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Chapter 2: Problem 20

Write the complex number in standard form.$$-4 i^{2}+2 i$$.

Short Answer

Expert verified
The complex number in standard form is \(4 + 2i\).

Step by step solution

01

Substitute the Value of \(i^2\)

We begin by replacing the value of \(i^2\). In the context of complex numbers, it's known that \(i^2 = -1\). Therefore, the expression becomes \(-4*(-1) + 2i\).
02

Simplify the Expression

By simplifying the expression we obtain: \(4 + 2i\).
03

Check the final form

Compare with the standard form. It can be seen that the form now fits the standard form \(a + bi\), with \(a = 4\) and \(b = 2\). Thus, the complex number in standard form is \(4 + 2i\).

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Key Concepts

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

Imaginary Unit
In mathematics, the imaginary unit is a key concept in understanding complex numbers. It is denoted by the letter \(i\) and has a unique property: when squared, it equals -1.
This means \(i^2 = -1\), a fundamental definition that distinguishes real numbers from complex numbers.
  • The imaginary unit allows us to extend the concept of numbers beyond the number line into a two-dimensional plane.
  • Using the imaginary unit, we can define complex numbers, which include a real and an imaginary part.
In our exercise, the imaginary unit helps transform \(-4i^2 + 2i\) into a form we can further simplify by recognizing \(i^2 = -1\). Understanding the imaginary unit \(i\) provides the foundation for working with all sorts of complex numbers.
Standard Form of Complex Numbers
The standard form of a complex number is essential for understanding how to express complex numbers in a convenient and useful way. This form is denoted as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
  • \(a\) is referred to as the real part of the complex number.
  • \(b\) is the coefficient of the imaginary part \(i\).
Putting a complex number in standard form makes it easier to perform arithmetic operations such as addition, subtraction, and comparison with other complex numbers.
By expressing \(-4i^2 + 2i\) as \(4 + 2i\), we have successfully rewritten the number in the form \(a + bi\) where \(a = 4\) and \(b = 2\), simplifying further operations and analysis.
Simplification of Expressions
Simplifying complex number expressions is an important skill that can make complex calculations easier and more intuitive. The process often involves substituting known values and performing basic arithmetic operations.
  • In our exercise, simplifying \(-4i^2 + 2i\) begins with substituting the value \(i^2 = -1\), allowing us to rewrite the expression.
  • Next, replace \(-4(-1)\) with \(4\), transforming the expression to \(4 + 2i\).
The illustration of these steps shows the importance of methodically substituting and calculating to reach a simplified form.
Simplification helps in recognizing the standard form of complex numbers, essential for clear communication and further studies in mathematics. Through these steps, the initial expression is simplified to \(4 + 2i\), offering a clearer understanding.

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Most popular questions from this chapter

Graphical Analysis From 1950 through 2005 , the per capita consumption \(C\) of cigarettes by Americans \(\begin{array}{lllllll}\text { (age } & 18 & \text { and } & \text { older) } & \text { can } & \text { be } & \text { modeled } & \text { by }\end{array}\) \(C=3565.0+60.30 t-1.783 t^{2}, 0 \leq t \leq 55,\) where \(t\) is the year, with \(t=0\) corresponding to 1950 . (Source: Tobacco Outlook Report) (a) Use a graphing utility to graph the model. (b) Use the graph of the model to approximate the maximum average annual consumption. Beginning in \(1966,\) all cigarette packages were required by law to carry a health warning. Do you think the warning had any effect? Explain. (c) In 2005, the U.S. population (age 18 and over) was \(296,329,000 .\) Of those, about 59,858,458 were smokers. What was the average annual cigarette consumption per smoker in 2005? What was the average daily cigarette consumption per smoker?

Find the values of \(b\) such that the function has the given maximum or minimum value. $$f(x)=-x^{2}+b x-75 ; \text { Maximum value: } 25$$

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$f(x)=x^{2}-6 x$$

Sketch the graph of a fifth-degree polynomial function whose leading coefficient is positive and that has a zero at \(x=3\) of multiplicity 2.

Use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). Then check your results algebraically by writing the quadratic function in standard form. $$f(x)=x^{2}+10 x+14$$
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