How this instrument works
The imaginary unit i is defined by one property: i² = −1. Multiply it by itself again and a four-step cycle appears — i¹ = i, i² = −1, i³ = −i, i⁴ = 1 — after which the pattern starts over exactly. Picture i sitting a quarter turn around the complex plane; multiplying by i again rotates a point another quarter turn counterclockwise, so four multiplications complete a full 360° turn and land back where they started.
Euler's formula ties that rotation to trigonometry: e^{iθ} = cos θ + i sin θ. Because i itself sits a quarter turn around the unit circle, i = e^{iπ⁄2}, and raising both sides to the power n gives iⁿ = e^{inπ⁄2} = cos(nπ⁄2) + i sin(nπ⁄2). That is exactly what this instrument evaluates — it reads the angle off the cycle directly through cosine and sine rather than multiplying i by itself n times, which is why it returns an exact answer instantly even for an exponent in the hundreds.
Every value iⁿ can take sits exactly on the unit circle, since multiplying by a number of magnitude one only rotates, never stretches. That is the quietly surprising part: unlike powers of an ordinary number bigger than one, which grow without bound, or a fraction, which shrinks toward zero, powers of the imaginary unit never grow or shrink at all — the real and imaginary parts stay locked to −1, 0, or 1 forever. This sheet steps its exponent as a whole number, but the same cosine-and-sine rule is defined for any real exponent, smoothly extending the four-point cycle into a continuous rotation used throughout complex analysis.
- Type the exponent into the Power, n field — any whole number from 0 up to 1000.
- Read Real part of iⁿ for the value's real-axis component.
- Read Imaginary part of iⁿ for its imaginary-axis component.
- Raise n by 1 four times in a row and watch the pair cycle through i, −1, −i, 1, then repeat exactly.
- Try n = 0 first to confirm the identity i⁰ = 1, the point every cycle both starts from and returns to.
Worked example — the third power of i
Set Power, n to 3. The formula gives real = cos(3π⁄2) = 0 and imag = sin(3π⁄2) = −1, so Real part of iⁿ reads 0.0 and Imaginary part of iⁿ reads −1.0 — together, i³ = −i. Direct multiplication checks it: i³ = i² × i = (−1) × i = −i, matching the trig computation digit for digit.
Picture it as rotation: start at 1 on the real axis, and each multiplication by i turns the point 90° counterclockwise. Three quarter-turns is 270°, landing exactly on the negative imaginary axis — the point −i — which matches the −1.0 this sheet reports for the imaginary part and the 0.0 it reports for the real part.
Questions
What is i to the third power?
i³ = −i. Multiply it out directly — i³ = i² × i = (−1) × i = −i — or read it off the four-step cycle i¹=i, i²=−1, i³=−i, i⁴=1. This sheet reaches the same answer by computing cos(3π⁄2) = 0 and sin(3π⁄2) = −1, giving real = 0.0 and imag = −1.0.
Why does iⁿ repeat every four powers?
Because cos(nπ⁄2) + i sin(nπ⁄2) has period 2π in its angle, and each step of n adds exactly π⁄2 to that angle. Four steps add 4 × π⁄2 = 2π, a full turn, so the pair of outputs returns to its starting values precisely every four powers, no matter how large n gets.
What is i to the power of 0?
i⁰ = 1, the same base case every nonzero number shares at the zero power. Plug n = 0 into the formula and cos(0) = 1 while sin(0) = 0, so Real part of iⁿ reads 1.0 and Imaginary part of iⁿ reads 0.0 — the point every four-step cycle both starts from and returns to.
Does iⁿ ever grow larger as n increases?
No. Every power of i has magnitude exactly one, since multiplying by a number that size only rotates a point around the origin rather than stretching it. Real and imaginary parts stay confined to −1, 0, or 1 for every n — even i¹⁰⁰, which lands back on 1 because 100 is an exact multiple of 4.
How is the formula iⁿ = cos(nπ⁄2) + i sin(nπ⁄2) derived?
It comes from Euler's formula e^{iθ} = cos θ + i sin θ together with the fact that i itself sits a quarter turn around the unit circle, i = e^{iπ⁄2}. Raising both sides to the power n gives iⁿ = e^{inπ⁄2}, and Euler's formula converts that back into cosine and sine of nπ⁄2.
Is this the same as converting a complex number to polar form?
Related but narrower. Polar form describes any complex number by a radius and angle, while this sheet fixes the radius at one and the starting angle at 90°, then asks only how a whole-number power of that single point behaves — where iⁿ lands, not what the radius and angle of an arbitrary a + bi are.