# Imaginary Unit

**Episode #9 of the c****ourse “Most important numbers in the world”**

**√-1**

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* ^{4}*i* = *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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