Problem 14 - A Graphical Approach to College Algebra

Question

Write each expression in standard form. Do not use a calculator. $$\frac{4+2 i}{i}$$

Step-by-Step Solution

Answer

The expression in standard form is \(2 - 4i\).

Step 1: Understand the problem

The expression \( \frac{4 + 2i}{i} \) needs to be written in standard form, which is \( a + bi \) where \( a \) and \( b \) are real numbers.

Step 2: Simplify by multiplying by the conjugate

Multiply both numerator and denominator of the expression by \( -i \) to eliminate the imaginary unit \( i \) in the denominator. \[ \frac{4 + 2i}{i} \times \frac{-i}{-i} = \frac{(4 + 2i)(-i)}{i(-i)} \]

Step 3: Compute the denominator

Calculate the denominator, which is \( i \times (-i) = -i^2 = 1 \) because \( i^2 = -1 \).

Step 4: Compute the numerator

Expand the numerator: \[ (4 + 2i)(-i) = 4(-i) + 2i(-i) = -4i - 2i^2 \] Remember that \( i^2 = -1 \), so:\[ -2i^2 = 2 \]Thus, the numerator becomes:\[ -4i + 2 \]

Step 5: Finalize the fraction

Combine the results:\[ \frac{-4i + 2}{1} = 2 - 4i \]So, the expression in standard form is \( 2 - 4i \).

Key Concepts

Standard Form of a Complex Number

Complex numbers consist of two parts: a real part and an imaginary part. They are often written in the standard form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, with \(i\) being the imaginary unit. This standard form allows for easy addition, subtraction, and comparison of complex numbers. It organizes the number in a way that's similar to polynomial terms, thus offering a straightforward structure.

• If you have a complex number like \(4 + 3i\), 4 is the real part, and \(3i\) is the imaginary part.
• The standard form helps in easily identifying and separating these components, making complex calculations more manageable.

In the given exercise, we've transformed the quotient \(\frac{4+2i}{i}\) into the standard form \(2 - 4i\), where 2 is the real part, and \(-4i\) is the imaginary part.

Simplifying Complex Expressions

Simplifying complex expressions involves eliminating unwanted elements such as the imaginary unit from denominators and expressing complex numbers in their simplest form. This often requires multiplying by the conjugate or similar techniques.

• For the expression \(\frac{4 + 2i}{i}\), simplifying meant removing \(i\) from the denominator.
• We achieved this by multiplying the expression by \(-i/-i\), simplifying the division effectively.

In the solution, multiplying the numerator and denominator by \(-i\) helps clear the imaginary number from the denominator, thereby achieving a more direct form. This approach relies on the identity \(i^2 = -1\), which converts complex manipulations into real-number calculations where possible.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers, representing the square root of \(-1\). It is not a real number but helps in expressing numbers that are capable of extending our numeric system beyond real numbers.

• By definition, \(i^2 = -1\).
• It allows us to work with equations and problems that have no real solutions, extending our ability to solve polynomial equations beyond real roots.
• In practical algebraic manipulations, \(i\) follows normal algebraic rules with the special note about \(i^2\).

In the exercise, the imaginary unit was initially part of the denominator, and through simplification, it's used not only to divide but also as a key to transform the given expression into its standard form.