Extending the Number System

In Grade 8, students discovered there are numbers that are not rational. They approximated them using rational numbers to locate them approximately on the number line. They also worked with integer exponents and became familiar with the basic exponent rules. Finally, in their work with quadratics, students encountered situations where they came to a negative under the square root sign, which was interpreted until now as meaning the equation had no solutions.

The theme of this unit is extending number systems and operations on numbers, first from rational to real numbers and then from real to complex numbers. Along the way students extend the operation of raising a number to a power to situations where the exponent is not an integer. They define the meaning of a numerical expression with rational exponents in terms of radicals, by reasoning about how to extend the exponent rules. They use this new understanding to solve equations involving rational exponents and radicals. Finally they extend the real number system and the operations on real numbers to include complex numbers and use them to solve quadratic equations with no real solutions.

In future units on exponential functions, students will extend their knowledge of rational exponents and apply them to even more complicated situations. Students pursuing careers in STEM fields might use imaginary and complex numbers in many of their future college courses.