Numbers {complex number}| can have real part x added to imaginary part i*y, where y is real number: x + i*y. Complex numbers can solve all polynomial equations, such as x^2 + 1 = 0. Complex numbers are roots of homogeneous polynomial equations with only positive factors: a*x^n + ... + C = 0. For example, quantum mechanics has only positive energy components, and so uses complex-number equations. Because polynomials can approximate all equations, complex numbers can approximately solve all equations. Because they have two independent components, complex numbers have no inequality equations.
The number i equals -1^0.5 {imaginary number}|. By DeMoivre's theorem, any power of i is expressible as a + b*i. For example, i^2 = -1, i^3 = -i, and i^4 = 1. i^0.5 = 1/(2^0.5) + (1/(2^0.5))*i. i^0.5 = -1/(2^0.5) - (1/(2^0.5))*i. i^0.333 = (3^0.5)/2 + (1/2)*i. Therefore, complex numbers need only a real part and an imaginary part, with no other components.
i^i = e^-pi / 2, for log i = 0.5 * pi * i. All i^i are real numbers. z = r * e^i*A. log z = log r + i*A. e^i*A = cos A + i*sin A.
All polynomial roots are expressible by at least one complex number (though not as radicals, by the Abel-Ruffini theorem [1824]).
Perhaps, a new complex-number type can use a factor of reals and imaginaries, but not be a hypercomplex number.
Complex numbers can be on planes {Argand diagram}, with real numbers on horizontal axis and imaginary numbers on vertical axis. Complex numbers can be on planes with polar coordinates: z = r * cos(A) + i * r * sin(A), where r equals length from point to origin {absolute value, complex number} {magnitude, complex number} {norm, complex number} {modulus, complex number}, and A equals angle to horizontal axis {argument, complex number} {phase, complex number} {amplitude, complex number}.
Complex numbers x + i*y have associated complex numbers {complex conjugate}|: x - i*y. Complex numbers multiplied by complex conjugates make real numbers whose magnitude is complex-number squared.
(cos(A) + i * sin(A))^n = cos(n*A) + i * sin(n*A) {DeMoivre's theorem} {DeMoivre theorem}.
e^i * pi = -1 {Euler's identity} {Euler identity}.
Numbers can have more than one imaginary component {hypercomplex number} {hypernumber}. Complex numbers are two-dimensional vectors, and hypernumbers are n-dimensional vectors. Hypernumbers can represent tensors, quaternions, matrices, determinants, and all number types. Hypernumbers are directed line segments {extension, calculus}.
magnitude
Magnitudes are the same as for vectors.
addition
Hypernumbers add corresponding parts, like complex numbers.
multiplication
Hypernumbers, like complex numbers, multiply like polynomials. Products are scalars or vectors. Product of same axis and itself makes scalars. When axis multiplies another axis, result is vector orthogonal to both original axes.
Hypercomplex numbers {quaternion} can be scalar plus three-dimensional vector: a + b*i + c*j + d*k, where a, b, c, and d are real numbers, and i, j, and k are orthogonal unit vectors.
operations
Quaternion addition is like translation. Multiplying quaternions is non-commutative: i*j = k, j*k = i, k*i = j, j*i = -k, k*j = -i, i*k = -j and describes quaternion rotations. Quaternions can divide.
space
Complex numbers map to two-dimensional space, and quaternions map to three-dimensional space.
spinor
Real-number spinors represent rotating quaternions.
Hypernumbers {biquaternion} can be real quaternion plus w times real quaternion: a + b*i + c*j + d*k + w * (e + f*i + g*j + h*k), where w^2 = 1. w commutes with all real quaternions. Biquaternion operations obey multiplication product law and are linear, associative, and non-commutative.
Hypernumbers {octonion} can have one real term and seven imaginary terms: N, i, j, k, l, m, n, p. Imaginary term multiplied by itself gives real term. Two different imaginary terms multiply to different third term, by cyclic ordering: i * j = k, for example. Octonions can divide. Figures {Fano plane} that represent octonions have seven points, each with two links.
Adding complex numbers is like adding vectors {parallelogram law}. Adding is translation. Triangle 0, 1, w is similar to triangle 0, z, wz.
Multiplying complex numbers is like multiplying vectors {similar triangles law}. Multiplying two complex numbers multiplies moduli and adds arguments. Arguments are like logarithms in this way.
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Date Modified: 2022.0225