The Powers of i Calculator provided by Hesapstan reduces any integer exponent of the imaginary unit i by the mod-4 cycle and explains why the result is 1, i, -1, or -i.
Powers of i repeat in a cycle of four values
The imaginary unit i is defined by the rule i² = -1. Because of this definition, integer powers of i do not keep producing new values; they repeat through four values: 1, i, -1, and -i.
The basic cycle is i⁰ = 1, i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. Since i⁴ returns to 1, every later integer exponent follows the same pattern.
The remainder after dividing the exponent by 4 determines the result: remainder 0 gives 1, remainder 1 gives i, remainder 2 gives -1, and remainder 3 gives -i.
Large exponents are reduced by taking the exponent modulo 4
For a power such as i¹⁰⁰, you do not need to multiply i one hundred times. The exponent 100 is divisible by 4, so the remainder is 0 and i¹⁰⁰ = i⁰ = 1.
For i²⁷, the remainder after division by 4 is 3. Therefore i²⁷ has the same value as i³, which is -i. The calculator shows this reduction step directly.
Negative exponents also follow the same four-value cycle
A negative exponent means a reciprocal. For example, i⁻¹ = 1/i, and this equals -i because i × (-i) = 1.
The calculator handles negative integer exponents by reducing them through the same cycle. For example, i⁻¹ = -i, i⁻² = -1, i⁻³ = i, and i⁻⁴ = 1.
This tool only computes integer powers of i. It does not perform general complex addition, multiplication, division, or exponentiation.
Decimal and fractional exponents are not supported
This calculator is built for integer exponents only. Zero, positive integers, and negative integers are supported, but exponents such as 1.5 or 2/3 are outside the scope of this tool.
Non-integer powers of complex numbers can involve more advanced topics and may not behave like the simple four-value cycle. This page therefore keeps the promise limited to integer powers of i.
The result is always one of 1, i, -1, or -i
For integer powers of i, there are exactly four possible results. No matter how large or negative the exponent is, reducing it modulo 4 selects one of those four values.
This is why powers of i are often quick to solve in algebra and complex-number exercises. The main step is to find the correct remainder after division by 4.
Examples show how the cycle is applied
Giriş / Input: n = 100 — Sonuç / Output: 1 — 100 mod 4 = 0, so i¹⁰⁰ = i⁰ = 1.
Giriş / Input: n = 27 — Sonuç / Output: -i — 27 mod 4 = 3, so the result is i³ = -i.
Giriş / Input: n = -1 — Sonuç / Output: -i — i⁻¹ = 1/i, and that value is -i.
Frequently Asked Questions
What is the imaginary unit i?
The imaginary unit i is defined by i² = -1. It is one of the basic building blocks of complex numbers.
Why do powers of i repeat every 4?
The sequence is i⁰ = 1, i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. Once i⁴ returns to 1, the same pattern repeats.
How do I find i to the 100th power?
Divide 100 by 4 and use the remainder. The remainder is 0, so i¹⁰⁰ = 1.
What is i to the power of -1?
i⁻¹ = 1/i, which equals -i because i multiplied by -i gives 1.
What is i²?
i² is -1 by the definition of the imaginary unit i.
Can I calculate i to a decimal power here?
No. This calculator supports integer exponents only. Decimal and fractional exponents are outside its scope.
Does this calculator do general complex arithmetic?
No. It only calculates integer powers of i. It does not add, divide, or multiply general complex numbers.