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The powers of i repeat in a cycle expressible with the following pattern, where n is any integer:. Making use of Euler's formula , i i is. Many mathematical operations that can be carried out with real numbers can also be carried out with i , such as exponentiation, roots, logarithms, and trigonometric functions. All of the following functions are complex multi-valued functions , and it should be clearly stated which branch of the Riemann surface the function is defined on in practice.

## What is the Square Root of i?

Listed below are results for the most commonly chosen branch. The imaginary-base logarithm of a number is:. As with any complex logarithm , the log base i is not uniquely defined. The cosine of i is a real number:. And the sine of i is purely imaginary:.

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From Wikipedia, the free encyclopedia. Because the real and imaginary parts are always separate, we regroup the terms: University of Toronto Mathematics Network: What is the square root of i?

URL retrieved March 26, The Interplay of Mathematics and Narrative illustrated ed. The Story of "i" [the square root of minus one]. Mathematical methods in the physical sciences 3. Retrieved from " https: Complex numbers Algebraic numbers Quadratic irrational numbers Mathematical constants.

## Question Corner -- What is the Square Root of i?

Views Read Edit View history. This page was last edited on 31 August , at Here is how he explained it: He also explained the laws of multiplying complex numbers: Minus by plus of minus, makes minus of minus. Plus by minus of minus, makes minus of minus. Minus by minus of minus, makes plus of minus. Plus of minus by plus of minus, makes minus.

## Imaginary Numbers

Plus of minus by minus of minus, makes plus. Minus of minus by plus of minus, makes plus. Minus of minus by minus of minus makes minus. These laws were directly copied from Wikipedia as the original text was taken directly from the book "Algebra. Imaginary numbers can be written as real numbers multiplied by the unit i imaginary number.

For example, the square root of a negative number could be an imaginary number.