Imaginary number
An imaginary number is a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary.
| All powers of i assume values from blue area |
| i−3 = i |
| i−2 = −1 |
| i−1 = −i |
| i0 = 1 |
| i1 = i |
| i2 = −1 |
| i3 = −i |
| i4 = 1 |
| i5 = i |
| i6 = −1 |
| i is a 4th root of unity |
Originally coined in the 17th century by René Descartes as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century).
An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number.