Abstract

The real number integer exponents of the imaginary unit, 'i' has 4 possible values—i, -1, -i, -1. Traditional method to find such values without the use of computational devices, such as calculators and computers, require splitting the exponent to simpler numbers, which can be quite burdensome if the exponent is of very large value, like 38127938127. This paper presents a simple three-step algorithm for quickly computing large exponents of the imaginary unit 'i,' explained in a way that even the non-mathematicians will be able to understand it. This method simplifies complex number exponentiation, turning minutes of hard calculation into seconds with minimal effort, and can serve as an extremely effective tool in mental mathematical calculations. The algorithm combines the real/imaginary determination, division by 2 (and subtraction by 1 if odd) and sign determination, using only last two digits of the exponent. We further prove our method using modular arithmetic and demonstrate it using examples.