Overview
This lecture introduces the imaginary unit "i", explains its definition, and explores how powers of "i" cycle through specific values.
Introduction to the Imaginary Unit "i"
- "i" is called the imaginary unit and is defined as the number whose square is -1.
- Unlike real numbers like pi or e, "i" does not have a tangible value.
- "i" can also be described as the principal square root of -1, but be cautious when using this term with negative numbers.
Powers of "i"
- ( i^0 = 1 ), because any number to the zeroth power is 1.
- ( i^1 = i ), by the definition of exponents.
- ( i^2 = -1 ), by the definition of "i".
- ( i^3 = i^2 \times i = -1 \times i = -i ).
- ( i^4 = i \times i^3 = i \times -i = -i^2 = -(-1) = 1 ).
- ( i^5 = i^4 \times i = 1 \times i = i ).
- ( i^6 = i \times i^5 = i \times i = i^2 = -1 ).
- ( i^7 = i \times i^6 = i \times -1 = -i ).
- ( i^8 = i^4 \times i^4 = 1 \times 1 = 1 ).
Pattern in Powers of "i"
- Powers of "i" cycle every four exponents: ( i, -1, -i, 1 ), and then repeat.
Key Terms & Definitions
- Imaginary unit ("i") — A number defined such that ( i^2 = -1 ).
- Principal square root — The standard (primary) square root of a number, for "i" it means the square root of -1.
Action Items / Next Steps
- Watch the next video to learn how to determine high powers of "i".
- Practice writing out powers of "i" to observe the cycle.