OpenLLM-France/wikimedia huggingface dataset:
# Music Theory/Indian/Madhyama

Madhyama/Madhyam is the 4th svara from the 7 svaras of Hindustani music and Carnatic music. Madhyama is the long form of the syllable म. For simplicity in pronouncing while singing the syllable, Madhyama is pronounced as Ma (notation - M). It is also called as मध्यम.

## More information and importance of Madhyama

The following is the information about Madhyama and its importance in Indian classical music:
* Madhyama is the fourth svara in an octave or Saptak.
* Ma is the immediate next svara of Gāndhāra (Ga).
* The svara of Madhyama is Tivra and Shuddha. In fact Madhyama is the only Tivra svara in the Saptak.
* It is said that Sadja is the basic svara from which all the other 6 svaras are produced. When we break the word Sadja then we get, Shad And Ja. It means that Shad is 6 and ja is 'giving birth' in Marathi. So basically the translation is :
So the svara Ma is formed from Sadja.
* The frequency of Madhyama is 320 Hz. The frequencies of the 7 svaras are also given below: Sa 240 Hz, Re 270 Hz, Ga 300 Hz, Ma 320 Hz, Pa 360 Hz, Dha 400 Hz, and Ni 450 Hz, Sa 480 Hz (Taar Saptak) ........ (and so on).
Consequently, the Ma after the Ga of 600 Hz (Taar Saptak) has a frequency of 640 Hz i.e. the double of the Lower octave Ma.
* There are 4 Shruti's of Madhyama. Previously the main Shruti not only for Ma but for all the other svaras was on the last Shruti but now it is considered to be on the 1st Shurti.
For example if these are the 4 Shruti's of Ma then,
* All the other svaras except Sadja(Sa) and Pancham (Pa) can be Komal or Tivra svaras but Sa and Pa are always Shuddha svaras. And hence svaras Sa and Pa are called Achal Svaras , since these svaras don't move from their original position. Svaras Ra, Ga, Ma, Dha, Ni are called Chal Svaras, since these svaras move from their original position.
Ma is a Shuddha Svara as well as a Tivra Svara. But Ma is the only Tivra Svara. This is denoted as म॓ .
* Ragas from Kalyan Thaat, Poorvi Thaat, Marwa Thaat, Todi Thaat have Tivra Madhyama, rest of the thaats have Shuddha Madhyama.
* Ragas where Ma is the Vadi svara - Raga Kedar, etc. Ragas where Ma is the Samvadi svara - Raga Malkauns.
* Hypothetically speaking, Ma is said to be the Mahipal - Lord Indra, Mahipal as in, the three main gods, Bhrama, Vishnu and Shiva were first created i.e. Sakar Bhrama (Sa) and then these three gods created Rishimuni i.e. Re and then Gandharvas were created for singing and then lord Indra or Raja Indra i.e. Mahipal was created. Ma is made the acronym of Mahipal for showing the importance of the syllable Ma.
* Madhyama is said to be sourced from call of the heron.
* Madhyama is associated with the planet Moon.
* Madhyama is associated with White colour.
# Music Theory/Indian/Ṛiṣabha

Ṛiṣabha/Risabha/Rishabh is the 1st Svara from the 7 svaras of Hindustani music and Carnatic music. Risabha is the long form of the syllables रे and री. For simplicity in pronouncing while singing the syllable, Risabha is pronounced as Re and Ri (notation - R). It is also called as ऋषभ.

## More information and importance of Risabha

Following are the information and importance of Risabha in the Indian classical music :
* Risabha is the second svara in an octave or Saptak.
* Re is the immediate next svara of Sadja (Sa).
* The svara of Risabha is Komal and Shuddha.
* It is said that Sadja is the basic svara from which all the other 6 svaras are produced. When we break the word Sadja then we get, Shad And Ja. It means that Shad is 6 and ja is 'giving birth' in marathi. So basically the translation is :
So the svara Re is formed from Ṣaḍja.
* The frequency of Risabha is 270 Hz. The frequencies of the 7 svaras are also given below: Sa 240 Hz, Re 270 Hz, Ga 300 Hz, Ma 320 Hz, Pa 360 Hz, Dha 400 Hz, and Ni 450 Hz, Sa 480 Hz (Taar Saptak) ........ (and so on).
Consequently, the Re after the Sa of 480 Hz (Taar Saptak) has a frequency of 540 Hz i.e. the double of the Lower octave Re.
* There are 3 Shruti's of Risabha. Previously the main Shruti, not only for Re but for all the other svaras, was on the last Shruti but now it is considered to be on the 1st Shrurti.
For example, if these are the 3 Shruti's of Re then,
* All the other svaras except Sadja(Sa) and Pancham (Pa) can be Komal or Tivra svaras but Sa and Pa are always Shuddha svaras. And hence svaras Sa and Pa are called Achal Svaras , since these svaras don't move from their original position. Svaras Ra, Ga, Ma, Dha, Ni are called Chal Svaras, since these svaras move from their original position.
* Ragas from Bhairav That, Poorvi That, Marwa That, Bhairavi That and Todi That, have Komal Risabha.
* Ragas where Re is the Vadi svara - Raga Brindabani Sarang, etc. Ragas where Re is the Samvadi svara - Raga Bhairav, etc.
* Hypothetically speaking, Re is said to be the Rishimuni, Rishimuni as in, the three main gods, Bhrama, Vishnu and Shiva were first created i.e. Sarkar (Sa) and then these three gods created Rishimuni i.e. Re. Re is made the acronym of Rishimuni for showing the importance of the syllable Re.
* Risabha is said to be sourced from the lowing of a bull.

## Difference in pronouncing

In the introductory part of Risabha it is given that, 'Risabha is the long form of the syllables रे and री. For simplicity in pronouncing while singing the syllable, Risabha is pronounced as Re and Ri'. This given because the pronouncing of the syllable Re is different in the Hindustani classical music and Carnatic classical music. In Hindustani classical music, the pronouncing of Risabha is done as Re and in Carnatic classical music, the pronouncing of Risabha is done as Ri.
So a Hindustani classical singer will sing the 7 svaras as : Sa Re Ga Ma Pa Dha Ni Sa.
While a Carnatic classical singer will sing these svaras like : Sa Ri Ga Ma Pa Dha Ni Sa.
# Music Theory/Indian/Ṣaḍja

Ṣaḍja/Sadja/Shadja is the 1st Svara from the 7 svaras of Hindustani music and Carnatic music. Sadja is the long form of the syllable सा. For simplicity in pronouncing while singing the syllable, Sadja is pronounced as Sa (notation - S). It is also called as षड्ज.

## More information and importance of Sadja

Following are the information and importance of Sadja in the Indian classical music :
* Sadja is the first svara in an octave or Saptak.
* Sadja is also the first Svara of a Raga. ( Not always but mostly it is the first svara.)
* The svara that is played on the Tanpura for the singer is Sadja. It is played to know that one is singing on the right pitch and octave.
* Shadja is the base or basic svara.
* It is fascinating to know that after one raga is sung or played, and when one more raga is to be performed, then the svara Sa is played so that there is no confusion in mixing of svaras in both the ragas that are performed. (Specifically in ragas of the same svaras but different Chalan or movement.)
* The svara of Sadja is never Komal or Tivra, it is always Shuddha in any given raga.
* It is said that Sadja is the basic svara from which all the other 6 svaras are produced. When we break the word Sadja then we get , Shad And Ja. It means that Shad is 6 and ja is 'giving birth' in marathi. So basically the translation is :
* The frequency of Shadja is 240 Hz. The frequencies of the 7 svaras are also given below: Sa 240 Hz, Re 270 Hz, Ga 300 Hz, Ma 320 Hz, Pa 360 Hz, Dha 400 Hz, and Ni 450 Hz, Sa 480 Hz (Taar Saptak) ........ (and so on).
Consequently the Sa after the Ni of 450Hz has a frequency of 480 Hz i.e. the double of the Lower octave Sa.
* There are 4 Shruti's of Sadja. Previously the main Shruti not only for Sa but for all the other svaras was on the last Shruti but now it is considered to be on the 1st Shurti.
For example if these are the 4 Shruti's of Sa then,
* All the other svaras except Sadja(Sa) and Pancham (Pa) can be Komal or Tivra svaras but Sa and Pa are always Shuddha svaras. And hence svaras Sa and Pa are called Achal Svaras , since these svaras don't move from their original position. Svaras Ra, Ga, Ma, Dha, Ni are called Chal Svaras, since these svaras move from their original position.
* It will be fascinating for you to know that Sa is the only svara which cannot be omitted in a particular raga. So Sa cannot be Varjit in any of the ragas. Hence Sa is the most important svara in the Saptak or in the Indian classical music.
* For some ragas where Sa is the Vadi or Samvadi svara, Sa is played repeatedly. But for ragas where Sa is non, Vadi or the Samvadi svara, it is still played repeatedly since it the main svara.
* Hypothetically speaking, Sadja is said to be the Sarkar, Sarkar as in the three main gods, Bhrama, Vishnu and Shiva. Sa is made the acronym of Sarkar for showing the importance of the syllable Sa.
* Sadja is said to be sourced from the cry of a peacock.
# Pinyin/Spirulina

Search Pinyin spirulina
# Pinyin/Spiritual

See also
# Pinyin/Spitted In

See also
# Pinyin/Spiritual Window

Spiritual window
Open spiritual window, let self learn to relax.
Lift the head to look at the outside world, let the spirit of self fly.
Open the spiritual window, follow the footsteps of others.
Look at the outside world, perhaps it would be better.
Open the spiritual window, walk through the window of others.
You will discover, their world would be more beautiful.
Xīnlíng zhī chuāng
Dǎkāi xīnlíng de chuāng, ràng zìjǐ xuéhuì fàngsōng.
Táitóu kànkàn wàimiàn de shìjiè, fàngfēi zìjǐ de xīnlíng.
Dǎkāi xīnlíng de chuāng, gēnsuí biérén de jiǎobù.
Kànkàn wàimiàn de shìjiè, yěxǔ huì gèng měi.
Dǎkāi xīnlíng de chuāng, zǒurù biérén de chuāng.
Nǐ huì fāxiàn, tāmen de shìjiè gèng měilì.
# Pinyin/Spleen

The spleen is an organ in vertebrates. In people, it is on the left side of the body, under the heart. The spleen helps fight infections and keep the blood cells healthy.
Pízàng shì jǐzhudòngwù de qìguān. Rén de pízàng zài tǐnèi zuǒcè, xīnzàng xiàmiàn. Pízàng bāngzhù duìkàng gǎnrǎn hé bǎochí xuèxìbāo de jiànkāng.
# Pinyin/Spoken Language

Search Pinyin language
# Pinyin/Spokesman

See also
# Pinyin/Spring

See also
Spring
Spring is like birds, fly freely.
Spring is like flowers, open beautifully.
Spring is like a brook, flows quickly.
Spring is like a rainbow, full of colors.
Chūntiān
Chūntiān xiàng xiǎoniǎo, fēi dé zìyóu.
Chūntiān xiàng xiǎohuā, kāi dé piàoliang.
Chūntiān xiàng xiǎoxī, liú dé fēikuài.
Chūntiān xiàng cǎihóng, chōngmǎn sècǎi.
# Passover Haggadah/Kiddush

Kadeish is Hebrew Imperative for Kiddush. This Kiddush is a special one for Passover, it refers to matzot and the Exodus from Egypt. Acting in a way that shows freedom and majesty, most Jews have the custom of filling each other's cups at the Seder table. The Kiddush is normally said by the father of the house.
The first cup is known as the cup of Sanctification, taken from the first "I Will" of Exodus 6:6-7: "Wherefore say unto the children of Israel: I am the LORD, and I will bring you out from under the burdens of the Egyptians" (JPS)
(Fill your cups)
Prayer:
(Drink the 1st cup).
# Passover Haggadah/Shulchan Orech

The meal is served.
(Literally, "The table is arranged".)
It is customary for the leader to start the meal by eating the egg from the seder plate, dipped in the salt water. Others present may also be given eggs to eat.
# Past LSAT Explained/February 2004 LSAT

The February 2004 LSAT

## Summary

As always, test forms for the February 2004 LSAT included an experimental (unscored) section. On most test forms, the experimental section appeared in Section 3. Be aware, however, that LSAC often administers a few selected forms with identical scored sections, but with their experimental sections in a different location. Ratings of this administration placed its difficulty on par with other recent exams. Princeton Review students report being pleased with their overall performance.

## Games (Analytical Reasoning)

Princeton Review students rated the difficulty of this section as similar to or slightly harder than games sections that have appeared on exams within the past year. The first game involved determining how many computers each of seven manufacturers had produced. In the second game, birds had to be selected for a sanctuary. The third game involved determining the order in which five of six colored pencils were used. In the last game, four houses of varying distance from a university had to be assigned to students or faculty for one-, two-, or three- semester leases. Test takers rated the first game as easiest, the second and third games as medium, and the fourth game as hardest. Our students found that their familiarity with the games and experience with making deductions helped them set up the games and work the questions efficiently.

## Arguments (Logical Reasoning)

One scored section had 26 questions, and the other had 25 questions. As usual, assumption, flaw, and inference questions were the most common question types. Test takers cited aggressive use of process of elimination (POE) as the most helpful technique. Overall, Princeton Review students reported being pleased with the arguments sections they saw, and they found the techniques they had learned in class worked well on the exam. Broad experience with past exams was invaluable to test-takers, and smart pacing choices were required to get the best possible results.

## Reading Comprehension

The Reading Comp section offered the usual breakdown of topics-science, law, social science, and arts/humanities. Examinees varied in their assessment of passage difficulty, but many said the social science and legal passages had the most difficult questions. The first passage had seven questions and argued for a reassessment of Nigerian writer Amos Toutala's work. The second passage had six questions, and it discussed two theories about the evolutionary advantage of kin recognition in animals. The third passage addressed the definition and division of personal property in divorce proceedings. Seven questions accompanied the last passage, which dealt with female education in France. Overall, this section demanded the skills of mapping the passage well, paraphrasing its contents, searching for specific evidence in support of answers, and eliminating choices judiciously.
# Passover Haggadah

The Haggadah (IPA: [həˈgɑdə]) (Hebrew: הגדה) contains the order of the Passover Seder held at home on the first night (and, outside Israel, also on the second night) of Passover. Haggadah, meaning "telling," is a fulfillment of the scriptural commandment to each Jew to "tell your son" about the liberation of the Israelites from slavery in Egypt, as described in the book of Exodus in the Torah. This wikibook is an English version of the Haggadah to assist in performing a Passover seder at home.

## Table of contents

* Kadesh (blessings and the first cup of wine)
* Ur'chatz (Hand washing)
* Karpas (appetizer)
* Yachatz (breaking of the middle matzah)
* Maggid (The telling)
** Ha Lachma Anya (invitation to the Seder)
** Mah Nishtanah (The Four Questions)
** The Four Sons
** "Go and learn"
** Kos Sheini (Second Cup of Wine)
* Rachtzah (ritual washing of hands)
* Motzi Matzah (blessings over the matzo)
* Maror (bitter herb)
* Korech (sandwich)
* Shulchan Orech (the meal [literally, "set table"])
* Tzafun (eating of the afikoman)
* Barech (Grace after Meals)
** Kos Shlishi (the Third Cup of Wine)
** Kos shel Eliyahu ha-Navi (cup of Elijah the Prophet)
* Hallel (songs of praise)
* Nirtzah (next year)
# Passover Haggadah/Ur'chatz

Everyone washes their hands from a vessel. Rings, watches and bracelets should be removed so that the whole hand including the wrist is washed. The water is poured over the hands. There are different customs; one is to pour water one hand thrice, and the other hand thrice again. The prayer for the washing of the hands is not said.
There is a custom for two young children to wash the hands of those seated around the table by bringing the water to them.
# Pashto/Salaam - سلام - Your first Pashto conversation

| You | Salaam |
| Your new friend | Salaam. Tsenga yee? |
| You | Za kha yam. Tsenga yee? |
| Friend | Za kha yam, manana. |

## Vocabulary

| Salaam | سلام | Hi |
| Tsenga yee? | څنگه يې؟ | How are you? |
| Za kha yam. | زه ښه يم | I am fine. |
| Manana | مننه | Thank you. |
# Passover Haggadah/Kadesh

A full cup of wine is lifted and the following is said:
(On Shabbat Sephardim add:)
יום הששי ויכלו השמים והארץ וכל ­צבאם ויכל אלהים ביום השביעי מלאכתו אשר עשה וישבת ביום השביעי מכל מלאכתו אשר עשה ויברך אלהים את יום השביעי ויקדש אתו כי בו שבת מכל מלאכתו אשר ברא אלהים לעשות
(On Shabbat Ashkenazim add:)
ויהי ערב ויהי בקר יום הששי ויכלו השמים והארץ וכל צבאם ויכל אלהים ביום השביעי מלאכתו אשר עשה וישבת ביום השביעי מכל מלאכתו אשר עשה ויברך אלהים את יומ השביעי ויקדש אתו כי בו שבת מכל מלאכתו אשר ברא אלהים לעשות
םברי מרנן
(Ashkenazim instead say)
םברי מרנן ורבנן ורבותי
On all other nights, begin here:
ברוך אתה יהוה אלהינו מלך העולם בורא פרי הגפן
# Notation

$\mathbb {N} \,$ is the set of positive integers, also called the natural numbers.
$\mathbb {Z} \,$ is the set of all integers.
$\mathbb {Q} \,$ is the set of all rational numbers.
$\mathbb {A} \,$ is the set of algebraic numbers.
$\mathbb {R} \,$ is the set of real numbers.
$\mathbb {C} \,$ is the set of complex numbers.
# Number Theory/Powers modulo 11

The following table shows the powers of integers modulo 11.
| x | x$^2$ | x$^3$ | x$^4$ | x$^5$ | x$^6$ | x$^7$ | x$^8$ | x$^9$ | x$^{10}$ |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 8 | 5 | 10 | 9 | 7 | 3 | 6 | 1 |
| 3 | 9 | 5 | 4 | 1 | 3 | 9 | 5 | 4 | 1 |
| 4 | 5 | 9 | 3 | 1 | 4 | 5 | 9 | 3 | 1 |
| 5 | 3 | 4 | 9 | 1 | 5 | 3 | 4 | 9 | 1 |
| 6 | 3 | 7 | 9 | 10 | 5 | 8 | 4 | 2 | 1 |
| 7 | 5 | 2 | 3 | 10 | 4 | 6 | 9 | 8 | 1 |
| 8 | 9 | 6 | 4 | 10 | 3 | 2 | 5 | 7 | 1 |
| 9 | 4 | 3 | 5 | 1 | 9 | 4 | 3 | 5 | 1 |
| 10 | 1 | 10 | 1 | 10 | 1 | 10 | 1 | 10 | 1 |
# Number Theory/Powers modulo 19

The following table shows the powers of integers modulo 19.
| x | x$^2$ | x$^3$ | x$^4$ | x$^5$ | x$^6$ | x$^7$ | x$^8$ | x$^9$ | x$^{10}$ | x$^{11}$ | x$^{12}$ | x$^{13}$ | x$^{14}$ | x$^{15}$ | x$^{16}$ | x$^{17}$ | x$^{18}$ |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 8 | 16 | 13 | 7 | 14 | 9 | 18 | 17 | 15 | 11 | 3 | 6 | 12 | 5 | 10 | 1 |
| 3 | 9 | 8 | 5 | 15 | 7 | 2 | 6 | 18 | 16 | 10 | 11 | 14 | 4 | 12 | 17 | 13 | 1 |
| 4 | 16 | 7 | 9 | 17 | 11 | 6 | 5 | 1 | 4 | 16 | 7 | 9 | 17 | 11 | 6 | 5 | 1 |
| 5 | 6 | 11 | 17 | 9 | 7 | 16 | 4 | 1 | 5 | 6 | 11 | 17 | 9 | 7 | 16 | 4 | 1 |
| 6 | 17 | 7 | 4 | 5 | 11 | 9 | 16 | 1 | 6 | 17 | 7 | 4 | 5 | 11 | 9 | 16 | 1 |
| 7 | 11 | 1 | 7 | 11 | 1 | 7 | 11 | 1 | 7 | 11 | 1 | 7 | 11 | 1 | 7 | 11 | 1 |
| 8 | 7 | 18 | 11 | 12 | 1 | 8 | 7 | 18 | 11 | 12 | 1 | 8 | 7 | 18 | 11 | 12 | 1 |
| 9 | 5 | 7 | 6 | 16 | 11 | 4 | 17 | 1 | 9 | 5 | 7 | 6 | 16 | 11 | 4 | 17 | 1 |
| 10 | 5 | 12 | 6 | 3 | 11 | 15 | 17 | 18 | 9 | 14 | 7 | 13 | 16 | 8 | 4 | 2 | 1 |
| 11 | 7 | 1 | 11 | 7 | 1 | 11 | 7 | 1 | 11 | 7 | 1 | 11 | 7 | 1 | 11 | 7 | 1 |
| 12 | 11 | 18 | 7 | 8 | 1 | 12 | 11 | 18 | 7 | 8 | 1 | 12 | 11 | 18 | 7 | 8 | 1 |
| 13 | 17 | 12 | 4 | 14 | 11 | 10 | 16 | 18 | 6 | 2 | 7 | 15 | 5 | 8 | 9 | 3 | 1 |
| 14 | 6 | 8 | 17 | 10 | 7 | 3 | 4 | 18 | 5 | 13 | 11 | 2 | 9 | 12 | 16 | 15 | 1 |
| 15 | 16 | 12 | 9 | 2 | 11 | 13 | 5 | 18 | 4 | 3 | 7 | 10 | 17 | 8 | 6 | 14 | 1 |
| 16 | 9 | 11 | 5 | 4 | 7 | 17 | 6 | 1 | 16 | 9 | 11 | 5 | 4 | 7 | 17 | 6 | 1 |
| 17 | 4 | 11 | 16 | 6 | 7 | 5 | 9 | 1 | 17 | 4 | 11 | 16 | 6 | 7 | 5 | 9 | 1 |
| 18 | 1 | 18 | 1 | 18 | 1 | 18 | 1 | 18 | 1 | 18 | 1 | 18 | 1 | 18 | 1 | 18 | 1 |
# Number Theory/Pythagorean Triples

Suppose a$^2$ + b$^2$ = c$^2$ in integers, what do we know? Note that by multiplying each term by a constant a new solution exists (ma)$^2$ + (mb)$^2$ = (mc)$^2$ So we may restrict our analysis to those (a,b,c) that have no common factors (we’ll call these triples “Primitive”). Also realize that it is sufficient that two of the numbers have no common factors, because if two have common factors then the third has the same factor in common as well. Since this is the case at least one of the legs is odd. Both “a” and “b” cannot be odd. First note that the square of any odd number is congruent to 1 (mod 4), verification, odd numbers are either congruent to 1 or 3 (mod 4), 1$^2$ = 1 (mod 4), and 3$^2$ = 9 = 1 (mod 4). Now any two odd numbers, “a” and “b”, have the following property a$^2$ + b$^2$ = 2 (mod 4), but this cannot be of the form “c$^2$ ” because “c” would have to be even for the square to be even, and the square of an even number is divisible by 4, so c$^2$ = 0 (mod 4), so in any primitive Pythagorean triple there one of the “legs” (“a” or “b”) must be odd, and the other must be even, forcing c to be odd.
Conclusion: if (a,b,c) is a primitive triple (a$^2$ + b$^2$ = c$^2$) then we may say that “a” is always odd, “b” is always even, and “c” is always odd, and gcd(a,b)=1
It would be nice if there were some sort of formula to generate all possible triples For this analysis a will be the odd leg, b will be the even leg and c the odd hypotenuse
c = b + x (note that x must be odd) a$^2$ + b$^2$ = (b + x)$^2$ a$^2$ = 2bx + x$^2$ or a$^2$ = x(2b + x)
gcd(x,2b + x)=1 because for any prime p (because x is odd, p cannot be 2) where p $\mid$ x, => p $\mid$ a$^2$ => p $\mid$ a, and p $\nmid$ (2b + x) because if p $\mid$ (2b + x) => p $\mid$ 2b => p $\mid$ b, but gcd(a,b)=1 so p $\nmid$ (2b + x).
Knowing this, we see that both x and (2b + x) must be square numbers, and that they must be odd if we rename these squares x = i$^2$ and (2b + x) =j$^2$ reveals that a = ij then substituting to find b gives (2b + i$^2$) =j$^2$ so b = (j$^2$ - i$^2$)/2 substituting to find c gives c = (j$^2$ - i$^2$)/2 + i$^2$ = (j$^2$ + i$^2$)/2 (note that i<j).
Conclusion: for any two odd numbers, with i<j and gcd(i,j) = 1, forms a pythagorean triple with the following form (ij , (j$^2$ - i$^2$)/2 , (j$^2$ + i$^2$)/2) = (a,b,c) where a$^2$ + b$^2$ = c$^2$
# Number Theory/Irrational Rational and Transcendental Numbers
### Definitions

Rational numbers are numbers which can be expressed as a ratio of two integers (with a non-null denominator).
This includes fractional representations such as ${\frac {3}{4}}\,,-{\frac {27}{3}}\,$ etc.
A rational number can also be expressed as a termininating or recurring decimal. Examples include
$1.25,-0.333333,0.999\ldots$
However, a decimal which does not repeat after a finite number of decimals is NOT a rational number.
One other representation that is sometimes used is that of a ratio e.g. $5:4\,$
The entire (infinite) set of rational numbers is normally referenced by the symbol $\mathbb {Q} \,$.
Irrational numbers are all the rest of the numbers - such as ${\sqrt {2}},\pi ,e\,$
Taken together, irrational numbers and rational numbers constitute the real numbers - designated as $\mathbb {R} \,$.
The set of irrational numbers is infinite - indeed there are "more" irrationals than rationals (when "more" is defined precisely).
Algebraic numbers are numbers which are the root of some polynomial equation with rational coefficients. For example, ${\sqrt {2}}$ is a root of the polynomial equation $x^{2}-2=0\,$ and so it is an algebraic number (but irrational).
Transcendental numbers are irrational numbers which are not the root of any polynomial equation with rational coefficients. For example, $\pi ,e\,$ are not the roots of any possible polynomial and so they are transcendental.
The set of transcendental numbers is infinite - indeed there are "more" transcendental than algebraic numbers (when "more" is defined precisely).
# Number Theory/Notation and Introduction to Proof

"Mathematics is the queen of the sciences and number theory is the queen of mathematics." — Gauss
"God invented the integers; all else is the work of man." — Kronecker
"Number is the within of all things." — Attributed to Pythagoras