ith root of 1  Summary and Q&A
TL;DR
The ith root of 1 leads to infinitely many real numbers, and the concept can be further explored through complex numbers and Euler's formula.
Key Insights
 🤙 Complex numbers can be represented using a coordinate system called polar coordinates.
 #️⃣ Euler's formula, e^(iθ) = cosθ + isinθ, connects complex numbers to trigonometry.
 😃 The ith root of 1 has infinitely many real number solutions, with the principle value being e^π.
 🪜 Additional solutions can be obtained by adding or subtracting multiples of 2π to the angle.
 🔬 Brilliant.org offers comprehensive courses on complex numbers and other math, science, and computer science topics.
 #️⃣ The ith root of 1 showcases the fascinating nature of complex numbers and their relationship to real numbers.
 #️⃣ Understanding the concept of Euler's formula is crucial in exploring complex numbers and their properties.
Transcript
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Questions & Answers
Q: How can the ith root of 1 be represented using complex numbers?
The ith root of 1 is expressed using polar coordinates, with the distance from the origin denoted as R and the angle denoted as θ. The representation is re^(iθ), where r is the distance and θ is the angle.
Q: What is Euler's formula and how does it help in understanding complex numbers?
Euler's formula, e^(iθ) = cosθ + isinθ, relates complex numbers to trigonometry. It allows for the conversion between exponential form and polar coordinates, making it easier to work with complex numbers.
Q: How many real number solutions are there for the ith root of 1?
The ith root of 1 has infinitely many real number solutions. The principle value is e^π, but additional solutions can be obtained by adding 2π or 2π to the angle.
Q: Can negative integers be plugged into the ith root of 1?
Yes, negative integers can be used. Plugging in negative integers results in additional solutions for the ith root of 1, such as e^(π) or e^(3π).
Summary & Key Takeaways

The ith root of 1 can be represented using complex numbers, utilizing a coordinate system called polar coordinates.

Euler's formula, e^(iθ) = cosθ + isinθ, helps to make sense of complex numbers.

The ith root of 1 has infinitely many real number solutions, with the principle value being e^π.