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real numbers have exactly one real cube root.” However, students don’t know about any numbers other than real numbers, so it doesn't make sense to make this distinction until they expand their concept of number to include imaginary and complex numbers in subsequent lessons.
The next few lessons introduce imaginary and complex numbers. The number line is renamed the real number line, and students observe that there are no real numbers that
square to make negative numbers. The symbol is used to deSne a number that
squares to make -1, i.e. is deSned as a solution to the equation . Since no real numbers are solutions to this equation, this number is represented as a point oV of the real number line. This leads to the construction of the imaginary axis and the complex
plane.Thesymbol isquicklyrenamed ,andstudentsSndthatnegativerealnumbers also have two square roots, one on the positive imaginary axis, and one on the negative imaginary axis. In other words, if is a positive number, then the equation has two solutions, and . Just as the symbol is deSned to describe the positive square
root of a real number, students learn the convention that if is a positive number, the notation is taken to mean the square root of on the positive imaginary axis, .
Students then use the fact that and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers to express them in the form
, where and are real numbers.
In the Snal few lessons, students use what they have learned about square roots and complex numbers to solve quadratic equations that have complex solutions. Students practice completing the square and using the quadratic formula. Sometimes these methods appear to produce diVerent solutions, for example one method may give a
solution that includes while another method produces instead. Students use what they know about square roots to determine whether solutions are equivalent.
A5 Exponential Equations and Functions
Students were introduced to exponential functions in an earlier course. This unit begins by activating students' prior knowledge. Students recall that an exponential function involves a change by equal factors over equal intervals and can be expressed as , where is the value of the function when is 0, and is the growth factor. They review the use of verbal descriptions, tables, and graphs to represent exponential functions.
The next few lessons extend students’ ability to write, interpret, and compare exponential functions. Previously, students saw exponential functions with mainly integers for the inputs. In this unit, they extend that work to include functions with rational numbers for
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