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Chapter 1: Problem 16

Write the given number in the form \(a+i b\). $$ \frac{(4+5 i)+2 i^{3}}{(2+i)^{2}} $$

Short Answer

Expert verified
The expression in the form \(a + bi\) is \(\frac{24}{25} - \frac{7}{25}i\).

Step by step solution

01

Simplify the Numerator

The given expression in the numerator is \((4 + 5i) + 2i^3\). First, calculate \(i^3 = i^2 \cdot i = (-1) \cdot i = -i\). Therefore, \(2i^3 = 2(-i) = -2i\). Now, substitute this back into the numerator to get \((4 + 5i) - 2i = 4 + 3i\).
02

Expand the Denominator

The denominator is \((2+i)^2\). Expand it using the formula \((a+b)^2 = a^2 + 2ab + b^2\), where \(a = 2\) and \(b = i\). This gives: \(2^2 + 2 \cdot 2 \cdot i + i^2 = 4 + 4i + (-1) = 3 + 4i\).
03

Divide the Complex Numbers

We have the fraction \(\frac{4 + 3i}{3 + 4i}\). Multiply both numerator and denominator by the conjugate of the denominator \(3 - 4i\) to get:\[\frac{(4 + 3i)(3 - 4i)}{(3 + 4i)(3 - 4i)}\]The denominator simplifies to \(3^2 - (4i)^2 = 9 + 16 = 25\).
04

Simplify the Numerator After Multiplication

Distribute in the numerator as follows:\((4 + 3i)(3 - 4i) = 4 \times 3 + 4 \times (-4i) + 3i \times 3 + 3i \times (-4i)\)Which simplifies to:\(12 - 16i + 9i - 12(-1) = 24 - 7i\)
05

Express in Standard Form

We divide each term in the numerator by the denominator:\[\frac{24 - 7i}{25} = \frac{24}{25} - \frac{7}{25}i\]So, the expression in the form \(a + bi\) is \(\frac{24}{25} - \frac{7}{25}i\).

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Key Concepts

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

Complex Number Division
When dividing complex numbers, the process involves finding the quotient of two complex expressions. A complex number has the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. To divide one complex number by another, you essentially multiply both the numerator and the denominator by the conjugate of the denominator. This is because you want to eliminate the imaginary component from the denominator, simplifying the division.
Here's a step-by-step breakdown of the approach:
  • Identify the complex conjugate of the denominator.
  • Multiply both the numerator and the denominator by this conjugate.
  • Perform the multiplication and simplify where possible.
  • After the imaginary unit has been eliminated from the denominator, express the result in its simplest form (\(a + bi\)).
This technique converts the division of complex numbers into a multiplication problem, dealing with simpler arithmetic.
Complex Conjugate
A complex conjugate is essentially two complex numbers, a mirror image over the real axis. If you have a complex number \(a + bi\), the complex conjugate is \(a - bi\). The conjugate is crucial when dividing complex numbers because multiplying a complex number by its conjugate results in a real number. This property makes the conjugate a valuable tool in simplifying division within complex numbers.
How to find and use the conjugate:
  • Change the sign of the imaginary part of the complex number (\(bi\) becomes \(-bi\)).
  • Use the conjugate to multiply with the denominator, thus eliminating the imaginary part.
  • This operation results in a real number in the denominator, simplifying the solution.
Imaginary Unit
The imaginary unit is a fundamental concept in complex numbers, represented by the symbol \(i\). It is defined by the property that \(i^2 = -1\). This unit is essential in the realm of complex numbers, where the square of \(i\) results in a real number. This is particularly useful when dealing with higher powers of \(i\).
Considerations involving \(i\):
  • Powers of \(i\) cycle in a pattern: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).
  • For expressions involving higher powers, break them down into one of these powers for ease of calculation.
  • Understanding \(i\) allows for the manipulation and simplification of complex equations.
Complex Number Simplification
Simplifying complex numbers often means expressing them in their standard form, \(a + bi\). This step involves several operations, including addition, subtraction, multiplication, and division of the complex numbers. Each process requires careful manipulation of the real and imaginary components.
Steps for simplifying:
  • Use arithmetic operations on real and imaginary parts separately. For addition and subtraction, combine like terms of real and imaginary components.
  • For multiplication, apply the distributive property, and replace \(i^2\) with \(-1\) when it appears.
  • In division, follow the process of multiplying by the conjugate to simplify the expression into the standard form.
By converting complex expressions into the form \(a + bi\), they become easier to interpret and manage.

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Most popular questions from this chapter

On page 31 we stated that if \(\rho_{1}>0\), then the set of points satisfying \(\rho_{1}<\left|z-z_{0}\right|\) is the exterior to the circle of radius \(\rho_{1}\) centered at \(z_{0} .\) In general, describe the set if \(\rho_{1}=0 .\) In particular, describe the set defined by \(|z+2-5 i|>0\).

Factor the given quadratic polynomial if the indicated complex number is one root. $$ 3 i z^{2}+(9-16 i) z-17-i ; z_{1}=5+2 i $$

Cubic Formula In this project you are asked to investigate the solution of a cubic polynomial equation by means of a formula using radicals, that is, a combination of square roots and cube roots of expressions involving the coefficients. (a) To solve a general cubic equation \(z^{3}+a z^{2}+b z+c=0\) it is sufficient to solve a depressed cubic equation \(x^{3}=m x+n\) since the general cubic equation can be reduced to this special case by eliminating the term \(a z^{2}\). Verify this by means of the substitution \(z=x-a / 3\) and identify \(m\) and \(n\). (b) Use the procedure outlined in part (a) to find the depressed cubic equation for \(z^{3}+3 z^{2}-3 z-9=0\) (c) A solution of \(x^{3}=m x+n\) is given by $$ x=\left[\frac{n}{2}+\left(\frac{n^{2}}{4}-\frac{m^{3}}{27}\right)^{1 / 2}\right]^{1 / 3}+\left[\frac{n}{2}-\left(\frac{n^{2}}{4}-\frac{m^{3}}{27}\right)^{1 / 2}\right]^{1 / 3}. $$ Use this formula to solve the depressed cubic equation found in part (b). (d) Graph the polynomial \(z^{3}+3 z^{2}-3 z-9\) and the polynomial from the depressed cubic equation in part (b); then estimate the \(x\) -intercepts from the graphs. (e) Compare your results from part (d) with the solutions found in part (c). Resolve any apparent differences. Find the three solutions of \(z^{3}+3 z^{2}-\) \(3 z-9=0\) (f) Do some additional reading to find geometrically motivated proofs (using a square and a cube) to derive the quadratic formula and the formula given in part (c) for the solution of the depressed cubic equation. Why is the name quadratic formula used when the prefix quad stems from the Latin word for the number four?

Suppose \(z_{1}, z_{2}, z_{3}\), and \(z_{4}\) are four distinct complex numbers. Interpret geometrically: $$ \arg \left(\frac{z_{1}-z_{2}}{z_{3}-z_{4}}\right)=\frac{\pi}{2}. $$

Solve the given quadratic equation using the quadratic formula. Then use (5) to factor the polynomial. $$ z^{2}+2 z-\sqrt{3} i=0 $$
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