Unlocking Imaginary Powers: Mastering 'i' with Your TI-84

Algebra 2 Grades High School 2:54 Video
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Lesson Description

Learn how to simplify and evaluate powers of the imaginary unit 'i' using your TI-84 calculator and understand the underlying mathematical principles.

Video Resource

Rewriting Powers of i Using Calculator Ti84 (Imaginary Numbers)

Mario's Math Tutoring

Duration: 2:54
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Key Concepts

  • Imaginary unit 'i' and its properties (i² = -1)
  • Cyclical nature of powers of 'i' (i, -1, -i, 1)
  • Using the TI-84 calculator to evaluate powers of 'i'
  • Simplifying powers of 'i' by hand using i² = -1 or dividing the exponent by 4

Learning Objectives

  • Students will be able to use the TI-84 calculator to evaluate powers of 'i'.
  • Students will be able to simplify powers of 'i' by hand using the cyclical nature of 'i'.
  • Students will be able to verify calculator results with manual calculations.
  • Students will be able to relate powers of i to remainders after division by 4.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the definition of the imaginary unit 'i' and its fundamental property (i² = -1). Discuss the importance of understanding complex numbers in mathematics.
  • Calculator Demonstration (10 mins)
    Show students how to use the TI-84 calculator to evaluate powers of 'i'. Demonstrate the 'second' and 'decimal point' keys to access 'i'. Input various powers of 'i' (e.g., i³, i²³, i³⁷) and observe the results.
  • Manual Simplification Techniques (15 mins)
    Explain two methods for simplifying powers of 'i' by hand: 1. **Using i² = -1:** Decompose the power of 'i' into multiples of i² and a remaining 'i' term (e.g., i²³ = (i²)¹¹ * i = (-1)¹¹ * i = -i). 2. **Dividing by 4:** Divide the exponent by 4 and focus on the remainder. A remainder of 0 corresponds to i⁰ = 1, a remainder of 1 corresponds to i¹ = i, a remainder of 2 corresponds to i² = -1, and a remainder of 3 corresponds to i³ = -i. Explain why this works due to the cyclical pattern.
  • Verification and Practice (15 mins)
    Have students verify the calculator results by manually simplifying the same powers of 'i' using the techniques learned. Provide additional practice problems of varying difficulty levels. Encourage them to work in pairs and check their answers.
  • Conclusion (5 mins)
    Summarize the key takeaways: how to use the TI-84 calculator to evaluate powers of 'i', the two manual simplification techniques, and the importance of understanding the underlying mathematical principles. Answer any remaining questions.

Interactive Exercises

  • Calculator Race
    Divide the class into teams. Provide a list of powers of 'i' (e.g., i¹⁰, i¹⁵, i²⁰, i²⁵, i³⁰, i³⁵). The first team to correctly evaluate all the powers of 'i' using their calculators wins.
  • Simplify and Verify
    Give each student a different power of 'i' to simplify by hand. After simplifying, they must verify their answer using the TI-84 calculator. They then explain their process to a partner.

Discussion Questions

  • Why is understanding the cyclical nature of powers of 'i' important?
  • How does using the calculator complement the manual simplification techniques?
  • Can you think of any real-world applications where complex numbers and powers of 'i' might be used?
  • What are the limitations of using the calculator for simplifying powers of i?

Skills Developed

  • Procedural fluency in using a calculator for mathematical operations
  • Conceptual understanding of imaginary numbers and their properties
  • Problem-solving skills in simplifying complex expressions
  • Verification and critical thinking skills in comparing calculator results with manual calculations

Multiple Choice Questions

Question 1:

What is the value of i²?

Correct Answer: -1

Question 2:

What is the simplified form of i⁵?

Correct Answer: i

Question 3:

Using a TI-84 calculator, what is i¹² equal to?

Correct Answer: 1

Question 4:

Which of the following is equivalent to i²⁷?

Correct Answer: -i

Question 5:

If you divide the exponent of 'i' by 4 and the remainder is 2, what is the value of i to that power?

Correct Answer: -1

Question 6:

Which expression is equal to i¹⁰⁰?

Correct Answer: 1

Question 7:

Simplify i³⁹

Correct Answer: -i

Question 8:

What is the result of i¹⁶?

Correct Answer: 1

Question 9:

The expression i⁴² is equal to:

Correct Answer: -1

Question 10:

What is the value of i⁰?

Correct Answer: 1

Fill in the Blank Questions

Question 1:

The imaginary unit 'i' is defined as the square root of _______.

Correct Answer: -1

Question 2:

The value of i⁴ is equal to _______.

Correct Answer: 1

Question 3:

When simplifying powers of 'i', dividing the exponent by _______ helps determine the result based on the remainder.

Correct Answer: 4

Question 4:

If i to the power of n equals i, and n is a whole number, then i to the power of n+4 equals _______.

Correct Answer: i

Question 5:

i¹⁷ simplified is _______.

Correct Answer: i

Question 6:

Using the calculator, i to the 10th power is _______.

Correct Answer: -1

Question 7:

The remainder when 25 is divided by 4 is _______. Therefore i²⁵ = i to the power of the remainder, simplified, i²⁵ = _______.

Correct Answer: 1, i

Question 8:

The first step to simplifying i³⁰ using the cyclical method is _______. The simplified answer to i³⁰ is _______.

Correct Answer: i³⁰=(i²)¹⁵, -1

Question 9:

When using the TI-84 calculator, press _______ then the _______ key to input i into the calculator.

Correct Answer: second, decimal point

Question 10:

When simplifying powers of i, the four possible values for i^n are i, -i, 1, and _______.

Correct Answer: -1

Educational Standards

A2.N.1:Extend the understanding of numbers and operations to include complex numbers, radical expressions, and expressions written with rational exponents. A2.N.1.1:Find the value of 𝑖ⁿ for any whole number 𝑛.

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