# Ithkuil/Mathematics

## Numbers

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. Use ultimate or monosyllabic stress for verbs. To save space, groups of ten are represented by a circle. Below are numbers are listed with Objective Specification (i).

Numbers attached to larger place values cannot be monosyllabic, as that would make them verbs.

gzil allal is 101, gzil aššal / gzil ars is 110, ksila gzalui / gzirt is 200, ksila (gzalui) llala / gzirt allal is 201, alẓilärs (agzalui) alẓilärs is 9999, apcal is 10000.

We use the Commitative u’ö for numbers greater than 100², 100⁴, or 100⁸, along with -iň for numbers greater than 100 multiplied by them.

cpilers acpalui ansilursu’ö (gzalui) cpilörs is 269766, llilärs gzalui apcal is 2100000, ksilors gzalui alẓilorsaň aẓkalui zalu’ö gzalui zalirsaň apcalui pšilersu’ö (gzalui) vralörs is 72793533460.

Got that? Great!

## Mathematics

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, however – oh boy – is the mediator. It is the intermediate between heaven and earth, the balancing leg of a stool, the strand of a braid, a knot tieable in only three spacial dimensions. It is the basis of this language, the stories we juggle, the divisions of time between past, present, and future; birth, life, and death; of existence between knowing, with the ability to know, of the known. Triangles have three sides and three angles and come in three types. The volumes of a cone, sphere, and drum are in a 1:2:3 ratio. The 3:1 and 3:2 ratios of musical tones represent fifth intervals, which I declare to be the most beautiful, but there is further complexity in numbers. The tetrad belongs to the domain of manifestation, the first thing born as a product of two twos. It begins the square numbers and defines the first polyhedron: the tetrahedron. It is associated with the elements phlegmatic water 🜄, sanguine fire 🜂, melancholic earth 🜃, and choleric air 🜁. The spring and autumn equinoxes and summer and winter solstices divide the year. The four segments of a cross define the compass directions and divide the square into four others. Everyday matter is composed of the three electr/prot/neutr-ons and the electron neutrino. The 4:3 musical ratio defines the fourth, the complement to the fifth in the octave. The pentad is the star shape a child instinctively draws. She paints the form of her hand's five digits or the pentagon whose angles match the bonds in water. It forms a liquid crystal lattice of icosahedra, one of the five Platonic solids, so it has the qualities of flow, dynamism, and life. It is also the fifth element, metal, quintessence, or qi. A rectangle of sides three and four has a diagonal of five. Fivefold symmetry is found in apples, flowers, the pattern of our neighbor Venus makes about Earth.The 5:4 ratio defines a major third in the modern musical scale, bute it also counts the notes in the universal pentatonic scale formed from a piano's black keys. The hexad is the perfect number, seen on the sides of a snowflake

Heptad is the standalone, the number of days in a week

Octad is the first cubic number, the nuber of electrons in a shell

Nonad is the first odd square number

Decad is the basis of the centesimal system, with us counting on ten digits

Undecad is the first number that helps us measure a circle

Duodecad is the first abundant number

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. For the longest time, the absence of a number was treated unseriously. 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 modifies 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 inverse: a negative number, which extends the natural numbers before zero. These types of numbers also took a long time before they were adopted. Together, these positive, negative and null numbers 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 invertable 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. Below are the degrees of the BAO affix -sd:

1 2 3 4 5 6 7 8 9
+ - × ÷ / -1 ± ² ³ ²√() ^()
= modulo inverse √() () !

This is assuming we are working with integers, of course. Division opens up the possibility of rational numbers which can be represented as a simple fraction or a centesimal fraction which can repeat periodically, as in the case of 1/3 or .3 . To convert 0.037 or 0.0135 to a fraction, take 1000x - x = 999x as 13.5135135135...-0.0135135135...=13.5 and say x=135/9990=5*9*3/9*3*74*5=1/74

Prime numbers can be discovered by the sieve of :eratosthenes:, Fermat Numbers, Mersenne Primes, Goldbach conjectures, Twin Primes, Perfect, Amicable, Irrational, Transcendental, 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

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=0.57721566490153286, 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,

¬

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

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.

Feigenbaums α=4.669201609 δ=2.50290787509589

Khinchin K=2.6854520010

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=6.2831853071795864769 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, c² is equal to the sum of the squares of the other two sides, a²+b². 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.

xm*xn=xm+n ; xm/xn=xm-n ; (xm)n=xmn ; (xy)m=xmym ; (x/y)n=xn/yn ; x-n=1/xn

(x/y)-n=(y/x)n ; x1=x ; x0=1 ; n√(x*y)=n√x*n√y ; n√(x/y)=n√x / n√y ; n√xm=xm/n

xlogₓz=z ; logₓ(xy)=y ; y√(xy)=x ; (y√z)y=z ; logₓz√z=x ; logy√z(z)=y ; zxzy=zx+y

logz(x)+logz(y) = logz(xy) ; 1/(1/logxz + 1/logyz)=logxyz ; x√zy√z=1/(1/x + 1/y)√z

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=[,,]

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

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 a, b, c and opposite sides a', b', c' (and spherically a", b", c"), 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(a) = sin(a+nτ) = cos(τ/4-a) and cos(a) = cos(a+nτ) = sin(τ/4-a) and tan(a+τ/8)=(1+tan(a))/(1-tan(a)). The tangent proportion graph repeats twice as much tan(a) = tan(a+n/2τ) = 1/tan(τ/4-a). 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(-a) = -sin(a) is odd and cos(-a) = cos(a) is even, so tan(-a) = -tan(a) is odd by definition of tan. Sine and cosine are related such that sin²(a) + cos²(a) = 1 so it follows that tan²(a) + 1 = 1/cos²(a) and 1 + 1/tan²(a) = 1/sin²(a).

(a'+b')/c' = cos((a-b)/2)/sin(c/2) and (a'-b')/c' = sin((a-b)/2)/cos(c/2)

radius of the circumcircle = a'/(2sin(a)) = b'/(2sin(b)) = c'/(2sin(c)), meaning that sin(a)/a = sin(b)/b' = sin(c)/c'

c'² = a'² + b'² -a'b'cos(c) or in spherical terms cos(a)=cos(b)cos(c)+sin(b)sin(c)cos(a)

(a'-b')/(a'+b') = tan((a-b)/2)/tan((a+b)/2)

For s = (a'+b'+c')/2, the radius of the inner circle = √((s-a')(s-b')(s-c')/s)

sin²a = 1/2*(1-cos²(a)) and cos²(a) = 1/2*(1+cos²(a)) so tan²(a) = (1-cos²(a))/(1+cos²(a))

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

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

sin³(a) = 3sin(a)cos²(a) - sin³(a) and cos³(a) = cos³(a) - 3sin²(a)cos(a) and tan³(a) = (3tan(a) - tan³(a)) / (1- 3tan²(a))

sin(a+b)*sin(a-b) = sin²(a) - sin²(b) = cos²(b) - cos²(a) and cos(a+b)*cos(a-b) = cos²(a) - sin²(b) = cos²(b) - sin²(a)

A natural way to define sin(a) is (eia - e-ia)/2i and similarly sinh(ia) = (eia - e-ia)/2, so sinh(ia)=isin(a) or sinh(a)=-isin(ia)

sinh(a) = sin(ia)*1/i = -sinh(-a) and cosh(v) = (eia + e-ia)/2 = cosh(-a) = cos(ia)

isina = (eia - e-ia)/2 and icosa = (eia + e-ia)/2 so itana = (eia - e-ia)/(eia + e-ia)

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

A catenary arch acosh(x/a)=a(ex/a + e-x/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

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

• 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