I Number Calculator - Powers, Cycle, and a+bi Form

An i number calculator is a quick tool that takes any integer n and returns i raised to that power, where i is the imaginary unit defined as the square root of -1. Type an integer and the result panel shows the value of i^n, the value written as a + bi, the position inside the four-value cycle, and the reduced exponent n mod 4. Powers of i repeat with period 4, so every integer exponent collapses to one of 1, i, -1, or -i.

• • Homework check for i^n: Verify a textbook or exam answer for a specific exponent, especially large ones like i^25 or i^100.
• • Reduce an exponent with the cycle: Confirm that any integer power lands in the same row of the 1, i, -1, -i cycle as its reduced exponent.
• • Work with the a+bi form: Read the value in standard a + bi notation for a downstream problem that needs the real and imaginary parts.
• • Spot the sign of the result: See whether i^n is a real number (1 or -1) or an imaginary number (i or -i) before using it.

The i in the calculator name is the imaginary unit, a fixed mathematical object that does not depend on the input. To combine i^n with another complex number, our Complex Number Calculator handles the addition, subtraction, multiplication, and division of a + bi pairs in a single result panel.

According to Britannica, the imaginary unit i is defined as the square root of -1 and serves as the basis of the standard complex-number form a + bi, in which the real part a and the imaginary part b sit side by side.

The calculator reduces the integer exponent n to a remainder in {0, 1, 2, 3} and looks up the value in a four-row table: i^0 = 1, i^1 = i, i^2 = -1, i^3 = -i. The cycle is the whole pattern that powers of i repeat with period 4.

• n: The integer exponent applied to i. May be positive, zero, or negative. The cycle reduces any n to a remainder between 0 and 3.
• i: The imaginary unit, defined as the square root of -1. Its defining property is i^2 = -1.
• i^n: The result of raising i to the nth power. Always one of 1, i, -1, or -i, regardless of how large n is.

The cycle of period 4 follows from i^2 = -1: i^4 = (i^2)(i^2) = (-1)(-1) = 1, so every subsequent power of i repeats the same four rows. Negative exponents use the mathematical remainder ((n mod 4) + 4) mod 4, so -1, -2, -3, -4 land on the rows for -i, -1, i, 1 with no case split needed.

According to Wolfram MathWorld, the imaginary unit i satisfies i^2 = -1, and the powers of i cycle through 1, i, -1, -i with period 4 because i^(n+4) = i^n for every integer n.

If the result has to be entered into a longer a + bi expression, our A+Bi Form Calculator returns the real part, the imaginary part, and the magnitude of any complex number in a single read.

Four small ideas explain why the imaginary unit behaves the way it does and why the cycle has exactly four rows. They connect the i^n rule to other tools on the site.

The same definition of i feeds into the polar form of a complex number, written as |z| e^(i phi). The symbol i in that expression is the same imaginary unit, which is why e^(i phi) describes rotations on the complex plane. When the powers of i have to be plotted as rotations on the plane, our Cartesian to Polar Calculator converts a + bi into the polar form r e^(i phi) and back.

Enter an integer n and read the four result rows on the right. The calculator updates as you type, so you can step through the cycle one exponent at a time.

1. 1 Type the integer exponent n: Enter any whole number. Positive, zero, and negative values all work. Decimal inputs are truncated toward zero.
2. 2 Read the value of i^n: Look at the highlighted Value of i^n row. The result is one of 1, i, -1, or -i.
3. 3 Read the a+bi form: Use the a+bi Form row for standard complex-number notation. Real-valued results have b = 0, imaginary-valued results have a = 0.
4. 4 Check the cycle position: The Cycle Position row labels the power as 1st, 2nd, 3rd, or 4th. The 4th power and the 1st power give the same value, the visible mark of the period-4 cycle.
5. 5 Read the reduced exponent: The Reduced Exponent row shows n mod 4, the value that picks the answer out of the four-row table.
6. 6 Step through the cycle: Type consecutive integers (for example, 5, 6, 7, 8) to see the four-value cycle play out in real time.

Example: a student is asked for i^7. They type 7, read -i in the Value row, see 0 - 1i in the a+bi Form row, see 4th power in the Cycle Position row, and confirm 3 in the Reduced Exponent row. Typing 11 gives the same -i result, since 11 mod 4 = 3 puts i^11 on the same row. When the exponent is a fraction rather than an integer, our Fractional Exponent Calculator handles the same kind of base-and-exponent expression for real bases and roots.

The i^n cycle is short, but using the calculator across a list of exponents saves time and removes the sign errors that happen when the cycle resets at a multiple of 4.

• • One tool for the four-value cycle: The calculator applies i^n = i^(n mod 4) in one step, so you do not work out the remainder by hand.
• • Value, a+bi, cycle, and reduced exponent together: The result panel shows all four rows at once, so the answer and the supporting detail are visible together without a second calculation.
• • Works for negative exponents: Inputs like -1, -2, and -3 land on the same rows of the cycle as 3, 2, and 1 because the formula uses the mathematical remainder.
• • Handles large exponents without growing: i^1000 and i^10000 collapse into one of the four cycle values, never larger than 1 in magnitude.
• • Real-time updates on typing: The result panel updates as you type, so stepping through consecutive exponents is three or four keystrokes.

Doing the i^2 = -1 step on paper, then i^3 = -i, and then the cycle reset at i^4 = 1 invites mistakes when the exponent is large or negative. The calculator applies the same four-row table to every integer, so i^2 and i^26 land on the same row.

When the value of i^n has to be checked against its magnitude, our Absolute Value Calculator returns |x| in one step, which is 1 for every power of i.

A few choices about the input change the meaning of the four result rows. The same exponent can be informative in one context and misleading in another if you are not careful about what n is meant to represent.

• • The four-value cycle rule is for integer n. Fractional or irrational exponents of i are not real numbers, so this calculator does not compute them.
• • The calculator takes one n and returns four rows for that n. For a list of exponents, run the calculator once per exponent.

When the question is about i raised to an integer power, the result rows are the complete answer. The next step (combining i^n with another complex number, plotting it on the plane, or checking its magnitude) is up to the workflow that called the power, and the other math-conversion calculators on the site cover those follow-up steps. According to Khan Academy, the imaginary unit i is defined so that i^2 = -1, and the powers of i repeat with period 4: i^0 = 1, i^1 = i, i^2 = -1, i^3 = -i, and then i^4 = 1 starts the cycle over.

When the imaginary unit appears inside a trigonometric or inverse-trig problem, our Arccos Calculator returns the angle whose cosine matches a given real input.