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Chapter 2: Problem 96

True or False? Determine whether the statement is true or false. Justify your answer. $$\begin{array}{l}{\text { The sum of two complex numbers is always a real }} \\\ {\text { number. }}\end{array}$$

Short Answer

Expert verified
The statement is false. The sum of two complex numbers is not always a real number.

Step by step solution

01

Understanding complex numbers

A complex number is a number that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, which satisfies \(i^2 = -1\). The real part of the complex number is \(a\) and the imaginary part is \(b\).
02

Adding two complex numbers

To add two complex numbers, simply add the real parts and the imaginary parts separately. For example, the sum of \(a + bi\) and \(c + di\) is \((a + c) + (b + d)i\).
03

Explanation of whether the sum is a real number

As seen in the previous step, the sum of two complex numbers results in another complex number. The sum is not always a real number, because the sum retains its imaginary part. The only case where the sum would be a real number is if the imaginary parts of both complex numbers cancelled each other out, which doesn't always happen. The claim in the exercise is therefore false.

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Key Concepts

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

Real Numbers
Real numbers are fundamental in mathematics. They include all the numbers we typically encounter in daily life. This includes whole numbers like 1, 2, and 3, fractions like 1/2 and 3/4, and irrational numbers like the square root of 2 or 蟺. Essentially, any number that can be represented on a continuous number line is a real number.

Real numbers are often denoted by the symbol \(a\) in the context of complex numbers. They form the building blocks of more complex number systems, including complex numbers themselves. Understanding real numbers is crucial, as they form the real part of a complex number, which is combined with an imaginary part to create the whole picture.
Imaginary Unit
The imaginary unit is a mathematical concept that extends our number system to incorporate solutions to equations like \(x^2 = -1\) which have no real solutions. The imaginary unit is denoted by \(i\), where \(i^2 = -1\).

Imaginary numbers are multiples of \(i\). For instance, \(2i\) is an imaginary number. They are not 'imaginary' in the colloquial sense of the word, they are very much used in real-world applications, especially in fields like electrical engineering and physics.
  • Example: \(3i\) is an imaginary number.
  • Origin: Imaginary numbers were first introduced to solve polynomial equations that could not be solved using real numbers alone.
Complex Addition
Complex addition is the process of adding two complex numbers. A complex number has a real part and an imaginary part, and it can be expressed as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.

When adding two complex numbers, \((a + bi)\) and \((c + di)\), you simply add the real parts together and the imaginary parts together:
  • The real parts: \(a + c\)
  • The imaginary parts: \(b + d\)
The result is a new complex number: \((a + c) + (b + d)i\). This process shows that complex addition is straightforward, resembling the addition of regular numbers but with two components to account for.
Sum of Complex Numbers
The sum of two complex numbers, as demonstrated through complex addition, typically results in another complex number. For a sum to be purely real, the imaginary parts must cancel each other out, meaning their sum is zero. For instance, if we add \(2 + 3i\) and \(4 - 3i\), the resulting sum would be \((2 + 4) + (3 - 3)i = 6 + 0i\), which simplifies to the real number 6.

Thus, the exercise statement asking if the sum of two complex numbers is always a real number is false. Usually, the sum retains the imaginary component unless specifically manipulated to cancel out. Understanding the behavior of complex sums is crucial for mastering complex calculations in mathematical contexts.

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Most popular questions from this chapter

Website Growth The number \(y\) of hits a new website receives each month can be modeled by \(y=4080 e^{k t},\) where \(t\) represents the number of months the website has been operating. In the website's thirdmonth, there were \(10,000\) hits. Find the value of \(k,\) and use this value to predict the number of hits the website will receive after 24 months.

Population The populations \(P\) (in thousands) of Horry County, South Carolina, from 1980 through 2010 can be modeled by $$P=20.6+85.5 e^{0.0360 t}$$ where \(t\) represents the year, with \(t=0\) corresponding to \(1980 .\) (Source: U.S. Census Bureau) (a) Use the model to complete the table. (b) According to the model, when will the population of Horry County reach \(350,000 ?\) (c) Do you think the model is valid for long-term predictions of the population? Explain.

Comparing Models If \(\$ 1\) is invested over a 10 -year period, then the balance \(A,\) where \(t\) represents the time in years, is given by \(A=1+0.06\|t\|\) or \(A=[1+(0.055 / 365)]^{1365 t ]}\) depending on whether the interest is simple interest at 6\(\%\) or compound interest at at \(5^{1} \%_{b}\) compounded daily. Use a graphing utility to graph each function in the same viewing window. Which grows at a greater rate?

Graphical Analysis In Exercises \(63-66,\) use a graphing utility to graph the rational function. State the domain of the function and find any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line. $$h(x)=\frac{12-2 x-x^{2}}{2(4+x)}$$

Solving an Inequality In Exercises \(67-72,\) solve the inequality. (Round your answers to two decimal places.)$$-0.5 x^{2}+12.5 x+1.6>0$$
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