Problem 24 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i\) \(2 i\left(\frac{1}{2}-i\right)\)

Step-by-Step Solution

Answer

The expression evaluates to \(2 + i\).

Step 1: Distribute the Imaginary Unit

Distribute the imaginary unit term: \(2i\) across the expression inside the parentheses, which is \(\frac{1}{2} - i\). This means you will multiply \(2i\) by \(\frac{1}{2}\) and by \(-i\).

Step 2: Multiply \(2i\) by \(\frac{1}{2}\)

First, multiply \(2i\) by \(\frac{1}{2}\):\[2i \times \frac{1}{2} = i\] This results from multiplying \(2\) (a real component) by \(\frac{1}{2}\), which equals \(1\), and keeping the \(i\) as is.

Step 3: Multiply \(2i\) by \(-i\)

Next, multiply \(2i\) by \(-i\):\[2i \times (-i) = -2i^2\] Since \(i^2 = -1\), \(-2i^2 = 2\). So, the multiplication gives you \(2\).

Step 4: Combine Real and Imaginary Parts

Combine the results from Steps 2 and 3. You have \(i\) (imaginary part from Step 2) and \(2\) (real part from Step 3). Hence, your expression becomes \(2 + i\).

Step 5: Write the Result in the Form \(a + bi\)

Finally, the result can be written in the form \(a + bi\), where \(a = 2\) and \(b = 1\). Thus, the expression evaluates to \(2 + i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in the study of complex numbers. It is defined by the property that \(i^2 = -1\). This is an important distinction from real numbers, which always have a positive square.

This special property allows us to delve into the world of complex numbers, offering solutions to equations that would otherwise not have real solutions. For instance, the equation \(x^2 + 1 = 0\) has no real solutions because no real number squared results in \(-1\). However, if we introduce \(i\), \(x = i\) becomes a solution.

• \(i\) serves as the "building block" for complex numbers.
• Its primary role is to enable square roots of negative integers.
• Understanding \(i\) is essential for working with more complex mathematical entities.

Complex Number Multiplication

Complex number multiplication may seem difficult, but it's straightforward once you understand the process. Multiplying complex numbers involves the distributive property and simplifying using the fact that \(i^2 = -1\).

Consider the example: \(2i\left(\frac{1}{2} - i\right)\). Multiplication here means we distribute \(2i\) across each term inside the parentheses. Basically, you treat \(i\) as you would a variable, applying the rules of algebra.

First, multiply \(2i\) by \(\frac{1}{2}\), yielding \(i\). Then, multiply \(2i\) by \(-i\). Remember, \(-i \times i = -i^2\), and since \(i^2 = -1\), this results in \(-(-2) = 2\).

• Use the distributive property to multiply each part separately.
• Replace \(i^2\) with \(-1\) during the simplification.
• Combine results by merging real and imaginary parts.

By understanding these steps, you simplify complex number multiplication effectively.

Standard Form a+bi

The standard form \(a + bi\) is how we write complex numbers, where \(a\) is the real part and \(b\) is the imaginary part, connected with \(i\). This form provides a clear and concise way to view and perform calculations with complex numbers.

• The part \(a\) (real number) shows the "horizontal" aspect on the complex plane.
• The \(b\) alongside \(i\) shows the "vertical" component.

Returning to our example, after performing the multiplications and simplifications, we end with \(2 + i\). Here, \(a = 2\) is the real component, and \(b = 1\) is the coefficient of \(i\).

This form allows easy addition, subtraction, and comparison of complex numbers, as both parts are plainly laid out for manipulation. Understanding this notation is crucial for any operation involving complex numbers.