Imaginary number

An imaginary number is the product of a real number and 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. 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. Imaginary numbers are often called purely imaginary to distinguish them from complex numbers more generally; the set of all imaginary numbers is sometimes denoted ⁠ i R {\displaystyle i\mathbb {R} } ⁠, where ⁠ R {\displaystyle \mathbb {R} } ⁠ denotes the set of real numbers.