Euclid

Living around 300BC, he is considered the Father of Geometry and his magnum opus: Elements, is one the greatest mathematical works in history, with its being in use in education up until the 20th century. Unfortunately, very little is known about his life, and what exists was written long after his presumed death. Nonetheless, Euclid is credited with the instruction of the rigorous, logical proof for theorems and conjectures. Such a framework is still used to this day, and thus, arguably, he has had the greatest influence of all mathematicians on this list. Alongside his Elements were five other surviving works, thought to have been written by him, all generally on the topic of Geometry or Number

theory. There are also another five works that have, sadly, been lost throughout history.

Pythagoras of Samos

Greek Mathematician Pythagoras is considered by some to be one of the first great mathematicians. Living around 570 to 495 BC, in modern day Greece, he is known to have founded the Pythagorean cult, who were noted by Aristotle to be one of the first groups to actively study and advance mathematics. He is also commonly credited with the Pythagorean Theorem within trigonometry. However, some sources doubt that is was him who constructed the proof (Some attribute it to his students, or Baudhayana, who lived some 300 years

earlier in India). Nonetheless, the effect of such, as with large portions of fundamental mathematics, is commonly felt today, with the theorem playing a large part in modern measurements and technological equipment, as well as being the base of a large portion of other areas and theorems in mathematics. But, unlike most ancient theories, it played a bearing on the development of geometry, as well as opening the door to the study of mathematics as a worthwhile endeavor. Thus, he could be called the founding father of modern mathematics.

Bust of Pythagoras of Samos in the Capitoline

Museums, Rome

Born c. 570 BC

Samos

Died c. 495 BC (aged around 75)

Metapontum

Era Ancient philosophy

Region Western philosophy

School Pythagoreanism

Main intere

sts

Metaphysics, Music,Mathematics, Et

hics, Politics

Notable ide

as

Musica universalis, Golden ratio[citation

needed],Pythagorean

tuning,Pythagorean theorem

Pythagoras of Samos (Ancient Greek: Πυθαγόρας ὁ Σάμιος [Πυθαγόρης in Ionian Greek] Pythagóras ho Sámios "Pythagoras the Samian", or simply Πυθαγόρας; b. about 570 – d. about 495 BC[1][2]) was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him. He was born on the island of Samos, and might have travelled widely in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical theories. The society took an active role in the politics of Croton, but this eventually led to their downfall. The Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said to have ended his days in Metapontum.

Pythagorean theorem

A visual proof of the Pythagorean theorem

Since the fourth century AD, Pythagoras has commonly been given credit

for discovering thePythagorean theorem, a theorem in geometry that states

that in a right-angled triangle the area of the square on the hypotenuse (the

side opposite the right angle) is equal to the sum of the areas of the

squares of the other two sides—that is,  .

While the theorem that now bears his name was known and previously

utilized by theBabylonians and Indians, he, or his students, are often said

to have constructed the first proof. It must, however, be stressed that the

way in which the Babylonians handled Pythagorean numbers implies that

they knew that the principle was generally applicable, and knew some kind

of proof, which has not yet been found in the (still largely

unpublished)cuneiform sources.[46] Because of the secretive nature of his

school and the custom of its students to attribute everything to their

teacher, there is no evidence that Pythagoras himself worked on or proved

this theorem. For that matter, there is no evidence that he worked on any

mathematical or meta-mathematical problems. Some attribute it as a

carefully constructed myth by followers of Plato over two centuries after the

death of Pythagoras, mainly to bolster the case for Platonic meta-physics,

which resonate well with the ideas they attributed to Pythagoras. This

attribution has stuck down the centuries up to modern times.[47] The earliest

known mention of Pythagoras's name in connection with the theorem

occurred five centuries after his death, in the writings

of Cicero and Plutarch.

Pythagoras started a secret society called the Pythagorean brotherhood devoted to the study of mathematics. This had a great effect on future esoteric traditions, such as Rosicrucianism and Freemasonry, both of which were occult groups dedicated to the study of mathematics and both of which claimed to have evolved out of the Pythagorean brotherhood. The mystical and occult qualities of Pythagorean mathematics are discussed in a chapter of Manly P. Hall's The Secret Teachings of All Ages entitled "Pythagorean Mathematics". [78] Pythagorean theory was tremendously influential on later numerology, which was extremely popular throughout theMiddle East in the ancient world. The 8th-century Muslim alchemist Jabir ibn Hayyan grounded his work in an elaborate numerology greatly influenced by Pythagorean theory.[citation

needed] Today, Pythagoras is revered as a prophet by the Ahl al-Tawhid or Druze faith along with his fellow Greek, Plato.

Aryabhatta (476-550) Ashmaka & Kusumapura (India)

Indian mathematicians excelled for thousands of years, and eventually even developed advanced techniques like Taylor series before Europeans did, but they are denied credit because of Western ascendancy. Among the Hindu mathematicians, Aryabhatta (called Arjehir by Arabs) may be most famous.

While Europe was in its early "Dark Age," Aryabhatta advanced arithmetic, algebra, elementary analysis, and especially trigonometry, using the decimal system. Aryabhatta is sometimes called the "Father of Algebra" instead of al-Khowârizmi

(who himself cites the work of Aryabhatta). His most famous accomplishment in mathematics was the Aryabhatta Algorithm (connected to continued fractions) for solving Diophantine equations. Aryabhatta made several important discoveries in astronomy; for example, his estimate of the Earth's circumference was more accurate than any achieved in ancient Greece. He was among the ancient scholars who realized the Earth rotated daily on an axis; claims that he also espoused heliocentric orbits are controversial, but may be confirmed by the writings of al-Biruni. Aryabhatta is said to have introduced the constant e. He used π ≈ 3.1416; it is unclear whether he discovered this independently or borrowed it from Liu Hui of China. Among theorems first discovered by Aryabhatta is the famous identity      Σ (k3) = (Σ k)2

A mathematician is a person with an

extensive knowledge of mathematics, a field that has been informally

defined as being concerned

with numbers, data, collection, quantity,structure, space, and change.

Mathematicians involved with solving problems outside of pure

mathematics are calledapplied mathematicians. Applied mathematicians

are mathematical scientists who, with their specialized knowledge

and professional methodology, approach many of the imposing problems

presented in related scientific fields. With professional focus on a wide

variety of problems, theoretical systems, and localized constructs, applied

mathematicians work regularly in the study and formulation of mathematical

models.

The discipline of applied mathematics concerns itself with mathematical

methods that are typically used in science, engineering, business, and

industry; thus, "applied mathematics" is a mathematical science with

specialized knowledge. The term "applied mathematics" also describes

the professional specialty in which mathematicians work on problems, often

concrete but sometimes abstract. As professionals focused on problem

solving, applied mathematicians look into the formulation, study, and use of

mathematical models inscience, engineering, business, and other areas of

mathematical practice.

Aryabhata  (476–550 CE) was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Āryabha ṭ īya  (499 CE, when he was 23 years old) and the Arya-siddhanta.

Statue of Aryabhata on the grounds

of IUCAA,Pune. As there is no known information

regarding his appearance, any image of Aryabhata

originates from an artist's conception.

Born 476 CE

Died 550 CE

Era Gupta era

Region India

Main interests Mathematics, Astronomy

Major works Āryabhaṭīya, Arya-siddhanta

Aryabhatiya

Direct details of Aryabhata's work are known only from the Aryabhatiya.

The name "Aryabhatiya" is due to later commentators. Aryabhata himself

may not have given it a name. His disciple Bhaskara I calls

it Ashmakatantra (or the treatise from the Ashmaka). It is also occasionally

referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because

there are 108 verses in the text. It is written in the very terse style typical

of sutra literature, in which each line is an aid to memory for a complex

system. Thus, the explication of meaning is due to commentators. The text

consists of the 108 verses and 13 introductory verses, and is divided into

four pādas or chapters:

1. Gitikapada: (13 verses): large units of time—kalpa, manvantra,

and yuga—which present a cosmology different from earlier texts

such as Lagadha's Vedanga Jyotisha (c. 1st century BCE). There is

also a table of sines (jya), given in a single verse. The duration of the

planetary revolutions during a mahayuga is given as 4.32 million

years.

2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra),

arithmetic and geometric progressions, gnomon / shadows (shanku-

chhAyA), simple, quadratic, simultaneous,

and indeterminate equations

3. Kalakriyapada (25 verses): different units of time and a method for

determining the positions of planets for a given day, calculations

concerning the intercalary month (adhikamAsa), kShaya-tithis, and a

seven-day week with names for the days of week.

4. Golapada (50 verses): Geometric/trigonometric aspects of

the celestial sphere, features of the ecliptic, celestial equator, node,

shape of the earth, cause of day and night, rising of zodiacal signs on

horizon, etc. In addition, some versions cite a fewcolophons added at

the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and

astronomy in verse form, which were influential for many centuries. The

extreme brevity of the text was elaborated in commentaries by his disciple

Bhaskara I (Bhashya, c. 600 CE) and byNilakantha Somayaji in

his Aryabhatiya Bhasya, (1465 CE). He was not only the first to find the

radius of the earth but was the only one in ancient time including the

Greeks and the Romans to find the volume of the earth.

Mathematics

Place value system and zero

The place-value system, first seen in the 3rd century Bakhshali Manuscript,

was clearly in place in his work. While he did not use a symbol for zero, the

French mathematician Georges Ifrah explains that knowledge of zero was

implicit in Aryabhata's place-value system as a place holder for the powers

of ten with null coefficients [7]

However, Aryabhata did not use the Brahmi numerals. Continuing

the Sanskritic tradition from Vedic times, he used letters of the alphabet to

denote numbers, expressing quantities, such as the table of sines in

a mnemonic form.[8]

Approximation of π

Aryabhata worked on the approximation for pi ( ), and may have come to

the conclusion that   is irrational. In the second part of

theAryabhatiyam (gaṇitapāda 10), he writes:

caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇāmayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ."Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached." [9]

This implies that the ratio of the circumference to the diameter is

((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate

to five significant figures.

It is speculated that Aryabhata used the word āsanna (approaching), to

mean that not only is this an approximation but that the value is

incommensurable (or irrational). If this is correct, it is quite a sophisticated

insight, because the irrationality of pi was proved in Europe only in 1761

by Lambert.[10]

After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation

was mentioned in Al-Khwarizmi's book on algebra.[3]

Trigonometry

In Ganitapada 6, Aryabhata gives the area of a triangle as

tribhujasya phalashariram samadalakoti bhujardhasamvargah

that translates to: "for a triangle, the result of a perpendicular with the

half-side is the area."[11]

Aryabhata discussed the concept of sine in his work by the name

of ardha-jya. Literally, it means "half-chord". For simplicity, people

started calling it jya. When Arabic writers translated his works

from Sanskrit into Arabic, they referred it as jiba. However, in Arabic

writings, vowels are omitted, and it was abbreviated as jb. Later writers

substituted it with jaib, meaning "pocket" or "fold (in a garment)". (In

Arabic, jiba is a meaningless word.) Later in the 12th century,

when Gherardo of Cremona translated these writings from Arabic into

Latin, he replaced the Arabic jaib with its Latin counterpart, sinus, which

means "cove" or "bay". And after that, the sinusbecame sine in

English.Alphabetic code has been used by him to define a set of

increments. If we use Aryabhata's table and calculate the value of

sin(30) (corresponding to hasjha) which is 1719/3438 = 0.5; the value is

correct. His alphabetic code is commonly known as the Aryabhata

cipher.

[12]

Indeterminate equations

A problem of great interest to Indian mathematicians since ancient times

has been to find integer solutions to equations that have the form ax +

by = c, a topic that has come to be known as diophantine equations.

This is an example from Bhāskara's commentary on Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest

value for N is 85. In general, diophantine equations, such as this, can

be notoriously difficult. They were discussed extensively in ancient

Vedic text Sulba Sutras, whose more ancient parts might date to 800

BCE. Aryabhata's method of solving such problems is called

the kuṭṭaka (कु� ट्टकु) method. Kuttaka means "pulverizing" or

"breaking into small pieces", and the method involves a recursive

algorithm for writing the original factors in smaller numbers. Today

this algorithm, elaborated by Bhaskara in 621 CE, is the standard

method for solving first-order diophantine equations and is often

referred to as the Aryabhata algorithm.[13] The diophantine equations

are of interest in cryptology, and the RSA Conference, 2006, focused

on thekuttaka method and earlier work in the Sulbasutras.

Algebra

In Aryabhatiya Aryabhata provided elegant results for the summation

of series of squares and cubes:[14]

and