Imaginary Unit

√-1

Episode #9 of the course “Most important numbers in the world”

Square roots are an essential mathematical concept, and as algebra advances and more complex concepts are needed to explain every mathematical situation, the number “i” becomes highly important. There are situations that call for enacting calculations that have reached limitations. The bridge between the gap, the number “i,” is so called because it is an “imaginary” number—that is, it is defined by a property that no real number possesses. “i” is defined exclusively by its property that:

i ² = -1

By taking the square root and establishing that i = √-1, an entire realm of “imaginary” numbers opens up to mathematical calculations. Therefore, the principles and rules that apply to real numbers can be extended to i through its treatment as a real value. Because no two negative numbers when multiplied together will produce a negative number, the very existence of i solves algebraic calculations that require the square root of a negative number.

i follows a regular pattern that makes it easy to predict and work with to solve and manipulate a number of theoretical math and algebraic calculations. The pattern of:

i ² = -1

i ³ = –i

i ⁴ = 1

i ⁵ =i ⁴i = i

repeats itself so that any exponent of i can be reduced to a simpler value.

In practical application, i is used in electrical engineering and electronics, as it can signify negative flows of electrical current. However, as i is commonly used in electrical engineering to signify the movement of electric current over time, the letter j is instead used to signify the concept of √-1 to avoid confusion.

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