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Chapter 4: Problem 46

In Exercises 45-54, write the quotient in standard form. $$ -\frac{14}{2 i} $$

Short Answer

Expert verified
The standard form of the given complex number is \( 7i \)

Step by step solution

01

Identify the given expression

The provided complex number is \( -\frac{14}{2i} \) where 'i' is an imaginary unit with the property \( i^2 = -1 \)
02

Multiply by \( \frac{-i}{-i} \)

To write this into standard form, we can multiply the denominator by -i (which is standard practice to remove 'i' from the denominator). Remember that we have to multiply the numerator by -i as well to maintain the equality, getting \( \frac{-14i}{2i*-i} \)
03

Simplify the expression

Simplify the expression, knowing \( i^2 = -1 \), so that \( \frac{-14i}{2*-1} \) simplifies to \( 7i \)

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Key Concepts

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

Imaginary Unit
The imaginary unit, denoted as 'i', is a fundamental concept in complex numbers. It is defined by the equation \( i^2 = -1 \). This property allows us to deal with the square roots of negative numbers, something that isn't possible within the realm of real numbers. Imaginary numbers are multiples of 'i', such as \( 2i \) or \( -3i \). When 'i' is involved in calculations involving complex numbers, it often requires special rules of algebra to manage the expressions effectively.
  • Imaginary numbers can be considered as perpendicular to real numbers in the complex plane.
  • The complex number system extends the real number system by incorporating these imaginary units.
Understanding 'i' is crucial, as it forms the basis for working with more complex numbers, enabling you to solve equations that have no real solutions.
Standard Form
Complex numbers are typically expressed in a 'standard form', which takes the shape \( a + bi \), where 'a' and 'b' are real numbers. The 'a' part is known as the real component, and the 'b' part, which is multiplied by 'i', is the imaginary component. Writing complex numbers in standard form is essential for effectively adding, subtracting, multiplying, or dividing them.
  • In standard form, it鈥檚 easy to see the real and imaginary parts at a glance.
  • It provides a uniform way to handle all complex numbers.
To convert a complex expression into standard form, you might need to simplify terms or rearrange them, as shown in the problem statement where the quotient was rewritten to eliminate 'i' from the denominator.
Quotient of Complex Numbers
Finding the quotient of complex numbers involves dividing one complex number by another. Since you often encounter imaginary units in the denominator, you need a technique to simplify such expressions. This is typically done by 'rationalizing the denominator'.To rationalize, you multiply both the numerator and the denominator by the conjugate of the denominator. For example, if the denominator is \( 2i \), its conjugate is \( -i \), as shown in the exercise. The process involves:
  • Multiplying the numerator and the denominator by the conjugate of the denominator.
  • Simplifying the resulting expression using \( i^2 = -1 \).
  • Writing the result in standard form.
Rationalizing helps remove the imaginary number from the denominator, making it easier to express and further manipulate the complex number.

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Most popular questions from this chapter

In Exercises 45-60, use the theorem on page 356 to find all the solutions of the equation and represent the solutions graphically. $$ x^{6}+1=0 $$

In Exercises 1-24, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$ \left[3\left(\cos 15^{\circ}+i \sin 15^{\circ}\right)\right]^{4} $$

In Exercises 97-100, solve the equation and check your solution. $$ 4(5 x-6)-3(6 x+1)=0 $$

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In Exercises 65-74, use the Quadratic Formula to solve the quadratic equation. $$ 4 x^{2}+16 x+15=0 $$
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