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Chapter 8: Problem 59

For Exercises 49-64, write each quotient in standard form. $$ \frac{2+3 i}{3-5 i} $$

Short Answer

Expert verified
The quotient in standard form is \(-\frac{9}{34} + \frac{19}{34}i\).

Step by step solution

01

Identify the Conjugate

To write the quotient \( \frac{2+3i}{3-5i} \) in standard form, first identify the conjugate of the denominator. The conjugate of \(3-5i\) is \(3+5i\).
02

Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate \(3+5i\): \[ \frac{(2+3i)(3+5i)}{(3-5i)(3+5i)}. \] This step ensures the denominator becomes a real number.
03

Multiply the Numerators

Apply the distributive property (FOIL) to the numerator: \[ (2+3i)(3+5i) = (2 \cdot 3) + (2 \cdot 5i) + (3i \cdot 3) + (3i \cdot 5i).\] Simplify this to: \[ 6 + 10i + 9i + 15i^2. \] Since \(i^2 = -1\), replace \(15i^2\) with \(-15\): \[ 6 + 10i + 9i - 15 = -9 + 19i. \]
04

Multiply the Denominators

Multiply the denominator using the difference of squares formula: \[(3-5i)(3+5i) = 3^2 - (5i)^2 = 9 - 25i^2.\]Since \(i^2 = -1\), replace \(-25i^2\) with \(-25(-1) = 25\):\[ 9 + 25 = 34. \]
05

Express the Quotient in Standard Form

Now, express the entire fraction in standard form \(a + bi\): \[ \frac{-9 + 19i}{34} = \frac{-9}{34} + \frac{19i}{34}. \] Simplify to get: \[ -\frac{9}{34} + \frac{19}{34}i. \]

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Key Concepts

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

Standard form of complex numbers
Complex numbers are an essential part of algebra and can be expressed in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit defined by \(i^2 = -1\). The term \(a\) is the real part, while \(bi\) is the imaginary part. This form is known as the standard form of complex numbers.

Writing complex numbers in this form allows for easy comparison, addition, subtraction, and further complex operations. For example, in the exercise where the fraction \(\frac{2+3i}{3-5i}\) was converted into standard form, the final result was displayed as \(-\frac{9}{34} + \frac{19}{34}i\). This setup reflects a cleaned-up version where we have distinct real and imaginary parts.
  • Ensure both real and imaginary components are separate and presented clearly.
  • Standard form helps in easily manipulating and understanding complex numbers.
  • Use the standard form to relate to real-world applications involving phase and amplitude calculations in engineering.
Conjugate of a complex number
A complex conjugate involves changing the sign of the imaginary part of a complex number. For a complex number \(a + bi\), its conjugate is \(a - bi\). This concept is fundamental when simplifying expressions involving complex numbers, especially when involved in division.

In the given example, the denominator \(3-5i\) was assigned a conjugate of \(3+5i\). By multiplying both the numerator and the denominator by this conjugate, a real number denominator was achieved during simplification.
  • The conjugate is crucial for removing complexity from the equation, especially when dealing with complex fractions.
  • It helps in achieving a result that fits neatly into the standard form.
  • It is particularly useful in engineering and physics for simplifying wave functions and expressions.
Multiplication of complex numbers
Multiplying complex numbers involves applying the distributive property, also known as the FOIL method (First, Outer, Inner, Last). This technique is used to multiply two binomials.

Consider the multiplication in the exercise:
  • First terms: \(2 \times 3 = 6\)
  • Outer terms: \(2 \times 5i = 10i\)
  • Inner terms: \(3i \times 3 = 9i\)
  • Last terms: \(3i \times 5i = 15i^2\)
Notice that \(15i^2\) becomes \(-15\) since \(i^2 = -1\). Combine like terms to simplify the expression to \(-9 + 19i\). This multiplication transforms the numerator into a new complex number that fits the standard form once the denominator is rationalized by using the conjugate.
  • FOIL method applies very effectively in complex number multiplication.
  • The imaginary unit's property \(i^2 = -1\) is pivotal in simplifying the resulting terms.
Rationalizing complex denominators
Rationalizing the complex denominator is a method to make denominators real numbers, often by using the conjugate. In complex arithmetic, ensuring the denominator is real aids in clearer interpretation and simplification, especially when expressing results in standard form.

In the problem, the expression \(\frac{2+3i}{3-5i}\) was rationalized by multiplying both the numerator and denominator by \(3+5i\). The process turned a complex denominator into a real number using the difference of squares formula:

\[(3-5i)(3+5i) = 3^2 - (5i)^2 = 9 + 25 \to 34\] The denominator becomes 34, a real number, enabling the final expression to be simplified easily into standard form.
  • Rationalizing denominators is crucial for simplifying and solving complex fractions.
  • It helps convert complex divisions to a more interpretable and usable form.
  • Understanding this process is valuable in fields requiring precise calculations, such as in signal processing.

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Most popular questions from this chapter

For Exercises 37-46, recall that the flight of a projectile can be modeled with the parametric equations $$ x=\left(v_{0} \cos \theta\right) t \quad y=-16 t^{2}+\left(v_{0} \sin \theta\right) t+h $$ where \(t\) is in seconds, \(v_{0}\) is the initial velocity in feet per second, \(\theta\) is the initial angle with the horizontal, and \(h\) is the initial height above ground, where \(x\) and \(y\) are in feet. Flight of a Projectile. A projectile is launched from the ground at a speed of 400 feet per second at an angle of \(45^{\circ}\) with the horizontal. How far does the projectile travel (what is the horizontal distance), and what is its maximum altitude? (Note the symmetry of the projectile path.)

In Exercises 13-28, express each complex number in polar form. $$ 3+0 i $$

In Exercises 21-40, convert each point given in polar coordinates to exact rectangular coordinates. $$ \left(2, \frac{3 \pi}{4}\right) $$

Fan Blade. The position on the tip of a ceiling fan is given by the parametric equations \(x=\sin (10 t)\) and \(y=\cos (10 t)\), where \(x\) and \(y\) are the vertical and lateral positions relative to the center of the fan, respectively, and \(t\) is the time in seconds. How long does it take for the fan, blade to make one complete revolution?

For Exercises 47 and 48 , refer to the following: Modern amusement park rides are often designed to push the envelope in terms of speed, angle, and ultimately \(g\)-force, and usually take the form of gargantuan roller coasters or skyscraping towers. However, even just a couple of decades ago, such creations were depicted only in fantasy-type drawings, with their creators never truly believing their construction would become a reality. Nevertheless, thrill rides still capable of nauseating any would-be rider were still able to be constructed; one example is the Calypso. This ride is a not-too-distant cousin of the better-known Scrambler. It consists of four rotating arms (instead of three like the Scrambler), and on each of these arms, four cars (equally spaced around the circumference of a circular frame) are attached. Once in motion, the main piston to which the four arms are connected rotates clockwise, while each of the four arms themselves rotates counterclockwise. The combined motion appears as a blur to any onlooker from the crowd, but the motion of a single rider is much less chaotic. In fact, a single rider's path can be modeled by the following graph: The equation of this graph is defined parametrically by $$ \begin{aligned} &x(t)=A \cos t+B \cos (-3 t) \\ &y(t)=A \sin t+B \sin (-3 t), 0 \leq t \leq 2 \pi \end{aligned} $$ Amusement Rides. Suppose the ride conductor was rather sinister and speeded up the ride to twice the speed. How would you modify the parametric equations to model such a change? Now vary the values of \(A\) and \(B\). What do you conjecture these parameters are modeling in this problem?
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