Problem 58 - Algebra 2 and Trigonometry

Question

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{1+i}{\frac{1}{2}-\frac{1}{2} i} $$

Step-by-Step Solution

Answer

The quotient in \(a + bi\) form is \(2 + 2i\).

Step 1: Identify the conjugate

To eliminate the imaginary part in the denominator, we'll multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\frac{1}{2} - \frac{1}{2}i\) is \(\frac{1}{2} + \frac{1}{2}i\).

Step 2: Multiply numerator and denominator

Multiply both the numerator \(1 + i\) and the denominator \(\frac{1}{2} - \frac{1}{2}i\) by the conjugate \(\frac{1}{2} + \frac{1}{2}i\), which results in\[\frac{(1+i)(\frac{1}{2}+\frac{1}{2}i)}{(\frac{1}{2}-\frac{1}{2}i)(\frac{1}{2}+\frac{1}{2}i)}.\]

Step 3: Expand the numerator

Expand the expression combined in the numerator:\[(1+i)(\frac{1}{2}+\frac{1}{2}i) = \frac{1}{2} + \frac{1}{2}i + \frac{1}{2}i - \frac{1}{2}i^2.\]Since \(i^2 = -1\), the expression simplifies to\[\frac{1}{2} + i + \frac{1}{2} = 1+i.\]

Step 4: Simplify the denominator

Expand the denominator:\[(\frac{1}{2} - \frac{1}{2}i)(\frac{1}{2} + \frac{1}{2}i) = (\frac{1}{2})^2 - (\frac{1}{2}i)^2 = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}.\]This is because the imaginary parts cancel out, leaving only real numbers.

Step 5: Put everything together

Now place the simplified expressions together:\[\frac{1+i}{\frac{1}{2}} = 2 + 2i.\]We multiply the entire expression by 2 to get each term in standard form.

Key Concepts

Conjugate in Complex Numbers

When working with complex numbers, the term *conjugate* refers to a specific transformation we apply to make the math easier. For a complex number given in the form \(a + bi\), its conjugate is \(a - bi\).

• It essentially means changing the sign of the imaginary component.
• This process helps eliminate the imaginary unit when needed, especially in division problems.

In the given problem, the denominator of the complex fraction is \(\frac{1}{2} - \frac{1}{2}i\). To simplify, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(\frac{1}{2} + \frac{1}{2}i\). By doing this, the denominator becomes a real number, making the division process straightforward. This approach takes advantage of the fact that the product of a complex number and its conjugate is always a real number. This property simplifies the process of converting complex fractions into their standard form \(a + bi\). It is a vital tool for anyone delving into complex arithmetic.

Imaginary Unit Basics

The imaginary unit, often denoted by \(i\), is a fundamental concept in complex numbers. It is defined as the square root of \(-1\).

• This means that \(i^2 = -1\).
• It allows us to work with numbers that square to negative values, something not possible with real numbers alone.

In the exercise, the presence of the imaginary unit is evident in both the numerator and the denominator. During the multiplication process, \(i^2\) appears, reminding us that it equates to \(-1\). This conversion is critical. When we expand expressions and simplify, recognizing \(i^2 = -1\) turns an otherwise complex situation into a manageable arithmetic problem.The imaginary unit extends our understanding of numbers beyond the real number system, enabling solutions to equations like \(x^2 + 1 = 0\), which have no real solution. Understanding the imaginary unit is therefore a key skill in both basic and advanced mathematical studies.

Understanding Complex Arithmetic

Complex arithmetic encompasses operations such as addition, subtraction, multiplication, and division involving complex numbers. Each operation has unique rules that emulate those of real numbers but also include considerations for the imaginary unit \(i\).

• Addition and subtraction: Simply combine like terms. Real parts combine with real parts, and imaginary parts with imaginary parts.
• Multiplication: Use distributive property or FOIL method while remembering that \(i^2 = -1\).
• Division: Resolve by multiplying numerator and denominator by the conjugate of the denominator to remove the imaginary part from the denominator.

In simpler terms, complex arithmetic may seem daunting due to the imaginary unit, but the principles are much like those for real numbers. In the exercise, after multiplying both the numerator and the denominator by the conjugate of the denominator, you simplify the expression. This involves applying the distributive law and simplifying terms by understanding that \(i^2 = -1\). Ensuring every outcome remains in the form \(a + bi\) allows effective comparison and utilization of these numbers in advanced applications, like engineering and physics.