Table of contents
- 0. Review of Algebra4h 16m
- 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 19m
- 10. Combinatorics & Probability1h 45m
1. Equations & Inequalities
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8:43 minutes
Problem 101
Textbook Question
Textbook QuestionWork each problem. Show that √2/2 + √2/2 i is a square root of i.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Numbers
Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for solving problems involving roots of negative numbers and performing operations like addition, multiplication, and finding square roots.
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Square Roots of Complex Numbers
Finding the square root of a complex number involves determining another complex number that, when squared, yields the original complex number. This process often requires converting the complex number to polar form and applying De Moivre's theorem, which states that for a complex number in polar form, its roots can be found by taking the root of the modulus and dividing the angle by the root's degree.
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Polar Form of Complex Numbers
The polar form of a complex number expresses it in terms of its magnitude (or modulus) and angle (or argument). A complex number z = a + bi can be represented as z = r(cos θ + i sin θ), where r = √(a² + b²) is the modulus and θ = arctan(b/a) is the argument. This form is particularly useful for multiplication, division, and finding roots of complex numbers.
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