Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Complex Numbers

College Algebra

1. Equations & Inequalities

Complex Numbers

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1:20 minutes

Problem 33

Textbook Question

Simplify each power of i. i^1001

Verified step by step guidance

Understand that the imaginary unit \(i\) is defined as \(i = \sqrt{-1}\), and it has a cyclical pattern in its powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then it repeats.

Recognize that the powers of \(i\) repeat every 4 terms. Therefore, to simplify \(i^{1001}\), find the remainder when 1001 is divided by 4.

Calculate \(1001 \div 4\) to find the quotient and remainder. The remainder will determine the equivalent power of \(i\).

Since the remainder is 1, \(i^{1001}\) is equivalent to \(i^1\).

Thus, \(i^{1001} = i\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Imaginary Unit (i)

The imaginary unit 'i' is defined as the square root of -1. It is a fundamental concept in complex numbers, allowing for the extension of the real number system to include solutions to equations that do not have real solutions, such as x^2 + 1 = 0. Powers of 'i' cycle through a predictable pattern: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1.

Modulo Operation

The modulo operation finds the remainder of division of one number by another. In the context of simplifying powers of 'i', we use modulo 4 to determine which power of 'i' corresponds to a larger exponent. For instance, calculating 1001 mod 4 gives a remainder of 1, indicating that i^1001 is equivalent to i^1, which simplifies to 'i'.