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Chapter 2: Problem 20

Find each product and write the result in standard form. $$(5-2 i)^{2}$$

Short Answer

Expert verified
The square of the complex number \(5 - 2i\) in standard form is \(21 - 20i\).

Step by step solution

01

Identify the real and imaginary parts

The given complex number is \(5 - 2i\). Here, the real part, \(a\), is 5 and the imaginary part \(b\), is 2.
02

Apply the formula

We use the formula \((a - bi)^2 = a^2 - 2abi - b^2\). Substituting \(a = 5\) and \(b = 2\) into the formula, we get \((5 - 2i)^2 = 5^2 - 2*(5 * 2)i - 2^2\). Simplify the terms to: \(25 - 20i - 4\).
03

Simplify the expression

Simplify the expression by combining the real numbers: \(25 - 4 = 21\), so the final equation is \(21 - 20i\) after simplifying.

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Key Concepts

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

Complex Conjugate
The complex conjugate of a complex number takes a given complex number and changes the sign of the imaginary part. If we have a complex number in the form of \( a + bi \), its conjugate is \( a - bi \). This process is useful in many applications, such as simplifying expressions involving complex numbers.
For instance, if the complex number is \( 5 - 2i \), the complex conjugate would be \( 5 + 2i \). Notice how the real part stays the same, but the imaginary part changes its sign from negative to positive.
  • Used to eliminate the imaginary part of a denominator when multiplying by its conjugate.
  • Helps in finding the magnitude of a complex number, utilizing the formula: \( |z| = \sqrt{z \cdot \overline{z}} \).
So, understanding complex conjugates can help solve and simplify problems with complex numbers more effectively.
Imaginary Unit
The imaginary unit, denoted as \( i \), is the core component of complex numbers. It is defined by the property \( i^2 = -1 \).
This special property allows calculations with complex numbers and extends real numbers into a more comprehensive number system. For example, in the expression \((5-2i)^2\), the term \(-2ai\) involved \(bi\), where \(b\) was the imaginary unit multiplier.
  • Understanding \(i\) can help grasp many concepts in complex arithmetic like the multiplication, division, and squaring of numbers.
  • \(i\) represents the square root of \(-1\), a concept not represented within real numbers.
Grasping the imaginary unit \(i\) empowers you to tackle more intricate mathematical problems efficiently.
Standard Form
The standard form of a complex number is given by \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. Writing in standard form allows for a uniform way to present complex numbers, making them easier to work with in calculations or comparisons.
In the exercise provided, when the complex number \((5 - 2i)^2\) was simplified, the result was \(21 - 20i\). Here, the number is in standard form with a real part of 21 and an imaginary part of \(-20i\).
  • Standardizing the form aids in performing additions, subtractions, and showing results clearly.
  • Beneficial in plotting complex numbers on the complex plane where the x-axis represents real parts, and the y-axis represents imaginary parts.
Mastering the idea of standard form brings clarity to the sometimes complex world of complex numbers, greatly aiding in solving mathematical problems.

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Most popular questions from this chapter

Use point plotting to graph \(f(x)=2^{x} .\) Begin by setting up a partial table of coordinates, selecting integers from \(-3\) to \(3,\) inclusive, for \(x .\) Because \(y=0\) is a horizontal asymptote, your graph should approach, but never touch, the negative portion of the \(x\) -axis.

Use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$(x-2)^{2} \leq 0$$

Write the equation of each parabola in standard form. Find the point on the line whose equation is \(2 x+y-2=0\) that is closest to the origin. Hint: Minimize the distance function by minimizing the expression under the square root.

Write a polynomial inequality whose solution set is \([-3,5]\)

Describe how to find a parabola's vertex if its equation is expressed in standard form. Give an example.
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