Imaginari

Imaginari is a term used in several languages to refer to imaginary numbers, the mathematical concept of numbers whose square is negative. Such numbers can be written in the form bi, where b is a real number and i is the imaginary unit satisfying i^2 = -1. The set of purely imaginary numbers includes numbers with zero real part, while complex numbers have both real and imaginary parts and can be written as a + bi.

The notion arose during the development of algebra in the 16th and 17th centuries, when mathematicians encountered solutions to equations that could not be expressed using only real numbers. The term imaginary was popularized by Descartes and later reconsidered, leading to formal acceptance of i as a mathematical object. In modern usage, imaginari (or its equivalents) is used as the plural form for imaginary numbers in some languages.

In the complex plane, imaginary numbers lie on the vertical axis. Multiplication by i rotates a number by 90 degrees; the modulus of bi is |b|. Complex numbers a + bi combine real and imaginary parts. Key operations include addition and multiplication, with conjugates a - bi and modulus sqrt(a^2 + b^2). Euler's formula e^{iθ} = cos θ + i sin θ and De Moivre's theorem relate imaginary numbers to trigonometry and powers of complex numbers.

Imaginaries are central in many areas of science and engineering, including Fourier analysis, signal processing, AC circuit theory, quantum mechanics, and control theory, where phasors and oscillations are modeled using complex numbers.