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Chapter 4: Problem 52

In Exercises 45-54, write the quotient in standard form. $$ \frac{8+16 i}{2 i} $$

Short Answer

Expert verified
The quotient in standard form is \(8-4i\).

Step by step solution

01

Identify the Complex Number

The complex number in this exercise is \(\frac{8+16i}{2i}\). It is not in the standard form, so we need to modify it.
02

Clear out the Imaginary part from the denominator

To remove the \(i\) from the denominator, multiply the numerator and the denominator by \(i\), which is equivalent to \(1\) and won't change the value of the fraction. Doing so, we get: \(\frac{(8+16i) \cdot i}{(2i) \cdot i} = \frac{8i+16i^2}{2i^2}\). And since \(i^2 = -1\), the fraction simplifies to: \(\frac{8i-16}{-2}\)
03

Simplify the Result

Now, divide both terms in the numerator by -2 to simplify the fraction. That gives us: \(\frac{8i-16}{-2}= -4i +8\).
04

Write the Result in Standard Form

The standard form for a complex number is \(a+bi\). Thus, the final answer would be \(8-4i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Imaginary Numbers
An imaginary number is essentially a number that yields a negative result when squared. The standard imaginary unit is denoted as 'i', and by definition, it satisfies the equation \(i^2 = -1\). Naturally, this concept might seem strange at first, because no real number can fulfill this condition.

Imaginary numbers play a crucial role when we need to take square roots of negative numbers. For instance, \(\sqrt{-4}\) is not possible in the realm of real numbers, but it can be expressed as \(2i\) using imaginary numbers. They are central to complex numbers and help in solving equations and understanding phenomena that cannot be described with real numbers alone.
Simplifying Complex Fractions
To simplify a complex fraction, the goal is to have a single fraction with a real number in the denominator and a complex number in the numerator. It's often useful to eliminate the imaginary part from the denominator, as seen in the exercise. Multiplying the numerator and denominator by the complex conjugate of the denominator achieves this.

Notice, in our exercise, we simply multiplied by 'i' to remove 'i' from the denominator as a first step. However, typically if you have a denominator like \((a+bi)\), you would multiply by its conjugate \((a-bi)\) to eliminate the imaginary part. This process results in a real number in the denominator, thus simplifying the complex fraction.
Complex Number Standard Form
The standard form of a complex number is expressed as \(a + bi\), where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. In this form, 'a' is known as the real part, and 'bi' is the imaginary part of the complex number.

For example, the outcome of our provided solution is in standard form, \(8 - 4i\), where the real part is 8 and the imaginary part is \(-4i\). The goal when converting any complex expression to standard form is clarity and simplicity鈥攅nsuring that anyone who reads the number can immediately distinguish between its real and imaginary components.

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Most popular questions from this chapter

In Exercises 59-64, write the complex number in standard form. $$ (2-\sqrt{-6})^{2} $$

In Exercises 75-82, simplify the complex number and write it in standard form. $$ \frac{1}{(2 i)^{3}} $$

In Exercises 1-24, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$ \left[3\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)\right]^{4} $$

In Exercises 71-74, determine whether the function has an inverse function. If it does, find its inverse function. $$ h(x)=\sqrt{4 x+3} $$

In Exercises 1-24, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$ \left[5\left(\cos 20^{\circ}+i \sin 20^{\circ}\right)\right]^{3} $$
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