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Chapter 3: Problem 45

Write the expression in standard form. $$ \frac{3}{-i} $$

Short Answer

Expert verified
The expression in standard form is \( 3i \).

Step by step solution

01

Recognize the Complex Denominator

The denominator in the fraction \( \frac{3}{-i} \) is a purely imaginary number \(-i\). To convert this into a standard form, which means the form of \( a + bi \), we need to eliminate the imaginary unit \( i \) from the denominator.
02

Multiply by the Conjugate

Multiply both the numerator and the denominator of the fraction \( \frac{3}{-i} \) by \( i \), the conjugate of \(-i\). This gives: \[ \frac{3 \cdot i}{-i \cdot i} \] which simplifies the denominator.
03

Simplify the Denominator

Calculate \(-i \cdot i = (-1)(i^2) = -(-1) = 1\) since \(i^2 = -1\). So, the denominator becomes \(1\). The fraction now looks like \( \frac{3i}{1} \).
04

Simplify the Fraction

Now, \( \frac{3i}{1} \) simplifies to \( 3i \). This is in the form \( 0 + 3i \), which is the standard form a complex number.

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Key Concepts

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

Imaginary Numbers
Imaginary numbers are numbers that can be expressed in the form of a real number multiplied by the imaginary unit, which is denoted by the letter \(i\). The imaginary unit is a mathematical concept used to represent the square root of \(-1\). This means that \(i^2 = -1\), which is a key property of imaginary numbers.
For instance, what makes \(-i\) in our original exercise a purely imaginary number is its form. It is essentially \(0 + (-1)i\), where the real part is zero, and the imaginary part is "\(-1i\)".
Standard Form of Complex Numbers
A complex number is typically expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This way of writing a complex number is known as its standard form.
In our context, put simply, the goal is to write a number in the form \(a + bi\), even if either \(a\) or \(b\) is zero. For example:
  • If \(a = 0\), then the complex number is purely imaginary.
  • If \(b = 0\), then the complex number is purely real.
In the problem at hand, the result \(3i\) can be seen as \(0 + 3i\), which perfectly adheres to this standard form.
Complex Conjugate
The complex conjugate of a complex number is found by changing the sign of its imaginary part. For a number \(a + bi\), the complex conjugate is \(a - bi\). Conjugates have a unique utility: when multiplied by each other, they yield a real number.
In the exercise given,
  • The conjugate of \(-i\) (which is \(0 - i\)) is \(i\) (or \(0 + i\)).
  • This means wherever we see \(-i\) in the denominator, multiplying by its conjugate \(i\) will help eliminate the imaginary unit.
By multiplying the denominator and the numerator by \(i\), we managed to transform the fractional expression into a standard form where the imaginary part is in the numerator.
Simplifying Fractions with Imaginary Denominators
Ensuring fractions with imaginary denominators are simplified into a recognizable form is essential. The method for doing so is to eliminate the imaginary number from the denominator by multiplication.
Using the complex conjugate is a systematic way to achieve this simplification. In the exercise example, the fraction \(\frac{3}{-i}\) was transformed by multiplying both the numerator and the denominator by \(i\). This yielded:
  • The denominator simplified from \(-i \cdot i\) to \(1\), as \(-i^2 = 1\).
  • The numerator became \(3i\), resulting in a fraction simplification to \(3i\), or \(0 + 3i\) in standard form.
In any fraction involving imaginary numbers as denominators, this technique of using the complex conjugate is vital in achieving the desired standard form.

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Most popular questions from this chapter

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=f(x+1)-1\)

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 9 x(x-4)=-36 $$

The table lists numbers of Wal-Mart employees \(E\) in millions, \(x\) years after 1987 . $$ \begin{array}{cccccc} x & 0 & 5 & 10 & 15 & 20 \\ \hline B & 0.20 & 0.38 & 0.68 & 1.4 & 2.2 \end{array} $$ (a) Evaluate \(E(15)\) and interpret the result. (b) Find a quadratic function \(f\) that models these data. (c) Graph the data and function \(f\) in the same \(x y\) -plane. (d) Use \(f\) to estimate the year when the number of cmployees may reach 3 million.

U.S. Home Ownership The general trend in the percentage \(P\) of homes lived in by owners rather than renters between 1990 and 2006 is modeled by $$P(x)=0.00075 x^{2}+0.17 x+44$$ where \(x=0\) comesponds to \(1990, x=1\) to \(1991,\) and so on. Determine a function \(g\) that computes \(P\), where \(x\) is the actual year. For example, \(P(0)=44,\) so \(g(1990)=44\)

Use transformations to sketch a graph of \(f\). \(f(x)=2(x-1)^{2}+1\)
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