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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 real number a can be regarded as a complex number a + 0i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. It is common to write a + 0i = a, 0 + bi = bi, and a + (−b)i = a − bi; for example, 3 + (−4)i = 3 − 4i.
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]
Argand diagram refers to a geometric plot of complex numbers as points z = x + iy using the horizontal x-axis as the real axis and the vertical y-axis as the imaginary axis. [3] Such plots are named after Jean-Robert Argand (1768–1822), although they were first described by Norwegian–Danish land surveyor and mathematician Caspar Wessel ...
Its graph is thus obtained by rotating the graph of ln(x) around the z-axis. In mathematics , a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers . The term refers to one of the following, which are strongly related:
The formula is still valid if x is a complex number, and is also called Euler's formula in this more general case. [1] Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2]
In November 2023, Desmos gave users the ability to bring sound to their graphs, allowing them to produce tones of a given frequency and gain. [14] Users can create accounts and save the graphs and plots that they have created to them. A permalink can then be generated which allows users to share their graphs and elect to be considered for staff ...
The formula is important because it connects complex numbers and trigonometry. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos nx and sin nx in terms of cos x and sin x. As written, the formula is not valid for non-integer ...