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Chapter 12: Problem 70

Let \(z=a+b i\) (a) Show that \((\bar{z})^{2}=\overline{z^{2}}\) (b) Show that \((\bar{z})^{3}=\overline{z^{3}}\)

Short Answer

Expert verified
Both parts are confirmed: \((\bar{z})^{2} = \overline{z^{2}}\) and \((\bar{z})^{3} = \overline{z^{3}}\).

Step by step solution

01

Understanding Complex Conjugates

To tackle this problem, we need a clear understanding of what a complex conjugate is. If we have a complex number \(z = a + bi\), its complex conjugate \(\bar{z}\) is \(a - bi\). The conjugate essentially reflects the complex number over the real axis on the complex plane.
02

Squaring the Complex Number

The first part of the problem asks us to show that \((\bar{z})^{2} = \overline{z^{2}}\). First, calculate the square of the complex number: \(z^{2} = (a + bi)(a + bi) = a^2 + 2abi - b^2\). Therefore, \(z^{2} = a^2 - b^2 + 2abi\).
03

Finding the Conjugate of the Square

Next, find the conjugate of \(z^{2}\): \(\overline{z^{2}} = (a^2 - b^2) - 2abi\).
04

Squaring the Conjugate

Now compute \((\bar{z})^{2}\): \((\bar{z})^2 = (a - bi)^2 = a^2 - 2abi + b^2\). Therefore, \((\bar{z})^2 = (a^2 - b^2) - 2abi\).
05

Comparing Results for Part A

Observe that \((\bar{z})^{2} = \overline{z^{2}}\). Both yield \((a^2 - b^2) - 2abi\), showing that the statement is true.
06

Cubing the Complex Number

For part (b), calculate \(z^{3} = (a + bi)^{3} = (a + bi)(a + bi)(a + bi)\). Begin by calculating \(z^{2}\), then multiply the result by \(a + bi\) again to get \(z^{3}\).
07

Expanding and Simplifying Cubic Expression

\(z^{3} = (a^2 - b^2 + 2abi)(a + bi) = a^3 - a b^2 + 3 a^2 bi - 3 a b^2 i + b^3 i\). Simplify to get \(z^{3} = (a^3 - 3ab^2) + (3a^2b - b^3)i\).
08

Finding the Conjugate of the Cube

Next, find the conjugate of \(z^{3}\): \(\overline{z^{3}} = (a^3 - 3ab^2) - (3a^2b - b^3)i\).
09

Cubing the Conjugate

Now, compute \((\bar{z})^{3}\): \((\bar{z})^3 = (a - bi)^3 = (a - bi)(a - bi)(a - bi)\). Expansion yields \((a^3 - 3ab^2) - (3a^2b - b^3)i\).
10

Comparing Results for Part B

After expanding and simplifying \((\bar{z})^3\), notice that \((\bar{z})^3 = \overline{z^{3}}\), confirming the correctness of the expression for the cube.

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Key Concepts

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

Complex Numbers
Complex numbers are an extension of real numbers that include imaginary numbers. They are usually expressed in the standard form \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, and \(i\) is the imaginary unit with the property \(i^2 = -1\). This allows us to encompass a broader range of numbers and solve equations that have no solutions in the real number system alone.

Key points to remember about complex numbers:
  • They can be added, subtracted, multiplied, and divided just like real numbers, with some additional rules for complex operations.
  • The conjugate of a complex number \(a + bi\) is \(a - bi\). This is crucial when dividing complex numbers as it helps eliminate the imaginary part from the denominator.
Understanding complex numbers is vital in fields such as engineering and physics because they allow for the representation and analysis of periodic functions, oscillations, and much more.
Mathematical Proof
A mathematical proof is a logical argument that establishes the truth of a theorem or statement. Proofs are essential in mathematics to confirm that a particular proposition is universally true.

Typically, a proof involves several steps that transform the premises of a statement into its conclusion using logical reasoning.

In the given exercise, we use algebraic manipulation to prove properties of complex conjugates.

While proving \((\bar{z})^2 = \overline{z^2}\), we:
  • Calculate \(z^2\) for a complex number \(z = a + bi\).
  • Determine its conjugate, leading to \(\overline{z^2}\).
  • Calculate \((\bar{z})^2\) directly, confirming both results' equality.
This systematic approach ensures that no assumptions are made, only facts derived from operations on complex numbers are used.
Complex Plane
The complex plane, also known as the Argand plane, is a way to visualize complex numbers. Here, the horizontal axis represents the real part, and the vertical axis represents the imaginary part. This is useful because it turns complex number operations into geometric transformations.

Here are some crucial aspects of the complex plane:
  • The position of a complex number \(z = a + bi\) on the plane is uniquely defined by the pair \((a, b)\).
  • The conjugate \(\bar{z} = a - bi\) is reflected across the real axis, which helps in understanding complex conjugates geometrically.
  • Operations like addition and multiplication can be visualized as shifting and rotating numbers in the plane.
This way of visualizing makes it easier to see things like magnitude (distance from the origin) and argument (angle with the positive real axis), which are foundational concepts in complex analysis and signal processing.

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Most popular questions from this chapter

Each equation has exactly one positive root. In each case, locate the root between successive hundredths. Determine the successive integer bounds by computing \(f(0), f(1), f(2),\) and so on, until you find a sign change. $$x^{3}-8 x^{2}+21 x-22=0$$

You need to know that a prime number is a positive integer greater than 1 with no factors other than itself and 1. Thus the first seven prime numbers are 2,3,5,7,11,13 and 17. Find all prime numbers \(p\) for which the equation \(x^{2}+x-p=0\) has a rational root.

Scipio Ferro of Bologna well-nigh thirty years ago discovered this rule and handed it on to Antonio Maria Fior of Venice, whose contest with Niccol貌 Tartaglia of Brescia gave Niccol貌 occasion to discover it. He / Tartaglial gave it to me in response to my entreaties, though withholding the demonstration. Armed with this assistance, I sought out its demonstration in /various / forms. - Girolamo Cardano, Ars Magna (Nuremberg, 1545 ) The quotation is from the translation of Ars Magna by T. Richard Witmer (New York: Dover Publications, 1993 ). In his book Ars Magna (The Great Art) the Renaissance mathematician Girolamo Cardano \((1501-1576)\) gave the following formula for a root of the equation \(x^{3}+a x=b\). $$x=\sqrt[3]{\frac{b}{2}+\sqrt{\frac{b^{2}}{4}+\frac{a^{3}}{27}}}-\sqrt[3]{\frac{-b}{2}+\sqrt{\frac{b^{2}}{4}+\frac{a^{3}}{27}}}$$ (a) Use this formula and your calculator to compute a root of the cubic equation \(x^{3}+3 x=76\) (b) Use a graph to check the answer in part (a). That is, graph the function \(y=x^{3}+3 x-76,\) and note the \(x-\) intercept. Also check the answer simply by substituting it in the equation \(x^{3}+3 x=76\)

\(\frac{x^{3}+4 x^{2}+16 x}{x^{4}+16}\) Hint: To factor the denominator, add and subtract the term \(16 x^{2}\)

Express the polynomial \(f(x)\) in the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\). Find a fourth-degree polynomial function that has zeros \(\sqrt{2},-\sqrt{2}, 1,\) and -1 and a graph that passes through (2,-20).
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