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The imaginary unit or unit imaginary number (i) is a mathematical constant that is a solution to the quadratic equation x 2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers , using addition and multiplication .
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i 2 = −1.
An imaginary number is the product of a real number and the imaginary unit i, [note 1] which is defined by its property i 2 = −1. [1] [2] The square of an imaginary number bi is −b 2. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary. [3]
In the four-dimensional space of quaternions, there is a sphere of imaginary units. For any point r on this sphere, and x a real number, Euler's formula applies: exp x r = cos x + r sin x , {\displaystyle \exp xr=\cos x+r\sin x,} and the element is called a versor in quaternions.
Quantities, Units and Symbols in Physical Chemistry, also known as the Green Book, is a compilation of terms and symbols widely used in the field of physical chemistry. It also includes a table of physical constants , tables listing the properties of elementary particles , chemical elements , and nuclides , and information about conversion ...
imaginary unit: unitless electric current: ampere (A) ^ Cartesian x-axis basis unit vector unitless current density: ampere per square meter (A/m 2) impulse: kilogram meter per second (kg⋅m/s) jerk: meter per second cubed (m/s 3)
Imaginary time is a mathematical representation of time that appears in some approaches to special relativity and quantum mechanics. It finds uses in certain cosmological theories. Mathematically, imaginary time is real time which has undergone a Wick rotation so that its coordinates are multiplied by the imaginary unit i .
where is position, is the wave function, is a periodic function with the same periodicity as the crystal, the wave vector is the crystal momentum vector, is Euler's number, and is the imaginary unit. Functions of this form are known as Bloch functions or Bloch states , and serve as a suitable basis for the wave functions or states of electrons ...