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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]
For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division: [ref 1] + ¯ = ¯ + ¯, ¯ = ¯ ¯, ¯ = ¯ ¯, ¯ = ¯ ¯, A complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the number is real.
Addition (usually signified by the plus symbol +) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. [2] The addition of two whole numbers results in the total amount or sum of those values combined. The example in the adjacent image shows two columns of three apples and two ...
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 ...
In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955 [1] [2]) or complex number (proposed by S. Khmelnik in 1964 [3] and Walter F. Penney in 1965 [4] [5] [6]).
where i is the imaginary unit, i.e., a (non-real) number satisfying i 2 = −1. Addition and multiplication of real numbers are defined in such a way that expressions of this type satisfy all field axioms and thus hold for C. For example, the distributive law enforces (a + bi)(c + di) = ac + bci + adi + bdi 2 = (ac − bd) + (bc + ad)i.
The imaginary unit i in the complex plane: Real numbers are conventionally drawn on the horizontal axis, and imaginary numbers on the vertical axis. 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 ...