123 1.3 Complex Numbers (a) Use the model to determine projected enrollment for fall 2022. (Round to the nearest tenth of a million.) (b) Use the model to determine the year in which enrollment is projected to reach 17.4 million. (c) How do the answers to parts (a) and (b) compare to the corresponding values shown in the graph? (d) The actual enrollment in fall 2005 was 15.0 million. The model here is based on data projections from 2020 to 2027. If we were to use the model for 2005, what would the projected enrollment be? (Round to the nearest tenth of a million.) (e) Compare the actual value and the value based on the model in part (d). Discuss the pitfalls of using the model to predict enrollment for years preceding 2020. 46. Baby Boom U.S. population during the years between 1946 and 1964, commonly known as the Baby Boom, can be modeled by the following linear equation. y = 2.8370x + 140.83 Here y represents the population in millions as of July 1 of a given year, and x represents number of years after 1946. Thus, x = 0 corresponds to 1946, x = 1 corresponds to 1947, and so on. (Data from U.S. Census Bureau.) (a) According to the model, what was the U.S. population on July 1, 1952? (b) In what year did the U.S. population reach 150 million? Basic Concepts of Complex Numbers The set of real numbers does not include all the numbers needed in algebra. For example, there is no real number solution of the equation x2 = −1 because no real number, when squared, gives -1. To extend the real number system to include solutions of equations of this type, the number i is defined. 1.3 Complex Numbers ■ Basic Concepts of Complex Numbers ■ Operations on Complex Numbers Imaginary Unit i i =!−1 , and therefore, i2 = −1. (Note that -i is also a square root of -1.) Complex numbers are formed by adding real numbers and multiples of i. Complex Number If a and b are real numbers, then any number of the form a +bi is a complex number. In the complex number a + bi, a is the real part and b is the imaginary part.* Square roots of negative numbers were not incorporated into an integrated number system until the 16th century. They were then used as solutions of equations and later (in the 18th century) in surveying. Today, such numbers are used extensively in science and engineering. Two complex numbers a + bi and c + di are equal provided that their real parts are equal and their imaginary parts are equal—that is, they are equal if and only if a = c and b = d. * In some texts, the term bi is defined to be the imaginary part.
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