# Extending the Number System

• Work with infinite decimal expansions of numbers on the number line.

• Reason about operations with rational and irrational numbers (N-RN.B.3).

• Extend properties of integer exponents to rational exponents and write expressions with rational exponents as radicals (N-NR.A.1, N-RN.A.2).

• Solve equations and real-world problems involving radicals and fractional exponents (A-REI.A.2).

• Note extraneous solutions and explain where they come from (A-REI.A.2).

• Discover a new type of number that is outside previously known number systems (N-CN.A.1).

• Perform operations with complex numbers (N-CN.A.2).

• Solve quadratic equations with complex solutions (N-CN.C.7).

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.

## Sections

#### Summary

Assess students’ ability to

• recall, from eighth grade, the definition of rational numbers and well-known examples of irrational numbers (8.NS.A.2);

• mentally compute and work with exponents, including 0, negative exponents, and negative bases (8.EE.A.1);

• solve a quadratic equation by various methods (A-REI.B.4).

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#### Summary

• Work with infinite decimal expansions of numbers on the number line.

• Reason about operations with rational and irrational numbers (N-RN.B.3).

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#### Summary

• Extend properties of integer exponents to rational exponents and write expressions with rational exponents as radicals (N-RN.A.1, N-RN.A.2).

• Solve real-world problems in which rational exponents arise (N-RN.A.1).

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#### Summary

• Solve radical equations and equations with fractional exponents (A-REI.A.2).

• Note extraneous solutions and describe where they came from(A-REI.A.2).

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#### Summary

Assess students’ ability to

• identify extraneous solutions when solving radical equations (A-REI.A.2);

• rewrite expressions with rational exponents and radicals (N-RN.A.2);

• demonstrate understanding about the outcomes of operations on rational and irrational numbers (N-RN.B.3).

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#### Summary

Discover a new type of number that is outside previously known number systems (N-CN.A.1).

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#### Summary

• Explore how the new number, i, behaves under certain operations (N-CN.A.2).

• Perform operations with complex numbers and draw conclusions about patterns that emerge (N-CN.A.2).

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#### Summary

Apply knowledge of complex numbers to solve quadratic equations with complex solutions (N-CN.C.7).

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#### Summary

Assess students’ ability to

• solve radical equations and demonstrate awareness of the possibility of extraneous roots resulting from an algebraic solution (A-REI.A.2);

• describe and show examples of how rational and irrational numbers behave under certain operations (N-RN.B.3);

• rewrite expressions involving rational exponents and radicals (N-RN.A.2);

• demonstrate understanding of the meaning of *i* (N-CN.A.1);

• perform operations with *i* and with complex numbers in the form *a* + *bi* (N-CN.A.2);

• solve quadratic equations with real coefficients that have complex solutions (N-CN.C.7).

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