Explore the properties of the imaginary unit i and calculate its powers. Understand the cyclic nature of imaginary arithmetic.
Power Configuration
Calculation Result
iResult
Evaluated Value
Pattern Index1
Remainder1
i
n leaves remainder 1
About this calculator
What is an Imaginary Number?
An imaginary number is defined by the property i² = -1. The unit imaginary number is i = √-1. Powers of i follow a repeating 4-step cycle: i, -1, -i, 1.
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Pro Tips
Simplify: To find iⁿ, divide n by 4 and look at the remainder.
Reality: Imaginary numbers are essential in AC circuit analysis and quantum mechanics.
Equality: Complex numbers consist of a real part and an imaginary part (a + bi).
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Fun Facts
"Euler introduced the symbol i in 1777, but it took decades for mathematicians to accept it as 'real'."
"The Mandelbrot Set, one of the most famous fractals, is built entirely on imaginary numbers."
"Negative numbers don't have real square roots, but they always have imaginary ones."
Formula: i^n = { 1 if n ≡ 0 (mod 4), i if n ≡ 1 (mod 4), -1 if n ≡ 2 (mod 4), -i if n ≡ 3 (mod 4) }
