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Chapter 6: Problem 4

What is \((a+b i)(a-b i)=?\)

Short Answer

Expert verified
The product of a complex number and its conjugate is \(a^2 + b^2\). So, \((a + bi) (a - bi) = a^2 + b^2\).

Step by step solution

01

Identify the Complex Numbers and their Conjugates

Let's denote \(a + bi\) as the complex number and \(a - bi\) as its conjugate. Remember that 'a' and 'b' are real numbers, and 'b' is the imaginary part.
02

Multiply the two Complex Numbers

Multiplying two complex numbers means applying the distributive property, equivalent to FOIL method in binomials. That gives us \((a + bi)(a - bi) = a^2 - abi + abi - b^2i^2\). 'i' is the imaginary unit with the property \(i^2 = -1\).
03

Simplify the Result

Simplify the equation after applying the property of 'i' and adding like terms. This results in \(a^2 - (-b^2) = a^2 + b^2\).

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Key Concepts

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

Understanding the Complex Conjugate
A complex conjugate pairs up with a complex number to assist in various mathematical operations. When we have a complex number, say \(a+bi\), its complex conjugate is represented as \(a-bi\). Essentially, the complex conjugate flips the sign of the imaginary part.
  • Real and Imaginary Parts: Here, \(a\) is the real part, and \(b\) is the imaginary part of the complex number.
  • Purpose of Conjugate: Complex conjugates help in simplifying algebraic expressions, especially in division and multiplication tasks.
  • Properties: Multiplying a complex number by its conjugate results in a real number, as seen in our original problem where \((a+bi)(a-bi)\) simplifies chiefly due to this reason.
Understanding this can help when dealing with complex equations and ensures your solutions remain real when needed.
The Role of the Imaginary Unit
The imaginary unit, denoted as \(i\), is fundamental to complex numbers. By definition, \(i\) is the square root of -1, and it follows a unique set of rules.
  • Properties of \(i\): One crucial aspect is that \(i^2 = -1\). This property is key to simplifying expressions involving imaginary numbers.
  • Cyclic Pattern: Powers of \(i\) repeat in a cycle: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and so forth.
  • Real World Interpretation: Although it represents an "imaginary" concept, \(i\) is used in fields such as engineering to solve real-world problems that involve oscillations and wave behaviors.
Being familiar with \(i\)'s behavior makes working with complex numbers less intimidating and more intuitive.
Applying the FOIL Method to Complex Numbers
The FOIL method, which stands for First, Outer, Inner, Last, is a strategy used to multiply two binomials. When extending this to complex numbers, the method works similarly.
  • Step-by-Step: For \((a+bi)(a-bi)\), you multiply as follows:
    • First: Multiply the first terms: \(a \times a = a^2\).
    • Outer: Multiply the outer terms: \(a \times -bi = -abi\).
    • Inner: Multiply the inner terms: \(bi \times a = abi\).
    • Last: Multiply the last terms: \(bi \times -bi = -b^2i^2\).
  • Simplification: By applying the property \(i^2 = -1\), the result \(-b^2i^2\) becomes \(+b^2\). The middle terms \((-abi + abi)\) cancel each other, leading to the simplified real result \(a^2 + b^2\).
Mastering the FOIL method with complex numbers enables you to handle even the most daunting algebraic expressions with confidence.

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Most popular questions from this chapter

In Example 6.8, the SFC for a Boeing 787 is \(0.587 \mathrm{lbm} / \mathrm{h} / \mathrm{lbf}\) leading to a fuel consumption rate of \(12,680 \mathrm{lbm} / \mathrm{h}\). If a standard aviation jet fuel (JP-4) costs \( 3.50 / \mathrm{US}\) gallon, has a density of \(6.84 \mathrm{lbm} / \mathrm{US}\) gallon, how much does fuel cost for each hour of level flight?

Simplify: \(\frac{(3+2 i)^{2}}{(-1+3 i)}\)

What is the total lift in steady level flight on an airfoil with a \(21.0 \mathrm{~m}\) wingspan traveling at \(150 . \mathrm{m} / \mathrm{s}\) if the air density is \(0.525 \mathrm{~kg} / \mathrm{m}^{3}\) and the circulation is \(55.0 \mathrm{~m}^{2} / \mathrm{s}\) ?

Carry out the multiplication \((3+2 i)(1+7 i)=\) ?

What are the real and imaginary parts of \(A=1 / z^{2}\), where \(z=x+i y\). Write your answer in the form \(A=B+i C=\operatorname{Re}(A)+i \operatorname{Im}(A)\).
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