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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.
Transcendental number: Any real or complex number that is not algebraic. Examples include e and π. Trigonometric number: Any number that is the sine or cosine of a rational multiple of π. Quadratic surd: A root of a quadratic equation with rational coefficients.
A negative real number −x has no real-valued square roots, but when x is treated as a complex number it has two imaginary square roots, + and , where i is the imaginary unit. In general, any non-zero complex number has n distinct complex-valued n th roots, equally distributed around a complex circle of constant absolute value .
This is the minimum number of characters needed to encode a 32 bit number into 5 printable characters in a process similar to MIME-64 encoding, since 85 5 is only slightly bigger than 2 32. Such method is 6.7% more efficient than MIME-64 which encodes a 24 bit number into 4 printable characters.
Binary coding systems of complex numbers, i.e. systems with the digits = {,}, are of practical interest. [9] Listed below are some coding systems , (all are special cases of the systems above) and resp. codes for the (decimal) numbers −1, 2, −2, i. The standard binary (which requires a sign, first line) and the "negabinary" systems (second ...
For n = 5, 10, none of the non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sum z + z = 2 Re z of each root with its complex conjugate (also a 5th root of unity) is an element of the ring Z[ 1 + √ 5 / 2 ] (D = 5). For two pairs of non-real 5th roots of unity these sums are inverse golden ratio ...
A real number can be expressed by a finite number of decimal digits only if it is rational and its fractional part has a denominator whose prime factors are 2 or 5 or both, because these are the prime factors of 10, the base of the decimal system. Thus, for example, one half is 0.5, one fifth is 0.2, one-tenth is 0.1, and one fiftieth is 0.02.
For example, the real cube roots of 8 and −8 are respectively 2 and −2. The real cube root of an integer or of a rational number is generally not a rational number, neither a constructible number. Every nonzero real of complex number has exactly three cube roots that are complex numbers.