THE PIONEERS OF THE NON- - Pioneered Non-Euclidean Geometry

EUCLIDEAN GEOMETRY
- His father, Farkas Bolyai, was a
GAUSS, KARL FRIEDRICH (1777- mathematician and educator who
1855) influenced his early interest in
mathematics
- Born on April 30, 1777, in Brunswick,
Germany. - Studied at the University of Göttingen,
where he was exposed to advanced
- Was a renowned mathematician and
mathematical ideas
physicist who made significant
contributions to various mathematical - Independently developed a consistent
fields. geometric framework where the parallel
postulate of Euclidean geometry did not
- Displayed exceptional mathematical
hold
talent from an early age, nurtured by his
teachers and supported by the Duke of - His key contribution was the formulation
Brunswick. of Hyperbolic Geometry along with
Gauss, and Jànos
- Major contributions include pioneering
Non-Euclidean Geometry; Fundamental
theorem of algebra; Gaussian elimination
EUGENIO BELTRAMI (1835-1900)
in linear algebra; method of least squares
in statistics; and his book Disquisitiones - An Italian mathematician
Arithmeticae (1801) which was
- Known for his significant contributions
considered the most influential book in
to non-Euclidean geometry in the 19 th
number theory.
century

- Born on February 16, 1835, in Cremona,

LOBACHEVSKY, NIKOLAI (1792-1856) Italy

- A Russian mathematician - Studied at the University of Pavia,

where he developed a strong interest in
- Known for his independent
mathematics, particularly in geometry
development of non-Euclidean geometry
- In 1868, He introduced the concept of a
- Born on December 1, 1792, in Nizhny
“Beltrami-Klein model”
Novgorod, Russia
- His work helped establish non-Euclidean
- Studied at Kazan University, located in
geometry as a legitimate field of study,
Russia where he later became a
influencing both mathematics and the
professor and eventually the rector
philosophy of geometry
- Developed a new system of geometry
that rejected Euclid’s parallel postulate
FELIX CHRISTIAN KLEIN (1849-1925)
- His ideas laid the groundwork for what
would become known as Hyperbolic - A German mathematician and
Geometry mathematics educator

- Published his work on non-Euclidean - His work was built up by using the
geometry in 1829-1830 in On the foundations laid by Nikolai Lobachevsky
Principles of Geometry and János Bolyai

- Although his contributions were rejected - His major contributions can be found in
in his time, it significantly influenced the the development of Hyperbolic Geometry
development of modern mathematics.
- Major contributions include the
Erlangen Program (1872);

JÀNOS BOLYAI (1802-1860) - Group Theoretic Approach;

- A Hungarian mathematician - Klein Disk Model;

- Unification of Geometries; - Al-Kharxmin

- Greatly influenced modern mathematics - Niels Henrik Abel

and physics
- Évariste Galois

JULES HENRI POINCARÉ (1854-1912)

Évariste Galois (1811-1832)
- A French mathematician
- Pioneered group theory, particularly
- Made significant contributions not only studying polynomial equations and their
in Non-Euclidean Geometry but also solutions.
Topology
- Introduced the concept of a Galois
- Contributions includes the Poincaré Disk group, fundamental for understanding
Model (1882); equation solvability.

- New methods for calculating distances

and angles in hyperbolic space;
Niels Henrik Abel (1802-1829):
- Introduced the concept of Hyperbolic
- Made significant contributions to group
Trigonometry.
theory and the theory of equations.
- He was considered as one of the
- Proved the unsolvability of the quintic
founders of Topology.
equation by radicals, a groundbreaking
- He developed the Poincaré conjecture, result.
which was later solved by Grigori
Perelman
MATHEMATICIANS IN THE BIRTH OF
MODERN ALGEBRA
HISTORY MODERN ALGEBRA AND
- Richard Dedekind
NUMBER THEORY
- Leopold Kronecker
WHAT IS MODERN ALGEBRA?
- David Hilbert
Modern algebra, also known as abstract
algebra, emerged from a gradual shift in
mathematical thinking throughout the
Richard Dedekind (1831-1916):
19th century. While earlier civilizations
developed practical forms of algebra, - Developed the theory of ideals in rings
modern algebra focuses on the structure and the concept of a field, crucial to
of mathematical objects rather than just abstract algebra.
solving equations.
- Introduced the notion of a module, a
fundamental concept in modern algebra.

Francois Viète- (1540- 1603)

- (Latin: Vieta), a great French Leopold Kronecker (1823-1891):

mathematician, is credited with the
- Advocated for the axiomatic approach
invention of modern algebraic notation.
to mathematics.
- The practice of using letters rather than
- Focused on algebraic numbers and
numbers to represent both known (but
made significant contributions to ring
unspecified) and unknown quantities
theory.
marked the beginnings of modern
algebra as we know it today. - Emphasized constructing mathematical
objects from more basic elements.

MATHEMATICIANS IN EARLY
FOUNDATION OF MODERN ALGEBRA David Hilbert (1862-1943):
- Made profound contributions to algebra, studied by number theorists include the
geometry, and mathematical logic. problem of determining the distribution
of prime numbers within the integers and
- Axiomatized Euclidean geometry,
the structure and number of solutions of
influencing the development of abstract
systems of polynomial equations with
algebra.
integer coefficients.
- Contributed significantly to the theory
of invariants and abstract algebraic
MATHEMATICIANS IN NUMBER
structures.
THEORY

- Pierre de Fermat

- Leonhard Euler

- Joseph-Louis Lagrange
MATHEMATICIANS IN 20 TH
CENTURY
- Carl Friedrich Gauss
OF MODERN ALGEBRA
- Adrien-Marie Legendre
- Emmy Noether

- Bartel Leendert van der Waerden

Pierre de Fermat (1607-1665)
- André Weil
- Fermat’s Last Theorem: No integer
Emmy Noether (1882-1935): solutions for xⁿ + yⁿ = zⁿ for n > 2
(proven by Andrew Wiles in 1994).
- Revolutionized abstract algebra with
contributions to ring theory and the - Proved (by infinite descent) that x⁴ + y⁴
theory of modules. = z⁴ has no non-trivial integer solutions.

- Developed “Noether’s Theorem,” - Fermat’s Little Theorem: If p is a prime

connecting symmetries in physical number and a is any integer, a^p a (mod
systems to conserved quantities p).

Bartel Leendert van der Waerden Leonhard Euler (1707-1783)

(1903-1996): - Euler product formula: Connects prime
- Published the landmark book “Modern numbers with the Riemann zeta function.
Algebra,” synthesizing and systematizing - Introduced Euler’s Totient Function φ(n),
the field. counting integers coprime to n.
- Introduced a rigorous axiomatic - Worked on sums of four squares,
approach, making the book a standard partitions, and prime number
textbook. distribution.

André Weil (1906-1998):

Joseph-Louis Lagrange (1736-1813)
- Made significant contributions to
algebraic geometry, number theory, and - Lagrange’s four-square theorem: Every
algebraic groups. positive integer can be written as the
sum of four perfect squares.
- His work on algebraic topology and the
Weil conjectures had a profound impact - Made important contributions to the
on modern mathematics. theory of quadratic forms.

- Proved key results of Fermat and Euler

What is the number theory? in number theory.

Number theory is the study of the

integers and related objects. Topics Carl Friedrich Gauss (1777-1855)
- Disquisitiones Arithmeticae: A
foundational text in number theory.

- Developed modular arithmetic and

introduced congruence notation.

- Proved the quadratic reciprocity law, a

central theorem in number theory.

- Worked on prime number distribution

and developed methods for factorization.

Adrien-Marie Legendre (1752-1833)

- Legendre’s symbol: A tool for

determining quadratic residues in
modular arithmetic.

- First to state the law of quadratic

reciprocity.

- Contributed to prime number theory

and conjectured the prime number
theorem.