify and solve expressions, including those with complex numbers. One such identity is \((a-b)^2 = a^2 - 2ab + b^2\), which helps expand binomials.
In our example, \((1-i)^2\) uses this identity with \(a = 1\) and \(b = i\). This allows breaking down and simplifying the expression step by step.

• The identity helps us expand \((1-i)^2\) into components that are easier to handle.
• By plugging in \(a\) and \(b\) correctly, we use this formula to find each part: \(1^2\), \(-2 \cdot 1 \cdot i\), and \(i^2\).

Mastering algebraic identities transforms complex multiplication into a more straightforward process by providing clear steps to follow, ultimately leading us to the solution of the problem.

Complex Multiplication

Complex multiplication is an essential operation involving complex numbers. The multiplication of complex numbers, such as \((1-i)^2\), involves both real and imaginary components, and it follows the distributive property.
In this case, we expand using an algebraic identity to handle the multiplication.

• We first simplify by expanding the expression using \((a-b)^2\).
• We handle terms like \(-2i\) and \(i^2\) separately, utilizing the property of \(i^2 = -1\) to transform imaginary terms into real components when necessary.
• Finally, we combine like terms involving both real and imaginary components to write the result in the form \(a + bi\).

Complex multiplication not only strengthens algebraic skills but allows for solving real-world problems in engineering and physics where such operations are common. Understanding it enriches one's mathematical toolkit.