Polynomial equations with rational coefficients have numbers {algebraic number} as roots, because polynomial functions involve only arithmetic operations. Algebraic numbers can have degrees, such as square root, depending on equation degree.
Kummer
If a is imaginary pth root of unity {algebraic number, Kummer}, f(a) = A(0) + A(1) * a + A(2) * a^2 + ... + A(p - 2) * a^(p - 2).
infinity
Algebraic and rational numbers have same infinity order.
denumerability
Integer-coefficient polynomial-equation roots are denumerable. Real-number-coefficient polynomial-equation roots are not denumerable. Algebraic numbers do not have unique factorization.
For number pairs {amicable number}, one number's factor sum, including 1, can equal other number. For example, 6 is amicable with 6: 3 + 2 + 1 = 6. 9 is amicable with 7: 3 + 3 + 1 = 7.
Number series {Bernoulli number} can be 1/6, 1/30, 1/42, 1/30, 5/66, 691/2730, 7/6, 3617/510, ....
Number series {Catalan number} can be 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ....
Numbers {cubic number} that are cubes of n (n^3) equal sum of n consecutive odd numbers starting with n * (n - l) + l and equal n * (n * (n - 1) + l) + 2 * (n - 1).
Number series {Euler number series} (En) can be 1, 5, 61, 1385, 50521, ....
e = 2.7182818284... {Euler's number} {Euler number} {Napier's constant}.
Whole numbers {even number} can be multiples of two.
Numbers {Fibonacci number} {Fibonacci sequence} can be sums of the two preceding numbers, starting with one: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, .... Fibonacci sequence is a geometric progression in which number ratios converge on golden ratio: 2/1, 3/2, 5/3, 8/5, 13/8, and so on.
Limit of 1/1 + 1/2 + 1/3 + ... + 1/n - log(n) is 0.5772... {Euler's constant} {gamma}, where n is integer that goes to infinity. Is Euler's constant rational or irrational?
Odd numbers {gnomonic number}, 2*n + 1, can add to squares of natural numbers n^2 to make squares of next natural number (n + 1)^2: n^2 + 2*n + l = (n + 1)^2.
Gödel numbering {Gödel number} can assign unique digit strings to statements, ideas, images, strings, and so on.
For Fibonacci sequence, sum of two numbers divided by larger number equals larger number divided by smaller number {golden ratio}| {golden section}. Golden ratio equals 1.618034..., reciprocal equals 0.618033..., and square equals 2.618034....
construction
First, bisect square side. Then draw circle with center at bisection point and radius from bisection point to square far corner. Extend one square side until extension meets circle. Extended side has length 1.618...
geometry
If right triangle has one side equal 1 and hypotenuse equal 1.618..., angle is one radian. Golden rectangle has sides in golden ratio and has central point angle of 58 degrees. Pentagram and decagram have sides in golden ratio. Golden ratio is ratio of rectangle {golden rectangle} sides in logarithmic spirals.
music
In music, ratio 2^0.67 = 1.59 ~ 1.618... is similar to major-sixth/octave = 1.67, octave/major-fourth = 1.6, and minor-seventh/major-second = 1.59. Golden ratio and its inverse can make all music harmonics.
If integer sets form mathematical fields, product of any two integers a b, plus product of any other two integers c d, is an integer e {ideal number}: a * b + c * d = e. Ideal numbers have unique factorization, primes, and maxima. Ideal numbers are algebraic-number classes.
To count numbers greater than zero {positive number} or less than zero {negative number} use signed whole numbers and zero {integer}, such as -9, 0, +9.
Integers can be function domain J, and positive integers can be function domain J+.
sums
Positive integers are sums of at most four squares. Integers are sums of at most 19 quarts. Positive even integers are sums of not more than four primes. Odd integers are sums of not more than three primes.
primes
Are there infinitely many positive integers, so both one less and one more are prime?
Numbers {irrational number}| can have infinite non-repeating decimals. Example is 2^0.5. Polynomial positive roots can have irrational numbers. Most irrational numbers are transcendental numbers, such as pi and e, not algebraic numbers. Transcendental-number infinity order is more than algebraic-number or rational-number infinity order. Irrational numbers in closed intervals are rational-number-series limits.
To count how many, use whole numbers, such as one, two, and so on {natural number}| {cardinal number}. Zero is natural number.
Real numbers {normal number} can have all digits, pairs, triples, and so on, in equal numbers.
Whole numbers {odd number} can be not multiples of two.
To count in order, use numbers {ordinal number}| like first, second, third, and so on.
Numbers {perfect cube} can be natural-number cubes, such as 1 = 1^3, 8 = 2^3, and 27 = 3^3.
Numbers {perfect number} can equal sum of prime factors: 6 = 1 + 2 + 3. If 2^n - 1 is prime number and n is integer, 2^(n - 1) * (2^n - 1) is perfect number. Number's prime-factor sum can be greater than number {deficient number} {defective number}. Number's prime-factor sum can be less than number {abundant number} {excessive number} {redundant number}.
Numbers {perfect power, number} can be smaller-number nth powers.
Numbers {perfect square} can be natural-number squares, such as 1 = 1^2, 4 = 2^2, and 9 = 3^2.
Only one and the number can divide into some whole numbers {prime number}| without remainder. For every prime p, factorial of p minus one, plus 1, is factorable by prime: ((p - 1)! + 1) / p.
General quadratic equation (a*x^2 + b*x + c = 0) solutions {quadratic irrational number} can have form a + b^0.5, where a and b are rational and b is not a perfect square. Ruler and compass constructions that do not result in rational numbers can most simply result in quadratic irrational lengths. Quadratic irrational numbers can be periodic continued fractions. For a = 0 and b = 14, 14^0.5 = 3 + (1 + (2 + (1 + (6 + (1 + (2 + (1 + (6 + (...)^-1)^-1)^-1)^-1)^-1)^-1)^-1)^-1)^-1. Solutions have form A + (B + (C + (D + (E + (B + (C + (D + (E + (...)^-1)^-1)^-1)^-1)^-1)^-1)^-1)^-1)^-1.
Numbers {rational number}|, such as 3/5 or 1/9, can have finite or repeating decimals. In one system, rational numbers derive from natural numbers, using reflexive, symmetric, and transitive axioms. Rational numbers are all numbers expressed as one integer divided by another integer.
Numbers {real number}| can include both irrational and rational numbers. Real numbers make all equation roots. In one system, real numbers derive from natural numbers using axioms of connection, calculation, order, and continuity. Real numbers and points on closed intervals have one-to-one correspondence. Real numbers and points in closed-interval squares have one-to-one correspondence, because square points are real-number pairs and line points are numbers that alternate pair digits.
N^2 = 1 + 3 + 5 + ... + (2*N - 1) {square number, theory}. (N^2 + N) / 2 = sum from 1 to N of i = (N/2) * (N + 1).
Non-algebraic numbers {transcendental number}| are roots of equations with trigonometric, inverse trigonometric, exponential, and logarithmic functions. Transcendental numbers can relate to circles, triangles, exponents, and logarithms, such as pi = 3.1415926535... and e = 2.7182818284... = Euler's number.
pi
pi is the limit of ratio between many-sided regular-polygon circumference and center-to-vertex line length.
e
e is base of expression e^-x, such that derivative of e^-x equals e^-x. Lower values make lower derivatives and higher make higher, so e is in middle. It also makes x^(1/x) maximum. It is the limit of (1 + 1/n)^n, when compound interest has many periods and interest is 1/n per period. Definite integral of (1/x) * dx from 1 to e equals 1.
e = 1/0! + 1/1! + 1/2! + 1/3! + ... = 1/1! + 2/2! + 3/3! + 4/4! + ...
e^a = 1 + a + a^2/2! + a^3/3! + ...
-1/(e^pi) = 1 + i - 1/2 - i/6 + 1/24 + i/120 + ...
-1/(e^pi) - 1/2 + 1/24 - 1/720 + ... = i*(1 - 1/6 + 1/120 + ...).
i = (-1/(e^pi) - 1/2 + 1/24 - 1/720 + ...)/(1 - 1/6 + 1/120 + ...).
integral from -infinity to +infinity of e^-x^2 * dx = pi^0.5.
trigonometry
sin(a) = a - a^3/3! + a^5/5! - a^7/7! + ...
cos(a) = 1 - a^2/2! + a^4/4! - a^6/6! + ...
i
i = e^(i*pi/2).
ln(i) = i*pi/2
e^(i*a) = cos(a) + i*sin(a), where a is in radians and is real.
sin(a) = (e^i*a - e^-i*a)/2i
cos(a) = (e^i*a + e^-i*a)/2.
e^i = 1 + i - 1/2 - i/6 + 1/24 + i/120 + ...
cos(i) = 1 + 1/2 + 1/24 + 1/720 + ...
i = arccos(1 + 1/2 + 1/24 + 1/720 + ...).
-e^-pi = e^i
i = ln(-e^-pi).
e^-pi = -e^i
pi = -ln(-e^i) = - ln(-1) - ln(i).
ln(i) = i*pi/2 = i/2 * (- ln(-1) - ln(i)).
ln(i) = -i*ln(-1)/2 - i*ln(i)/2
i*ln(i) = ln(-1)/2 + ln(i)/2
2*i*ln(i) = ln(-1) + ln(i).
2*i*ln(i) - ln(i) = ln(-1).
ln(i) * (2*i - 1) = ln(-1).
2^0.5 is minimum of rectangle-diagonal-length to average-side-length ratio.
The Hebrew-alphabet first letter, aleph, with subscript from 0 to infinity, represents infinity orders {transfinite number}|. The infinity orders are infinite. Finite-number sets have a greatest number.
10^12 is a number {trillion}|.
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.
N! = N * (N - 1) * ... * 1 {factorial}|. Factorial symbol {factorial sign} is exclamation point.
Factorial is true even if number n is not integer: n! = product from k = 1 to k = infinity of ((k + 1) / k)^n * k / (k + n) {interpolation problem}. Eulerian integrals, like gamma function or beta function, can be for interpolation.
pi = 4 * (1/1 - 1/3 + 1/5 - 1/7 + ...) {Leibniz formula}.
pi = 2 * (2/1) * (2/3) * (4/3) * (4/5) * (6/5) * (6/7) * ... pi/2 is product {Wallis's product} {Wallis' product} {Wallis product} of terms (2 * r / (2*r - 1)) * (2 * r / (2*r + 1)), from r = 1 to infinity.
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Date Modified: 2022.0225