Ithkuil/Mathematics

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Numbers[edit]

Counting rod horizontal black 5.svgCounting rod vertical black 5.svgCounting rod horizontal red 5.svg

Ithkuil uses a centesimal number system with the base roots 0-10 and the TNX affix. Stem 1 refers to quantity, stem 2 refers to aspects, and stem 3 refers to ordinals. To save space, groups of ten are represented by a circle. Below are the noun and verb forms of the numbers 1-30 with Objective Specification.

For nouns, the Vc value, and for verbs, the Vk value, may be omitted if phonotactically permissible and the following word is unambiguous. The Vv value may be omitted and indicated through modifying the Vr value if phonotactically permissible and the previous word is unambiguous. If the number is a noun, the previous word must be an adjunct indicating nominal distinction. Monosyllabic words are interpreted as having EXS Context.

Counting rod vertical black 1.svg Counting rod vertical black 2.svg Counting rod vertical black 3.svg Counting rod vertical black 4.svg Counting rod vertical numeral -4.svg Counting rod vertical numeral -4.svgCounting rod vertical black 1.svg Counting rod vertical numeral -4.svgCounting rod vertical black 2.svg Counting rod vertical numeral -4.svgCounting rod vertical black 3.svg Counting rod vertical numeral -4.svgCounting rod vertical black 4.svg Counting rod numeral 0.svg
(a)llol(a)

ulloil(a)

(a)ksol(a)

uksoila

(a)zol(a)

uzoila

(a)pšol(a)

upšoila

(a)stol(a)

ustoila

(a)cpol(a)

ucpoila

(a)nsol(a)

unsoila

(a)čkol(a)

učkoila

(a)lżol(a)

ulżoila

(a)vrolalm(a) = (a)ššol(a)

uvroilalm(a) = uššoila

Counting rod vertical black 1.svgCounting rod numeral 0.svg Counting rod vertical black 2.svgCounting rod numeral 0.svg Counting rod vertical black 3.svgCounting rod numeral 0.svg Counting rod vertical black 4.svgCounting rod numeral 0.svg Counting rod vertical numeral -4.svgCounting rod numeral 0.svg Counting rod vertical numeral -4.svgCounting rod vertical black 1.svgCounting rod numeral 0.svg Counting rod vertical numeral -4.svgCounting rod vertical black 2.svgCounting rod numeral 0.svg Counting rod vertical numeral -4.svgCounting rod vertical black 3.svgCounting rod numeral 0.svg Counting rod vertical numeral -4.svgCounting rod vertical black 4.svgCounting rod numeral 0.svg Counting rod numeral 0.svgCounting rod numeral 0.svg
(a)llolalm(a)

ulloilalm(a)

(a)ksolalm(a)

uksoilalm(a)

(a)zolalm(a)

uzoilalm(a)

(a)pšolalm(a)

upšoilalm(a)

(a)stolalm(a)

ustoilalm(a)

(a)cpolalm(a)

ucpoilalm(a)

(a)nsolalm(a)

unsoilalm(a)

(a)čkolalm(a)

učkoilalm(a)

(a)lżolalm(a)

ulżoilalm(a)

(a)vrolälm(a) = (a)ššolalm(a)

uvroilälm(a) = uššoilalm(a)

Counting rod vertical black 1.svgCounting rod numeral 0.svgCounting rod numeral 0.svg Counting rod vertical black 2.svgCounting rod numeral 0.svgCounting rod numeral 0.svg Counting rod vertical black 3.svgCounting rod numeral 0.svgCounting rod numeral 0.svg Counting rod vertical black 4.svgCounting rod numeral 0.svgCounting rod numeral 0.svg Counting rod vertical numeral -4.svgCounting rod numeral 0.svgCounting rod numeral 0.svg Counting rod vertical numeral -4.svgCounting rod vertical black 1.svgCounting rod numeral 0.svgCounting rod numeral 0.svg Counting rod vertical numeral -4.svgCounting rod vertical black 2.svgCounting rod numeral 0.svgCounting rod numeral 0.svg Counting rod vertical numeral -4.svgCounting rod vertical black 3.svgCounting rod numeral 0.svgCounting rod numeral 0.svg Counting rod vertical numeral -4.svgCounting rod vertical black 4.svgCounting rod numeral 0.svgCounting rod numeral 0.svg Counting rod numeral 0.svgCounting rod numeral 0.svgCounting rod numeral 0.svg
(a)llolälm(a)

ulloilälm(a)

(a)ksolälm(a)

uksoilälm(a)

(a)zolälm(a)

uzoilälm(a)

(a)pšolälm(a)

upšoilälm(a)

(a)stolälm(a)

ustoilälm(a)

(a)cpolälm(a)

ucpoilälm(a)

(a)nsolälm(a)

unsoilälm(a)

(a)čkolälm(a)

učkoilälm(a)

(a)lżolälm(a)

ulżoilälm(a)

(a)vrolelm(a) = (a)ššolälm(a)

uvroilelm(a) = uššoilälm(a)

agzol(a) / gzola is 100, agzol(a) allal(a) / gzola llala is 101, aksol(a) agzalui / ksola gzalui / agzalk(a) / gzalka is 200, aksol(a) (agzalui) allal(a) / ksola (gzalui) llala / agzalk(a) allal(a) / gzalka llala is 201. We use the Commitative eyë for numbers greater than 100², 100⁴, or 100⁸, along with -iň for numbers greater than 100 multiplied by them.

Mathematics[edit]

A numerical expression can represent the physical quantity or form of existing phenomena, dimensional relations, or abstract patterns, and how they can change. We often represent values based off of a unit through symbols such as strokes or natural numbers. The Monad is God, a spirit, a circle, eternity, permanence, the limit, the purest tone. One is the mold that shapes all things and the thing shaped by all molds. It is the source, center, destination, and entirety of the universe. To speak of one is to imply separation from it, thus misrepresenting it. Multiplying or dividing by one results in the same object. Unity permeates everything. There is nothing without it and nothing within it. One has majestic unconditional love. One is unique. The dyad is the other side of the coin. The shadow, the opposite, the other. Self vs. non-self is the first distinction we make. Two is the basis of comparison. There is odd-even, limited-unlimited, right-left, male-female, resting-moving, straight-curved, positive-negative, real-imaginary. Two points define a line and our bodies are bilaterally symmetric. We breath in and breath out, opposite charges attract, and good has some evil while evil has some good. Duality is complementary. The Chinese call this yin-yang. The 2:1 ratio of musical tones represent the difference of the same pitch raised or lowered by an octave. In day we live under a sun and at night under a moon of the same apparent size. The triad is .... the third leg of a stool balances it. The volumes of a cone, sphere, and drum are in a 1:2:3 ratio.

Quadrad, pentad, hexad, heptad, octad, nonad, decad, undecad, duodecad. Ancient Egyptians, Sumerians, Chinese, Greeks, Mayans, Romans, Indians and Persian and modern people and computers use positional notation. The earliest part of a natural number signifies the largest non-empty component and later parts of it are exponentially smaller. In Ithkuil, nine digits and nine additional extensions can fill an empty slot in a counting number. This lets one count to 100 in a slot, which is the factor by which preceding slots multiply by to be part of a big number. E2 marks one empty slot, E4 marks two empty slots, E8 marks four empty slots, and E16 marks eight empty slots. Addition and subtraction are functions that can be performed on two numbers to calculate a result. In practice, one can count the number of books they possess over time. Addition makes the result of one number and a second number into a larger number, and subtraction changes the second number to make the result a smaller or equal number to the first. Addition ordering has no regard on the result of two counting numbers. Subtraction ordering can give either a representable natural number or its opposite. The opposite is one of the negative numbers, which extend the natural numbers before zero. Together, these number sets create the sequence of integers which can be represented with a number line in two infinite directions. Distances between two number-points on a number line can also be represented by a number. If two points are directly next to each other, the conventional distance to the right for bigger numbers is one and to the left for smaller numbers is negative one. Addition and subtraction are functions that change numbers by several unities into results. Multiplication and division are functions that can also be performed on two numbers to calculate a result. In practice, one can count the possibilities of outfits from the wardrobe. Multiplication makes one number and a second number into a result further from zero, and division inverts the second number into a number smaller than one so the result is closer to zero. Like addition, multiplication ordering has no regard on the result of two counting numbers. Multiplication is the repeated addition of a number, like how three times four is really three plus three plus three plus three. For example, four times three is four plus four plus four. Both of these repeated sums equal twelve, so three times four equals four times three by an axiom of transitive equality. Complementary to multiplication, division is repeated subtraction, and like subtraction, division ordering changes the result of two numbers. The results of one number then another versus the reverse will always multiply to one. The inverse of a function is a function that turns it into no total change. The inverse of adding one is subtracting one because if you start with a number and do both, the process results in the starting number. The inverse of multiplying by two is dividing by two because if you start with a number and do both, the process results in the starting number. Division with integers is not directly invertible because remaining number from the result is often ignored. For example, nine divided by two is four with a leftover one, so four times two plus one is nine, but without the one four times two is eight.

1 2 3 4 5 6 7 8 9 Ɛ
∴l∵ ∴k∵ ∴ţ∵ ∴p∵ ∴s∵ ∴q∵ ∴n∵ ∴f∵ ∴x∵ ∴m∵ ∴t∵
0 1000 . τ (=2π) + × - -1 ( )
∴ň∵ ∴oţ∵ ∴š∵ ∴ẋ∵ ∴tw∵ ∴kw∵ ∴r∵ ∴ř∵ ∴t’∵ ∴ut’∵ ∴ļ∵

One can use numbers that can be spoken more quickly under a form that counts to twelve in a slot. A hundred is written as 84 because eight dozens with four units add up to ten tens. A hundred and forty-four is written as 100 because a dozen dozens is one hundred and forty-four. This is the same as one of a twelve times twelve. Place values can continue for bigger or smaller numbers. Parts systematically split into twelve are written after the dozenal point, like the number six plus three tenths plus four hundredths plus nine ten-hundredths plus four ten-thousandths. For addition, 6+6 is 10 because six plus six equals twelve. The digit of unity is moved before where it was and a place-value zero is used to show twelve instead of one. Ɛ7+8 is 103 because 7+8 of units is 3 units and 1 dozen, and Ɛ+1 of dozens is 10 dozens, so there is one dozen-of-dozens digit, then a zero dozens digit, and then three one digits. For the inverse function, the subtraction sign is written prior to the regular number. 6+-6 means six plus the opposite of six, which equals nothing.

Telling time by o'clock

Egyptian multiplication, Russian peasant multiplication, Modular arithmetic, Rounding, Sunzi division, Galley division

Types of numbers: In 5-8=-3 the number after the equals sign is an integer, as opposed to a natural number. Technically it is a negative integer instead of a positive integer because it is less than 0. 1/2 is represented as 0.6 because two halves is 2×0.6=1.0=0.6+0.6=1/2+1/2=2/2. In 5/8=.76 the number is rational, as opposed to an integer because it contains fractions of unity. Cantor's diagonal. The dozenal point represents the boundary between numbers formed as components smaller than unity coming later and numbers formed as components of unity coming earlier. The irrational -√2=-1.4Ɛ79170ᘔ07Ɛ85... can not be represented as a ratio of two integers and it continues unpredictably after the dozenal point.

= + × modulo inverse -1 exponent 2√() √() () !
∴mw∵ ∴ḍy∵ ∴tw∵ ∴kw∵ ∴zy∵ ∴řw∵ ∴ř∵ ∴hl∵ ∴qw∵ ∴ky∵ ∴py∵ ∴řy∵
∴â∵ ∴öa∵ ∴i∵ ∴a∵ ∴üa∵ ∴öu∵ ∴ô∵ ∴e∵ ∴ê∵ ∴ou∵ ∴u∵ ∴öi∵

Rational, Sieve of Eratosthenes, Fermat Numbers, Mersenne Primes, Goldbach conjectures, Twin Primes, Perfect, Amicable, Irrational, Transcendental, τ=6.349416967Ɛ635..., i, Complex, Summation, Product, Modular arithmetic, Least Common Multiple, Greatest Common Factor, Continued fractions, Polynomial, Partial fraction, Complex number, Constant, Variable, Coefficient, Arithmetic axioms, x=1; x=y; xy=y2; xy-x2=y2-x2; x(y-x)=(x+y)(y-x); x=x+y; Logs, Exponents, Roots, Binomial theorem, Root Extraction, Cube roots, Peano Axioms, Approximation, Truncation, Rounding, Boundary values

The counting board (abacus, uses beads on rods to represent place values.

Rod calculus, Napier's bones, Slide Rule, Combinatorics, Permutation, Inversion, Cyclic permutation, 15 Puzzle, Identical elements, Latin Square, Combination, Sample with Replacement

Graph theory, Euler Path, Hamilton Path, Four-Color Map, Magic square, Composite, Magic shapes, Symbolic logic, Proposition, Tautology, Syllogism and Proof, Logic Circuit

{} ℕ1 from Pʰ to Qʰ for Pʰ iterations
čw cw pʰw lw ly ny nw
awe* ara* awo* ayo* *oye* / *oy owe*
2nd level parentheses 3rd level parentheses 4th level parentheses
p’ up’ k’ uk’ q’ uq’

Set Theory, Russel's Paradox, Proper subset, Ordered complement, Venn Diagram, Set Algebra, Boolean Algebra, Transfinite numbers, Continuum Hypothesis, Arithmetic, A sequence of numbers can be patterned to enable one to predict further items, Harmonic Series, Geometric mean, Infinite series, Convergence, Euler-Mascheroni constant γ=0.6Ɛ15188ᘔ6760Ɛ3, Infinite Geometric series, Convergence test, Alternating Series and Completeness, Tower of Hanoi, Fibonacci: 1, 1, 2, 3, 5, 8, 11, 19, 2ᘔ, Golden ratio φ, Lucas sequence, Binet formula, Pell number sequence, Figurate numbers,

¬
∴u’a*∵ ∴i’a*∵ ∴a’i*∵ ∴iwa*∵ ∴ora*∵ ∴o’a*∵ ∴aru*∵ ∴a’u*∵ ∴owi*∵ ∴oyu*∵

Factoring, Remainder theorem, Root-Coefficient Relationships, Linear equations, Quadratic equations, Inequalities, Cubics, Quartics

¯ëveqʰavepʰâvet’ëqʰípʰ; ¯ëveqʰařvepʰâvet’ëqʰírpʰ; ¯ët’ëveqʰut’epʰâvet’ëqʰápʰ;

¯ët’ëvḍut’eqʰâveqʰaḍéqʰ; ¯ët’ëvôḍut’eqʰâveqʰôḍéqʰ; ¯ët’ëvôḍut’erqʰâḍeqʰôvéqʰ;

¯ëverpʰâřvépʰ; ¯ëvèlâv; ¯ëvèňâl; ¯ëveḍâžöàhrëžüvâḍ;

¯ëhrëvüžihrëḍüžâhrëvḍüüž; ¯ëhrëvüžirëhrëḍüžâhrëvôḍüüž;

¯ëřt’ëřhrëžüviřhrëžüḍùt’âhrëžüvḍë; ¯ëhrëpʰüçâhrëëpʰ; ¯ëhrëpʰüpʰââl; ¯ëhrëçââl; ¯ëhrëlââň;

¯ëhrëvüžâḍöařwëhrëḍüžââv;

¯ëkyëvḍupʰâkyëvupʰakyëḍúpʰ; ¯ëkyëvôḍupʰâkyëvupʰôkyëḍúpʰ; ¯ëkyëveqʰupʰâkyëvùt’ëpʰôqʰ;

¯ëkyëvùlâv; ¯ëkyëvùkâqwëv; ¯ëkyëvupʰâvelôpʰ; ¯ëkyëveqʰupʰât’ouvupʰut’éqʰ; ¯ët’ouvupʰut’eqʰâvèt’ëqʰôpʰ;

¯ëkyëžuvakyëžuḍâkyëžuřt’ëřvařḍút’; ¯ëkyëžult’öžüvùt’âv; ¯ëhrëžüt’oužuḍùt’âḍ;

¯ëkyëveḍuḍâv; ¯ëkyëžuḍeḍâž; ¯ëvuhrëžüvâž; ¯ëhrëvuḍüvâḍ;

Systems of equations, Diophantine Equations, Pythagorean number theorem, Fermat's Last Theorem

II FUNCTIONS

GEOMETRY

TRIGONOMETRY and HYPERBOLICS

ANALYTIC GEOMETRY

VECTOR ANALYSIS

FRACTALS

MATRICES

III CALCULUS

POWER SERIES

INDETERMINATE LIMITS

COMPLEX NUMBERS

EXTREMA and CRITICAL POINTS

ARC LENGTH

CENTROIDS

AREA and VOLUME

MOTION

HARMONIC ANALYSIS

APPROXIMATION

PROBABILITY

i≡√-1 extends the real number line into a plane, making the full representation of functions complex. For example, 3i lies perpendicular to distances over the real number line. It has the same distance from 0 on the number line as 3 and -3, but lies 3√2 units away from them.

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211 is a look-and-say number sequence where Conway's λ≡lim n→∞ Mn+1/Mn =

Feigenbaums α=4.80446936ᘔ2149 δ=2.605036ᘔ50951Ɛ

Kinchin K=

Points are the source for and reference of everything. A path between two points travels through infinitely many others. Each point can represent a distance scalar, location, or idea. Infinite even arrangements of points in a single dimension that extend infinitely in two opposite directions but can also be bounded are lines or loops. Lines are the path along which lies the shortest distance between two points. Loops are the path along which every point is away from the center by a radius r. Line and loop segments can be bounded to certain regions of spacetime. The intersection of two lines forms four angles. An angle < τ/4 radians is acute. An angle > τ/4 radians is obtuse. Angles < 0 or > τ can be simplified to be between 0 and τ. Angles of 0 or (n/2)τ are straight lines.

The circle constant τ=6.349416967Ɛ635... is equal to the ratio between a loop's radius and its edge. A loop can be drawn using all points of constant length from the center. Likewise, an ellipse can be drawn using all points of constant length from one center to it to the other center. There are just as many points along the edge of the loop as there are points contained within. Three points of a 2D shape define a plane, where if a straight line falling on two other straight lines creates two angles on one side of it that add up to less than t/2, then the two lines extend to a point where they intersect on that side. If the sum of the internal angles of a triangle < τ/2, such as on the surface of a saddle, the geometry is hyperbolic. If the angles sum > τ/2, such as on a ball, the geometry is elliptic. The planar surface may also be curved in higher dimensions. If the perimeter of a circle does not equal τ times a multiple of the radius, the lines defining the geometry are non-Euclidean. However, most shapes are Euclidean-based.

△ ⊿

Most forms can be decomposed into three-side 2D-shape components on a fundamental level. Such triangles have three angles. If a triangle's largest side length j is opposite to a τ/4 right angle, j2 is equal to the sum of the squares of the other two sides, p2+q2. This is the theorem of Pythagoras. Non-right triangles can be split into two right triangles. The sum of the internal angles of a triangle is τ/2. Its perimeter is given by the sum of the lengths of the sides. The area is given as the base*height/2 or the average of two sides*the sine of the angle in between. The center of an incircle bounded by a triangle meets at the intersection of the lines bisecting each angle of the triangle. The radius r of the circle going passing through the vertices of a triangle is equal to a side length divided by twice the sine of its opposing angle. The center of this circumcircle bounding a triangle meets at the intersection of the perpendicular lines bisecting each line of the triangle.

Trigonometric proportions. We can define ratios of circle chords or angles. Sine is the opposite side over the longest side. Cosine is the short adjacent side over the longest side. Tangent is sine over cosine or otherwise the opposite side over the short adjacent side.

Thales' theorem. Cèyuán hǎijìng. No matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant.

▭ ⬜ ♢

Quadrilaterals are composed of two triangles, so the internal angles sum to t and the perimeter is the sum of the non-touching component perimeters. A shape with an internal angle greater than t/2 is concave with area equal to the sum of its component triangles.

[The area of a quadrilateral is the sum of the squares of two opposing sides minus the sum of the squares of the other two opposing sides multiplied by 1/4 times the tangent of the line segment lengths connecting opposite ends of the shape or otherwise 1/2 times the sine of that angle multiplied by the two line segments. Trapezoids have a set of parallel lines, so the area simplifies to 1/2 times the sum of the two parallel line segment lengths times.] Kites. Parallelograms. Diamonds.

⬠ ⬡

Pentagons. Hexagons. Pentadecagons. The perimeter of an n-gon generalizes to the sum of the side lengths. The area of a regular convex n-gon is the sum of the squares of the side lengths over 2tan(τ/2n). The perimeter and area of other shapes is like that of a quadrilateral.

If a line extends from a shape's at an angle greater than the largest angle between two intersection lines, then it extends into a third dimension to potentially occupy volume. Block. Parallelepiped. Sphere. Plane. Cylinder. Cone. Pyramid. Cavalieri's principle.

Tetrahedron. Simplex. Tesseract. Platonic Solids. Archimedean solids.

Gödel's Theorems

Algorithms and Recursion

Evaluating infix order of operations

Ratio and Mean Proportional

Translational/Rotational Symmetry

e≡Σn=[0,∞]1/n! which evaluates =1/1+1/1+1/2+1/6+1/20+1/80+... =

Approximating angles

Conic Section vertices

Triangular and tetrahedral numbers

Completing the Square

Gaussian Elimination

The equations 2x+1y+1z=3; 1x-1y-1z=0; 1x+2y+1z=0; are a matrix D=[[2,1,1],[1,-1,-1],[1,2,1]] Y=[[3],[0],[0]]

with Dx=[[3,1,1],[0,-1,-1],[0,2,1]]; Dy=[[2,3,1],[1,0,-1],[1,0,1]]; Dz=[[2,1,3],[1,-1,0],[1,2,0]]

det(D)=3; det(Dx)=3; det(Dy)=-6; det(Dz)=0; and x=Dx÷D; y=Dy÷D; and z=Dz÷D;

Linear Programming

Quadratic formula

Horner's scheme (Sharaf al-Dīn al-Ṭūsī)

Cubic and Quartic polynomial factoring

Pythagorean Triples and Fermat's Last Theorem

Representation of a function

Coordinate plane

Line slopes/intercepts

Distance function

Domain & Range

Function characteristics (monotonicity, extrema, inflection points)

Complex Roots

Given triangles with angles v, ḍ, ž and opposite sides v', ḍ', ž' (and spherically v'', ḍ'', ž''), we can use the trigonometric functions. The graphs of sine and cosine repeat over a period τ as they trace out the horizontal vertical components of a point traveling counterclockwise along a circle.

sin(v) = sin(v+nτ) = cos(τ/4-v) and cos(v) = cos(v+nτ) = sin(τ/4-v) and tan(v+τ/8)=(1+tan(v))/(1-tan(v)). The tangent proportion graph repeats twice as much tan(v) = tan(v+n/2τ) = 1/tan(τ/4-v). At (n+1/2)*τ/2 the absolute value of tangent is infinite. The graph of cosine is horizontally symmetric, but sine and tangent are rotationally symmetric. sin(-v) = -sin(v) is odd and cos(-v) = cos(v) is even, so tan(-v) = -tan(v) is odd by definition of tan. Sine and cosine are related such that sin2(v) + cos2(v) = 1 so it follows that tan2(v) + 1 = 1/cos2(v) and 1 + 1/tan2(v) = 1/sin2(v).

(v'+ḍ')/ž' = cos((v-ḍ)/2)/sin(ž/2) and (v'-ḍ')/ž' = sin((v-ḍ)/2)/cos(ž/2)

radius of the circumcircle = v'/(2sin(v)) = ḍ'/(2sin(ḍ)) = ž'/(2sin(ž)), meaning that sin(v)/v = sin(ḍ)/ḍ' = sin(ž)/ž'

ž'2 = v'2 + ḍ'2 -v'ḍ'cos(ž) or in spherical terms cosa=cosbcosc+sinbsinccosa

(v'-ḍ')/(v'+ḍ') = tan((v-ḍ)/2)/tan((v+ḍ)/2)

For j = (v'+ḍ'+ž')/2, the radius of the incircle = √((j-v')(j-ḍ')(j-ž')/j)

sin2(v) = 1/2*(1-cos(2v)) and cos2(v) = 1/2*(1+cos(v)) so tan2(v) = (1-cos(2v))/(1+cos(2v))

sin2(v/2) = (1-cos(v))/2 and cos2(v)/2 = (1+cos(v))/2 so tan2(v)/2 = (1-cos(v))/(1+cos(v)) = sin(v)/(1+cos(v)) = (1-cos(v))/sin(v)

sin(2v) = 2sin(v)cos(v) and cos(2v) = cos2(v) - sin2(v) = 2cos2(v) - 1 = 1 - 2sin2(v) and tan(2v) = 2tan(v)/(1-tan2(v))

sin(3v) = 3sinacos2(v) - sin3(v) and cos(3v) = cos3(v) - 3sin2(v)cos(v) and tan(3v) = (3tan(v) - tan3(v)) / (1- 3tan2(v))

sin(v+ḍ)*sin(v-ḍ) = sin2(v) - sin2(ḍ) = cos2(ḍ) - cos2(v) and cos(v+ḍ)*cos(v-ḍ) = cos2(v) - sin2(ḍ) = cos2(ḍ) - sin2(v)

A natural way to define sin(v) is (eiv - e-iv)/2i and similarily sinh(iv) = (eiv - e-iv)/2, so sinh(iv)=isin(v) or sinh(v)=-isin(iv)

sinh(v) = sin(iv)*1/i = -sinh(-v) and cosh(v) = (eiv + e-iv)/2 = cosh(-v) = cos(iv)

isin(v) = (eiv - e-iv)/2 and icos(v) = (eiv + e-iv)/2 so itan(v) = (eiv - e-iv)/(eiv + e-iv)

We know eix=cos(v)+isin(v), and similarily ev=cosh(v)+sinh(v)

A catenary arch acosh(v/a)=a(ev/a + e-v/a)/2 is different from a parabola y=ax2+bx+c because it naturally forms under the suspension of gravity

The catenoid is a minimal surface when bounded in a closed space, like the shape of soap film between two empty circles..

Inverse trig functions

Polar coordinates

Cylindrical coordinates

Spherical coordinates

Euler angles

John Wallis

Subsequence

Cauchy Sequence

Logic & Probability

A set is a group of ideas, typically numbers, or other sets of numbers. Repeated set members are irrelevant. A tuple is an ordered set. Both have cardinality, but members of tuples have ordinality, so repeated members are relevant.

Assume the existence of numbers irrespective of an applied context defined by set-ideas. Zero is the number represented by the empty set. To define the extent to which natural numbers can reach, we say ∞ is the thing with all numbers such that it also contains their successors. Extensionality, empty set, pairing, union, infinity, schema of replacement, power set, regularity, specification. Power set, complement. Russel's Paradox. Continuum hypothesis. Ring/Field

Yáng Huī's triangle

Converse, inverse, contrapositive, syllogism

Symmetry and Chirality

Chinese remainder theorem

A set of numbers has an average value equal to the sum of each item over the quantity of items. The range is the absolute value of the difference between the largest and smallest items.

A distribution can be discrete or continuous.

Fair representation of data though a bar chart, pie chart, line chart.

Nash equilibrium

Mean is μ. median. mode. Standard deviation σ=√[(∑(x-μ)2)/(n-1)] and z-score z=(x-μ)/σ

Simpson's paradox

Probability density function f(x)=1/(σ√τ)e-½z² is positive for all x.

Knowing the probability of two events A and B, and the conditional probability of B given A fully probable, the conditional probability of A given B fully probable is P(A|B) = P(B|A)P(A)/P(B).

Box and whiskers plot

T Distribution

Binomial Distribution

Scatter plot

Linear regression

The null hypothesis is an assumption that statistical observations are due to chance alone. A type I error rejects the null hypothesis when it is true and a type II error accepts the null hypothesis when it is false. The significance level is the probability of a type I error. The power of the test is the probability of correctly rejecting the null hypothesis and decreasing the false-negative rate. By convention, if the p-value i below 5%, the results of rejecting the null hypothesis are deemed significant.

Chi-Squared Test X2=∑i=[1,j]⌊(Oi-Ei)2/Ei

Poisson Distribution f(k;λ)=λke/k!

Cauchy Sequence

e≡lim n→∞ (1+1/n)n;

Squeeze theorem

tangent line lim v->j (f(v)-f(a))/(v-a); f(1)(v)= lim j->∞ (f(v+j)-f(v))/h

d/dx(xn) = nxn-1; d/dx(lnx) = x-1; d/dx(emx) = memx; d/dx(f(x)g(x))=f'(x)g(x)+f(x)g'(x); d/dx(f(x)/g(x))=(f'(x)g(x)-f(x)g'(x))/(g(x)2)

Discriminant: D=fxxfyy-fxy2 trigonometric formulae

Definite integral: ∫[a,b]xdx=[a,b]1/2x2=1/2(b2-a2)

∫xndx=x(n+1)/(n+1); ∫x-1dx=ln|x|; ∫enxdx=e(nx)/n

By parts: given ∫xe(x^2)dx; substitute x2 with m and d/dx(x2)=2xdx with dm; this means 1/2dm=xdx;

Rewrite ∫xe(x^2)dx as ∫1/2emdu and solve as 1/2em+c which equivalently is 1/2e(x^2)+c; generally ∫qdp=pq-∫pdq; where p=∫dp;

For definite integrals ∫[a,b]qdp=[a,b]pq-∫[a,b]pdq; For ∫x(x+1)dx with m=x, dm=dx, dn=√(x+1)dx, v=2/3(x+1)3/2;

Taylor Series approximates a polynomial function with trigonometric sums to the nth degree:

Pn(x) =f(a) +f(1)(a)(x-a) +f(2)(a)(x-a)2/2! +f(3)(a)(x-a)3/3! +f (4)(a)(x-a)4/4! +... +f (n)(a)(x-a)n/n!

Parametric

Vectors

Representation of a line

Dot Product

Cross Product

Normal

Curvature

Partial Derivative

Lagrange Multiplier ∇f(x,y,z)=λ∇g(x,y,z); g(x,y,z)=k

Surface Area

Vector Field

Line Integral

Greene's Theorem

Stoke's Theorem

Linear Algebra

Kronecker delta

Basis

Eigenfunction

Field

Abstract Space

Differential

First Order

Root

Laplace Transform

System

Linear Homogeneous Equation

Fourier Series

Black-Scholes model ∂V/∂t+½S22V/∂S2+∂V/∂S-rV=0

Group Theory

At the beginning, we assumed addition and multiplication are commutative.

Permutation

Manifold

Cauchy's integral formula

Topology

Banach Tarski theorem

Exercises[edit]

  • Write what is 2+2=?
  • Give the distance of (4,3) from the origin.
  • Evaluate ∫sin3xdx
  • Solve dT/dt=-k(T-Ta) for Ta=20, T(0)=80, and T(2)=60
  • Let ∆⊂ℝ2 be a planar bounded closed convex set with nonempty interior U and prove that ∆ is homeomorphic to the closed disk D2

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