OpenLLM-France/wikimedia huggingface dataset:
# Numbers
## Nā Helu

* 0 ʻole
* 1 ʻekahi
* 2 ʻelua
* 3 ʻekolu
* 4 ʻehā
* 5 ʻelima
* 6 ʻeono
* 7 ʻehiku
* 8 ʻewalu
* 9 ʻeiwa
* 10 ʻumi
* 11 ʻumikūmākahi (10 + 1)
* 12 ʻumikūmālua (10 + 2)
* 13 ʻumikūmākolu (10 + 3)
    You may notice that you write the tens place number (ʻumi), then kūmā (the adding word), then the ones place number without the ʻe- prefix. This rule is always the same.
* 14 ʻumikūmāhā
* 15 ʻumikūmālima
* 16 ʻumikūmāono
* 17 ʻumikūmāhiku
* 18 ʻumikūmāwalu
* 19 ʻumikūmāiwa
* 20 iwakālua
    Again, same rule.
* 21 iwakāluakūmākahi (20 + 1)
* 22 iwakāluakūmālua (20 + 2)
* 23 iwakāluakūmākolu (20 + 3)
    Youʻve got the hang of it...
* 30 kanakolu
* 40 kanahā
* 50 kanalima
* 60 kanaono
* 70 kanahiku
* 80 kanawalu
* 90 kanaiwa
* 100 hoʻokahi hanele (or hoʻokahi haneli)
* 126 hoʻokahi hanele iwakāluakūmāono
* 173 hoʻokahi hanele kanahikukūmākolu
* 200 ʻelua hanele
* 300 ʻekolu hanele

# School Materials
## Nā 'Ikamu o Ke Kula

| Hawaiian | English |
| Ke Kula | School |
| Ke Kulanui | College |
| Ka Pākaukau | Desk |
| Ka Pepa | Paper |
| Ka Penikala | Pencil |
| Ka Hulu Kākau/Ka Peni | Pen |
| Ka Puke | Book |
| Ke Uaki | Clock |
| Ka Papa | Class |
| Ke Kumu | Teacher |
| Ka Haumana | Student |
# Hawaiian/Lesson 1
## Pāpāʻōlelo / Conversation

It is before school, and two students meet for the first time. They introduce themselves in Hawaiian.
| Ikaika: | Aloha kakahiaka! | Ikaika: | Good morning! |
| Kauʻi: | Aloha kakahiaka. ʻO wai kou inoa? | Kauʻi: | Good morning. What is your name? |
| Ikaika: | ʻO Ikaika koʻu inoa. ʻO wai kou inoa? | Ikaika: | My name is Ikaika. What is your name? |
| Kauʻi: | ʻO Kauʻi koʻu inoa. Pehea ʻoe? | Kauʻi: | My name is Kauʻi. How are you? |
| Ikaika: | Maikaʻi au. A ʻo ʻoe? | Ikaika: | I'm good. And you? |
| Kauʻi: | Maikaʻi loa au, mahalo. Aloha, e Ikaika! | Kauʻi: | I'm very good, thanks. Goodbye, Ikaika! |
| Ikaika: | Aloha a hui hou. | Ikaika: | Goodbye, until we meet again. |

## Greetings

| ʻŌlelo Pelekane | ʻŌlelo Hawaiʻi |
| Hello! | Aloha! |
| Good morning! | Aloha kakahiaka! |
| Good mid-day! | Aloha awakea! |
| Good afternoon! | Aloha ʻauinalā! |
| Good evening! | Aloha ahiahi! |
| Goodbye! | Aloha! |
| Until we meet again. | A hui hou. |

## Names
### What's your name?

To ask a name in Hawaiian, start with ʻO wai, add one of the words in the table below, and finish with inoa?. So, to ask "What is your name?", you should say "ʻO wai kou inoa?".
| ʻŌlelo Pelekane | ʻŌlelo Hawaiʻi |
| your | kou |
| my | koʻu |
| his/her | kona |
| our (2 people, inclusive) | kō kāua |
| our (3+ people, inclusive) | kō kākou |
| our (2 people, exclusive) | kō māua |
| our (3+ people, exclusive) | kō mākou |
| your (2 people) | kō ʻolua |
| your (3+ people) | kō ʻoukou |
| their (2 people) | kō lāua |
| their (3+ people) | kō lākou |
Don't worry if this looks like a lot to learn; we'll cover it in more detail in another lesson.
Kane ikaika Pu’u wai piha o kaloha

### My name is...

To give someone's name in Hawaiian, start with ʻO, add the name, add one of the phrases in the table for What's your name? section, and finish with inoa. Here are some examples:
| ʻŌlelo Pelekane | ʻŌlelo Hawaiʻi |
| My name is Kaniela. | ʻO Kaniela koʻu inoa. |
| Your name is Awapuhi. | ʻO Awapuhi kou inoa. |
| His/her name is Keanu. | ʻO Keanu kona inoa. |
| Our names (2 people) are Mele and Kapua. | ʻO Mele lāua ʻo Kapua kō kāua inoa. |
| Our names are Kawika, Leiʻala, and Malia. | ʻO Kawika, Leiʻala, a ʻo Malia kō kākou inoa. |
| Their names are Leimaile and Uʻilani. | ʻO Leimaile lāua ʻo Uʻilani kō lāua inoa. |
# Hawaiian/About Hawaiian

The Hawaiian language (Ka ‘ōlelo Hawai‘i) is a member of the Austronesian language family. This family is spread throughout the Pacific and Southeast Asia and stretches from Madagascar to Rapa Nui. The languages in this family are closely related to each other.
The language takes its name from the Hawai'i archipelago. Hawaiian, along with English, is an official language of the State of Hawaii (United States).
For various reasons, the number of native speakers of Hawaiian gradually decreased during the period from the 1830s to the 1950s. Hawaiian was essentially displaced by English on six of the seven inhabited islands. Nevertheless, from about 1949 to the present, there has been a gradual increase in attention to, and promotion of, the language. Public Hawaiian-language immersion pre-schools called Pūnana Leo were started in 1984; other immersion schools followed soon after. The first students to start in immersion pre-school have now graduated from college and many are fluent Hawaiian speakers.
As of 2000, native speakers of Hawaiian amount to under 0.1% of the statewide population. According to the recent U.S. Census, 27,160 speak the language at home.
Hawaiian grammar is relatively simple, and complex sentences can be created from combining simple rules. There are a couple of points to note, however, that make Hawaiian distinctively different in grammar from English:
* The common word order is Verb Subject Object (e.g. Hele au i ka hale, Go-I-to-the-house).
* Adjectives always follow nouns (e.g. ke kanaka pono, the-person-good).
* All nouns (even abstract) take an article (e.g. ke Akua, ka po'e, the God, the people)
# Famous Theorems of Mathematics/Algebra

Algebra is a branch of mathematics concerning the study of structure, relation and quantity. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.
Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.
Proofs are arranged in the following sections:

## Linear Algebra

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. Linear algebra is also intimately connected with matrix theory. Proofs in linear algebra are also sub-divided into these portions:
* Matrix Theory
* Vector Spaces
* Linear Transformations

## See also

* Algebra
* Linear Algebra
# Famous Theorems of Mathematics/Algebra/Linear Transformations
## Lemma for the eigenspace

All eigenvectors of the linear transformation A that correspond to the eigenvalue λ form a subspace L$^{(λ)}$ in L.

### Proof by Shilov (1969)

In fact, if Ax$_1$ = λx$_1$, and Ax$_2$ = λx$_2$, then
    A(αx$_1$ + βx$_2$) = αAx$_1$ + βAx$_2$ = αλx$_1$ + βλx$_2$ = λ (αx$_1$ + βx$_2$)
with which the statement in the lemma is proven.

## Lemma for linear independence of eigenvectors

Eigenvectors x$_1$, x$_2$, ... , x$_n$ of the (linear) transformation A with respective pairwise distinct eigenvalues λ$_1$, λ$_2$, ... , λ$_n$, are linearly independent.

### Proof by Shilov (1969)

This statement is proved by induction to number n. It is obvious that for n = 1 the lemma is true. Suppose that the lemma is true for all n – 1 eigenvalues of the transformation A; it remains to show that it is true for all n eigenvectors of the transformation A. Suppose a linear combination of n eigenvectors of the transformation A is 0:
    α$_1$x$_1$ + α$_2$x$_2$ + ... + α$_n$x$_n$ = 0.
Applying transformation A to this identity, one has
    α$_1$λ$_1$x$_1$ + α$_2$λ$_2$x$_2$ + ... + α$_n$λ$_n$x$_n$ = 0.
Multiply the first equation by λ$_n$ and subtract from the second one; one obtains
    α$_1$(λ$_1$ – λ$_n$)x$_1$ + α$_2$(λ$_2$ – λ$_n$)x$_2$ + ... + α$_{n–1}$(λ$_{n–1}$ – λ$_n$)x$_{n–1}$ = 0,
from where by induction all coefficients must be zero. Distinct eigenvalues have nonzero difference, so each α$_i$ = 0 for i < n; the first equation reduces to
    α$_n$x$_n$ = 0
which means α$_n$ = 0, too. Consequently, all coefficients α$_i$ are 0. Therefore, the vectors x$_1$, x$_2$, ..., x$_n$ are linearly independent.
# Famous Theorems of Mathematics/Applied Mathematics
## -div is adjoint to d

The claim is made that −div is adjoint to d:
    $\int _{M}df(X)\;\omega =-\int _{M}f\,\operatorname {div} X\;\omega$
Proof of the above statement:
    $\int _{M}(f\mathrm {div} (X)+X(f))\omega =\int _{M}(f{\mathcal {L}}_{X}+{\mathcal {L}}_{X}(f))\omega$
       $=\int _{M}{\mathcal {L}}_{X}f\omega =\int _{M}\mathrm {d} \iota _{X}f\omega =\int _{\partial M}\iota _{X}f\omega$
If f has compact support, then the last integral vanishes, and we have the desired result.

## Laplace-de Rham operator

One may prove that the Laplace-de Rahm operator is equivalent to the definition of the Laplace-Beltrami operator, when acting on a scalar function f. This proof reads as:
    $\Delta f=\mathrm {d} \delta f+\delta \,\mathrm {d} f=\delta \,\mathrm {d} f=\delta \,\partial _{i}f\,\mathrm {d} x^{i}$
       $=-*\mathrm {d} {*\partial _{i}f\,\mathrm {d} x^{i}}=-*\mathrm {d} (\varepsilon _{iJ}{\sqrt {|g|}}\partial ^{i}f\,\mathrm {d} x^{J})$
       $=-*\varepsilon _{iJ}\,\partial _{j}({\sqrt {|g|}}\partial ^{i}f)\,\mathrm {d} x^{j}\,\mathrm {d} x^{J}=-*{\frac {1}{\sqrt {|g|}}}\,\partial _{i}({\sqrt {|g|}}\,\partial ^{i}f)\mathrm {vol} _{n}$
       $=-{\frac {1}{\sqrt {|g|}}}\,\partial _{i}({\sqrt {|g|}}\,\partial ^{i}f),$
where ω is the volume form and ε is the completely antisymmetric Levi-Civita symbol. Note that in the above, the italic lower-case index i is a single index, whereas the upper-case Roman J stands for all of the remaining (n-1) indices. Notice that the Laplace-de Rham operator is actually minus the Laplace-Beltrami operator; this minus sign follows from the conventional definition of the properties of the codifferential. Unfortunately, Δ is used to denote both; reader beware.

## Properties

Given scalar functions f and h, and a real number a, the Laplacian has the property:
    $\Delta (fh)=f\,\Delta h+2\partial _{i}f\,\partial ^{i}h+h\,\Delta f.$

### Proof

    $\Delta (fh)=\delta \,\mathrm {d} fh=\delta (f\,\mathrm {d} h+h\,\mathrm {d} f)=*\mathrm {d} (f{*\mathrm {d} h})+*\mathrm {d} (h{*\mathrm {d} f})\;$
          $=*(f\,\mathrm {d} *\mathrm {d} h+\mathrm {d} f\wedge *\mathrm {d} h+\mathrm {d} h\wedge *\mathrm {d} f+h\,\mathrm {d} *\mathrm {d} f)$
          $=f*\mathrm {d} *\mathrm {d} h+*(\mathrm {d} f\wedge *\mathrm {d} h+\mathrm {d} h\wedge *\mathrm {d} f)+h*\mathrm {d} *\mathrm {d} f$
          $=f\,\Delta h$
             $+*(\partial _{i}f\,\mathrm {d} x^{i}\wedge \varepsilon _{jJ}{\sqrt {|g|}}\partial ^{j}h\,\mathrm {d} x^{J}+\partial _{i}h\,\mathrm {d} x^{i}\wedge \varepsilon _{jJ}{\sqrt {|g|}}\partial ^{j}f\,\mathrm {d} x^{J})$
             $+h\,\Delta f$
          $=f\,\Delta h+(\partial _{i}f\,\partial ^{i}h+\partial _{i}h\,\partial ^{i}f){*\mathrm {vol} _{n}}+h\,\Delta f$
          $=f\,\Delta h+2\partial _{i}f\,\partial ^{i}h+h\,\Delta f$
where f and h are scalar functions.
# Famous Theorems of Mathematics/Euclid's proof of the infinitude of primes

The Greek mathematician Euclid gave the following elegant proof that there are an infinite number of primes. It relies on the fact that all non-prime numbers --- composites --- have a factorization into primes.
Euclid's proof shows that for any finite set S of prime numbers, one can find a prime not belonging to that set. (Contrary to what is asserted in many books, this need not be the first n prime numbers for some n, nor did Euclid assume it to be the set of all prime numbers. For example, the finite set could be { 2, 7, 31 }.)
Euclid considered the number ΠS, the result of multiplying all members of the finite set S. (For example, if S is { 2, 7, 31 } then ΠS is 2 × 7 × 31 = 434.) Then he added 1 to this number, getting 1 + ΠS. (For example, if S is { 2, 7, 31 } then 1 + ΠS is 1 + (2 × 7 × 31) = 435.) Euclid claimed that this number 1 + ΠS cannot be divisible by any of the primes in the finite set S that we started with. (For example, if S is { 2, 7, 31 } then the number 1 + ΠS, which is 1 + (2 × 7 × 31) = 435, is not divisible by 2, 7, or 31; in fact it is 435 = 3 × 5 × 29.) But 1 + ΠS, like every number, must either be prime itself or be divisible by some prime number other than itself. Either way we have a prime number that is not in our initial finite set S. (For example, if S is { 2, 7, 31 } then the resulting new prime numbers not in S are 3, 5, and 29.)
In Euclid's phrasing of the proof, rather than multiplying all of the primes in S, the smallest common multiple was considered. But the smallest common multiple of distinct prime numbers is the same as their product.
Many books erroneously state that Euclid's proof was by contradiction, starting with the assumption that only finitely many prime numbers exist.

## References

* ↑ Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52.
# Famous Theorems of Mathematics/Analysis

Analysis has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function. It also includes the theories of differentiation, integration and measure, infinite series, and analytic functions. These theories are often studied in the context of real numbers, complex numbers, and real and complex functions. However, they can also be defined and studied in any space of mathematical objects that is equipped with a definition of "nearness" (a topological space) or more specifically "distance" (a metric space).
Proofs are arranged in the following sections:
* Metric Spaces
* Real Analysis
* Functional Analysis
* Harmonic Analysis
* Complex Analysis
* Differential Geometry
* Topology
* p-adic Analysis
* Non-standard Analysis or model theory
* Numerical Analysis
Most of the standard proofs of analysis studied at the undergraduate level are in the metric spaces section. This is so because proofs of such results in real analysis, complex analysis and even in topology are similar to them.

## See also

* Topology
* Real Analysis
* Functional Analysis
* Complex Analysis
* Differential Geometry
* Numerical Methods
# Famous Theorems of Mathematics

Not all of mathematics deals with proofs, as mathematics involves a rich range of human experience, including ideas, problems, patterns, mistakes and corrections. However, proofs are a very big part of modern mathematics, and today, it is generally considered that whatever statement, remark, result etc. one uses in mathematics, it is considered meaningless until is accompanied by a rigorous mathematical proof. This book is intended to contain the proofs (or sketches of proofs) of many famous theorems in mathematics in no particular order. It should be used both as a learning resource, a good practice for acquiring the skill for writing your own proofs is to study the existing ones, and for general references.
It is not however intended as a companion to any other wikibook or wikipedia articles but can complement them by providing them with links to the proofs of the theorems they contain.
One note here. There are usually many ways to solve a problem. Many times the proof used comes down to the primary definitions of terms involved. We will follow the definition given by the first major contributor.

## Table of contents
#### High school

* Pythagoras theorem
* Euclid's proof of the infinitude of primes
* √2 is irrational
* sin$^2$Θ+cos$^2$Θ=1

#### Undergraduate

* e is irrational
* π is irrational
* Fermat's little theorem
* Fermat's theorem on sums of two squares
* Sum of the reciprocals of the primes diverges
* Bertrand's postulate
* Law of large numbers
* Spectral Theorem
* L'Hôpital's rule
* Four color theorem
* e$^{πi}$+1=0
* Gaussian integral
* Gamma function
* Zeta function

#### Postgraduate

* Fermat's last theorem
* Brouwer fixed-point theorem
* Prime number theorem
* Riemann hypothesis

## Old table of contents

Proofs and definitions will be arranged according to the fields of mathematics:
* Algebra
* Analysis
* Applied Mathematics
* Geometry
* Logic
* Number Theory
* Set Theory
* Boy's surface

## Further reading

* Mathematical Proof - about the theory and techniques of proving mathematical theorems

## Resources
### Manual of style

* Proof style - Style guide for proofs.

### External links etc

* Proofs on the wikipedia project
* List of mathematical proofs
* List of theorems
* Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review
* The Hundred Greatest Theorems
* ProofWiki.org - The online compendium of mathematical proofs
# Final Fantasy/Gauntlets

Gauntlets are armored gloves worn by warriors to protect their hands in battle. Each adventurer may carry up to four pieces of armor (beware, this includes armor, helmets, and shields as well) but may only equip one piece of body armor.

## Gloves

* Gloves (Leather Gloves)
# Final Fantasy/Gauntlets/Gloves

| Armor Type | Gloves (Gauntlets) |
| Bought At | Pravoka |
| Price | 60 G |
| Absorb Bonus | 1 |
| Evade % Bonus | 1 |
Gloves are simple gauntlets made of leather. Any hero is able to wear them, but they provide minimal protection from attack. They do not interfere with the gestures mages need to cast spells.
Gloves may be worn by:
* Fighter / Knight
* Thief / Ninja
* Black Belt / Master
* Red Mage / Red Wizard
* White Mage / White Wizard
* Black Mage / Black Wizard
# Final Fantasy/Helmets

Helmets are considered by many heroes to be necessary to survival when adventuring, as they protect the warrior's head from attacks. Most characters cannot wear stronger helmets than the Cap, though Fighters and Ninja have access to three helmets and the Knight to five. Each adventurer may carry up to four pieces of armor (beware, this includes armor, shields, and gauntlets as well) but may only equip one helmet.

## Headgear

* Cap (Leather Cap)

## Helmets

* Wooden Helmet (Helmet)

## Heavy Helmets

In Final Fantasy Origins, Helmet refers specifically to the Wooden Helmet.
# Final Fantasy/Inns

Inns are places in which heroes are able to rest and recover from their adventures. Any adventurer who stays at an inn will heal all of his wounds as well as getting a good night's sleep. Magical powers will also be fully restored. The game can be saved after staying at an Inn.
Later in the game, after obtaining the airship, it is best to save money by visiting the inn at Coneria.
Inn Locations and Prices:
| City | Price |
| Coneria | 30 G |
| Pravoka | 50 G |
| Elfland | 100 G |
# Final Fantasy/Item Shops

Item stores are convenient locations for heroes to stock up on restorative items that can assist them in surviving the trials of the wilderness. Some of the item stores have very limited selections, while others will sell nearly all of the items.
The item stores and their wares are:
* Coneria
** Heal Potion (Potion)
** Pure Potion (Antidote)
** Tent (Sleeping Bag)
* Pravoka
** Heal Potion (Potion)
** Pure Potion (Antidote)
** Tent (Sleeping Bag)
** Cabin (Tent)
* Elfland
** Heal Potion (Potion)
** Pure Potion (Antidote)
** Cabin (Tent)
** House (Cottage)
** Soft Potion (Gold Needle)
# Final Fantasy/Healer

Healers are locations at which fallen heroes may be revived. If a hero's health drops to 0, that hero will die. There are two ways of reviving the dead hero: a powerful mage can use a Life spell or the party can visit a healer.
Healer Locations and Prices:
| City | Price |
| Coneria | 40 G |
| Pravoka | 80 G |
| Elfland | 200 G |
# Final Fantasy/Helmets/Cap

| Armor Type | Helmet (Headgear) |
| Bought At | Elfland |
| Price | 80 G |
| Absorb Bonus | 1 |
| Evade % Bonus | 1 |
The Cap is a light helmet that most characters can use to protect their heads. Caps are usually made of leather or cloth and provide minimal protection. However, most heroes are capable of wearing them and so they are a common choice of headgear until magical ribbons can be found and equipped.
Caps may be found in:
Caps may be worn by:
* Fighter / Knight
* Thief / Ninja
* Red Mage / Red Wizard
* Black Mage / Black Wizard
* White Mage / White Wizard
Final Fantasy Origins: Leather Cap
Return to Helmet Return to Final Fantasy
# Final Fantasy/Helmets/Wooden Helmet

| Armor Type | Helmet |
| Bought at | Elfland |
| Price | 100 G |
| Absorb Bonus | 3 |
| Evade % Bonus | 3 |
The Wooden Helmet is a simple piece of protective headgear made of wood. It protects a hero's head from enemy attacks. However, wood is not the optimal material for such protection and the Wooden Helmet does not provide much protection. They are also heavy and have a tendency to impede vision. As such, they are an unpopular piece of armor, especially for spellcasting adventurers.
Wooden Helmets may be worn by:
* Fighter / Knight
* Ninja
Final Fantasy Origins: Helmet
# Final Fantasy/Intelligence

Intelligence is the statistic that determines the effectiveness of magical attacks. All forms of mages and wizards have high Intelligence scores, with the Black Mage and Black Wizard possessing exceptionally great quantities of Intelligence. The Fighter and, to a lesser extent the Knight, has a terrible score.
It is widely believed that, due to a design mistake, Intelligence score has no actual effect on gameplay.
# Historical Rhetorics/Cicero's Public and the Greek Tradition/Meador, Prentice A. "Rhetoric and Humanism in Cicero." ''Philosophy & Rhetoric'' 3 (1970): 1–12.

Meador, Prentice A. "Rhetoric and Humanism in Cicero." Philosophy & Rhetoric 3 (1970): 1–12.
Meador begins with an explanation of the most common criticism of Cicero—that he “presents no original views in his treatises” (1)—and seeks immediately to refute this claim, asserting that Cicero is a humanist and a proponent of rhetoric. It is Cicero’s belief, according to Meador, that man is incapable of attaining “perfect knowledge” (3), because all knowledge is based on sensory perception and “perception [cannot] yield knowledge…[because] subjective limitations prevent reliable receptibility”(4). He believes, therefore, that rhetoric is “situationally oriented and founded upon probability” (9). However, even in the absence of absolute knowledge, it is possible for men to behave morally because it is in their nature to be virtuous since, although “[e]ach man will by nature look after his own interest” he is also “dependent by nature on others around him” (5) thereby making it in man’s nature to act for the good of the larger community. He builds on this idea when he defines the idea of a “universal law” as that in which the principle factor is “respect for the common bond of mankind” (7), a respect which he defines as instinctual (part of our nature). And seeking the cooperation of a community requires the use of rhetoric, hence the link between humanism and rhetoric.