Mathematics during Prehistoric Times
Our ancestors from the past would have possessed a broad idea of
quantity and would have been able to point out, by instinct, one or two
antelopes in a herd. Notched bones found in Africa between 20, 000 and 25,
000 years ago give some of the first class of human thought around digits.
Mathematics is an old discipline and records of its beginning date back
as far as humanity itself. It has been part of the development of other fields
and has progressed rapidly from simple measurement of an object to a more
complex branch of academic study.
Babylonian Mathematics
The rich valley located between the Euphrates and Tigris rivers was
Mesopotamia, the home of Babylonians. Sumer (a region of Mesopotamia
modern day Iraq) is frequently referred to as a Cradle of Civilization.
Sumerians developed the oldest known writing system, the
pictographic cuneiform script, which makes used of wedge-shaped symbols
inscribed on baked clay tablets, we know more about ancient Sumerian and
Babylonian Mathematics than we do about early Egyptian mathematics.
SEXAGESIMAL SYSTEM
Sumerian and Babylonian Mathematics used the sexagesimal, or base
60, mathematical system, and could be physically numbered using the
twelve knuckles on one hand and the five fingers on the other.
The reason that we have 60 seconds in a minute, 60 minutes an hour
and 360 degrees for a full rotation is all due to the Babylonians basing their
number system on 60.
Despite having sixty base numbers, the Babylonians never used any
form of zero.
BABYLONIAN NUMERICAL CHART
Multiplication Formulas
To multiply numbers only the squares of the numbers needed to be known.
ab=[(a+b)² - a² - b²]/2
ab = [(a + b)² - (a - b)²/4
Division Formulas
The Babylonians could not create a system of long division and instead had
to be used.
a/b = a x (1/b)
Egyptian Mathematics
The ancient Egyptian settled along Nile valley around 6000 BCE. They
developed early mathematical concepts for agricultural and religious
purposes, which lay the groundwork for later developments in mathematics.
Egyptian use body parts for measurement a palm for with an acute for
elbow to fingertips. They develop a decimal system based on these
measurements.
NUMERATION SYSTEM
Their base 10 system uses symbols like stroke for units, heel bones for
tens, coils of rope for hundreds and lotus plants for thousands. Large
numbers were cumbersome due to lack of place value.
Multiplication method - The Egyptians use the method of repeating doubling
for multiplication. This method is like modern binary multiplication.
Division method - The Egyptians use the method of repeated having a
subtraction to perform division. This method is like breaking down the
division into a series of steps that involves subtracting the divisor or doubled
amounts like the modern binary division.
Unit Fractions - Unit fractions were used intensively. They represented
fractions like 1/2 1/4 1/8 etc. The eye of a Horus symbolizes fractions.
Circle Area Approximation - the Egyptians approximated the area of circle
using a square. Do you find the area of a circle with diameter of 9 units to be
close to a square with sides of eight units.
3-4-5 Triangle - the Egyptians use the 3-4-5 right angles in construction. This
principle predates Pythagoras.
Influence on mathematics - Egyptian mathematics influence later cultures
and developments. Their methods lay the groundwork for future
mathematical advancements.
Abu'l - Hasan Ibn Yunus (950 - 1009)
a prominent Egyptian astronomer and mathematician whose works are
known for being innovative because they were founded on tedious
calculations and close attention to detail. On the moon a crater bearing his
name is called Ibn Yunus.
Ahmes (1680 BCE- 1620 BCE)
one of the oldest mathematical texts is the Rhind Papyrus. It is named after
Scottish Egyptologist Alexander Henry Rhind who purchase the papyrus in in
Egypt in 1858. The papyrus was copied by describe Ahmes and according to
him the knowledge is from an ancient study from around 2000 BCE. Our
primary resource for knowledge about Egyptian mathematics is the papyrus.
The retro contains the division of two by the unit fractions of the odd integers
3 to 101 as well as division of the digits 1 through 9 by 10. The progression
granary volumes and the four operations.
Islamic, Hindu, and Chinese Mathematics
Mayan Mathematics
At least 36 BCE, the Maya and their neighbors independently created
the concept of zero and worked with large sums and dates. They measured
the solar year and lunar month with greater accuracy than Europeans, but
their numbering systems and mathematics were unaffected by Mayan and
Meso American Mathematics due to physical distance.
Base 20 Vigesimal Number System
The Mayans and other Meso American tribes used a base 20 vigesimal
number system, possibly derived from finger and toe counting. They used
shell-shaped zeros, dots, and bars for numbers, making addition and
subtraction easy. However, they assigned the third position a value of 360
rather than 400 (higher position revert to multiples of 20).
Chinese Mathematics
Rods Numerals
Small bamboo rods were used to represent the numbers 1 through 9 which
were then placed in columns to represent units, tens, hundreds, thousands,
etc.
Lo Shu Magic Square
an order three squares with rows, columns, and diagonals that add up to 15.
Jiuzhang Suanshu or Nine Chapters on rye Mathematical Art
The book exhibits correct rules that were itilized for the areas of triangles,
rectangles and trapezoids.
Yang Hui
he discovered the existing Chinese magic square, created a triangular
representation of binomial coefficients which was called Pascal's Triangle.
Carl Friedrich Gauss
He discovered the method involves deducing an unknown number from other
known information which is now known as Gaussian Elimination.
Liu Hui
A known commentator of tye Nine Chapters created a method that
approximated the value of 3.14159 and created a very early version of
integral and differential calculus using a regular polygon with 192 sides.
Chinese Remainder Theorem
a method used to find the smallest value of an unknown number by using
remainders after dividing by smaller numbers, such as 3,5 and 7.
Qin Jiushao
He explored solutions to quadratic and even cubic equations using a
technique of repeated approximations.
Zhu Shijie
The last and the greatest among the Sung mathematicians.His works are
Suanxue Qeming or Introduction to Mathematical Studies which was written
in 1299 and Siyuan Yujian or the Jade Mirror of the Four Origins which was
written in 1303 and it marks. great contribution in the development of
Chinese Algebra.
Zu Chongzhi
For 15 Century, he discovered the values 3.1415927, which he considered
an “excess value” and 3.1415926, a “deficit value.”
Islamic Mathematics
From the eight-century onward, the Islamic Empire spread across Persia, the
Middle East, Central Asia, North Africa, Iberia, and parts of India made major
contributions of Mathematics.
Muhammad Al-Khwarizmi
A famous Persian mathematician was one of the best early Muslim
Mathematicians and a director of the House of Wisdom in the ninth century.
His ardent support of the Hindu numerical system (1-9 and 0). Algebra was
another significant addition made by Al-Khwarizmi. He introduced the basic
algebraic techniques of “reduction and balancing”, and he give a thorough
explanation of how to solve polynomial equations with second degree.
Muhammad Al-Karaji
A Persian mathematician who lived in the tenth century, labored to further
develop algebra by releasing it from its geometrical roots and developing the
theory of algebraic calculus. Al-Karaji was the first to use the method of proof
by mathematical induction to support his claims. He employed mathematical
reasoning among other techniques to demonstrate the binomial theorem.
Omar Khayyam
Famous for his poetry, was also a great mathematician and astronomer. He
improved on Indian methods for finding square and cube roots, figuring out
how to find roots for higher numbers too. He also studied different types of
cubic equations, which are more complicated than quadratic equations.
While he did solve these equations, he couldn't fully separate algebra from
geometry, which is why a completely algebraic solution took another 500
years and was done by mathematicians in Italy.
Nasir Al-Din Al-Tusi
A Persian scientist and mathematician, was one of the first to make
trigonometry its own subject, separate from astronomy. He built on the work
of Greeks and Indians, and wrote about spherical trigonometry, which is
about triangles on a sphere. He also figured out the law of sines for plane
triangles, which says that the ratio of a side to the sine of its opposite angle
is the same for all sides. This law for spherical triangles had already been
discovered by other Persians earlier.
Thabit ibn Qurra
An Arab mathematician from the 9th century, figured out a way to find
amicable numbers. These are pairs of numbers where the sum of the
divisors of one number equals the other number. For example, 220 and 284
are amicable. It's interesting that both Fermat and Descartes later found this
same formula.
Abul Hasan al-Uqlidisi
The 10th Century Arab mathematician, who wrote the earliest surviving text
showing the positional use of Arabic numerals, and particularly the use of
decimals instead of fractions
Ibn al-Haytham
a Persian scholar known as Alhazen, was a big name in science and math.
He did important work in optics and physics, but also connected algebra and
geometry. He came up with a problem that is now called "Alhazen's
problem". He was also the first to figure out the formula for adding together
the fourth powers of numbers, using a method that could be used for other
problems too.
Kamal al-Din al-Farisi
The 13th Century Persian, who applied the theory of conic sections to solve
optical problems, as well as pursuing work in number theory such as on
amicable numbers, factorization and combinatorial methods
Ibn al-Banna al-Marrakushi
The 13th Century Moroccan, whose works included topics such as computing
square roots and the theory of continued fractions, as well as the discovery
of the first new pair of amicable numbers since ancient times (17,296 and
18,416, later re-discovered by Fermat) and the first use of algebraic notation
since Brahmagupta.
Ibrahim Ibn Sinan
The 10th Century Arab geometer, who continued Archimedes’ investigations
of areas and volumes, as well as on tangents of a circle
Brahmagupta
Important works on mathematics and astronomy. In his book on
mathematics, Brahmagupta provides guidelines for computing squares and
square roots as well as instructions on how to determine an integer’s cube
and cube-root. He also provided guidelines for dealing with five different
kinds of fractional combination. The Brahmasphutasiddhanta, is the thought
to be the earliest text to treat zero as a number rather than just a
placeholder digit, as the Babylonians did, or as a symbol for a lack of
quantity, as the Greeks and Romans did.
Mathematics of Ancient Greece
As the Greek empire began to expand its sphere of influence into Asia
Minor, Mesopotamia, and further, the Greeks were astute enough to
recognize the importance of assimilating and modifying fundamental aspects
of the communities they conquered. By the time of the Hellenistic period, the
Greeks had led one of the biggest and most dramatic changes in
mathematical thinking.
Pythagoras of Samos (570 BCE to 495 BCE)
A mathematician and scientist, he was credited with formulating
Pythagorean theorem. His work earned him many followers, and he
established a community of learners who were devoted to the study of
religion and philosophy.
Socrates (470 BCE to 399 BCE)
He was considered the foremost philosopher of ancient times and made
great contributions to the field of ethics. Socrates was a known critic of
intellectuals during his time, but he himself did not claim to be “wise” and
merely considered himself a “midwife” that helped inquiring minds achieve
wisdom. He also believed that philosophy could enable a man to live a life of
virtue. He was credited with formulating the Socratic Method – a means of
examining a topic by devising a series of questions that let the learner
examine and analyze his knowledge and views regarding the topic.
Democritus (c. 460–370 BCE)
“Nothing exists except atoms and empty space; everything else is opinion.”
Democritus, the so-called “laughing philosopher,” was an influential ancient
Greek philosopher and one of the first advocates of democracy, equality and
liberty. He was also the first person, along with his mentor Leucippus, to
advance the hypothesis that all matters are composed of small invisible
particles called atoms. Many consider Democritus to be the “father of
modern science”. Apart from that, Democritus was one of the first known
critics and a proponent of the just theory—the idea that people should take
up arms to defend themselves from tyrants.
Plato (c. 428–348 BCE)
“We can easily forgive a child who is afraid of the dark; the real tragedy of
life is when men are afraid of the light.”
Plato, a student of Socrates, is regarded as the father of political science and
the founder of one of the world’s first-known institutions of higher learning,
the Academy in Athens. The primary groundwork of Plato’s philosophy is a
threefold approach – dialects, ethics and physics, the central point of unison
being the theory of forms. For him, the highest of forms was that of the
‘good’, which he took as the cause of being and knowledge. Plato wrote one
of the first and most influential works on politics, The Republic, which
described an ideal or Utopian society. Like his mentor Socrates, Plato was a
critic of democracy.
Zeno of Elea's
The famous paradox, "Achilles and the Tortoise," illustrates the concept of
infinity. Imagine Achilles racing a tortoise, giving the tortoise a head start.
Even though Achilles is faster, every time he reaches the tortoise's previous
position, the tortoise has moved a little further. This creates an infinite
sequence of ever-decreasing distances, making it seem like Achilles can
never overtake the tortoise, even though logically he must. This is
analogous to summing an infinite series of fractions (like 1/2 + 1/4 + 1/8...),
which appears to never reach a whole number, even though it
mathematically approaches 1. Aristotle later addressed this paradox,
arguing that infinity is a concept, not a reality.
Menelaus of Alexandria (1st-2nd Century CE)
He was the first person to realize that straight lines on a flat level are the
natural counterpart to geodesics on a curved surfaced. In his book
Sphaerica, he talked about the idea of a spherical triangle, which is a shape
made of three great circle arcs that he called trilaterals.
Diophantus (c. 200-284 CE)
He was the first person to see fractions as numbers. He is considered a
founder of a science that would become known as algebra. He worked on
several very hard algebra problems, such as what is now called Diophantine
analysis. Diophantus’ “Arithmetica” is the most important work on algebra in
all of Greek mathematics.
Euclid (c. 330-275 BCE)
He was a Greek mathematician who lived and worked in Alexandria, Egypt,
around 300 BCE, during the rule of Ptolemy I. Euclid’s Stoicheion or Elements
is thought to be the most important and influential math book ever written.
Thus, he is known to be the “Father of Geometry”.
Archimedes (c. 287-212 BCE)
Archimedes’ main interest was pure mathematics, and the discovery in 1906
of writings that had never been found before, called the Archimedes
Palimpsest, has revealed additional details about how he arrived at his
mathematical conclusions. He created Archimedes’ Principle, which states
that an object submerged in a fluid can be lifted by a force equal to the
weight of the fluid has pushed out of the way.
Apollonius of Perga (c. 240-190 BCE)
There was other educational institution in the Hellenistic Greek empire
besides Alexandria. Apollonius of Perga, who’s late third century BCE work on
geometry-particularly on conics and conic sections-significantly impacted
later European mathematicians. Perga is a city in a modern-day southern
Turkey. The ellipse, parabola, and hyperbola all received their names from
Apollonius, who’s also demonstrated how they might be formed in numerous
ways through a core.
Hipparchus (c. 240-190 BCE)
He restored the use of arithmetic methods that the Chaldeans and
Babylonians had initially established, and he is typically attributed to the
invention of trigonometry. He measured the various sections of the moon
visible from multiple points. He then created the first chord table (side
lengths corresponding to different triangle angles). This happening allowed
Ptolemy to include in its Almagest table of trigonometric chords in a circle for
steps of 1/4 degree that is accurate to about five decimal places.
The Birth of Roman Numerals
Roman numerals are widely used today, and for the better part of a
thousand years, they dominated trade and administration throughout
Europe. It was a decimal (base 10) system, but because it was not directly
positional and did not include a zero, it was awkward and ineffective when
used for arithmetic and mathematics. It was based on the Roman alphabet,
specifically, the letters I, V, X, L, C, D, and M, which combine to represent the
sum of their values (for example, VII = V + I + I = 7).
Medieval Period and the Renaissance
Medieval Mathematics
The development of mathematics was sluggish during the medieval era. In
other aspects, it was a time of significant philosophical change. On the
surface, the Roman Church still controlled much of philosophy and all of
religion, but on the inside, the traditional Aristotelian viewpoints started to
lose ground. Despite dominating education for many more centuries, some
ideas started to gain acceptance. Most notably, we witness an animated
discussion about the infinite, real, and potential.
Leonardo of Pisa
He was better known by his pseudonym Fibonacci, is regarded as Europe’s
first truly exceptional medieval mathematician. In a treatise titled Liber
Abacci, he helped spread the use of the Hindu-Arabic number system
throughout Europe. This may have been his most important mathematical
contribution, even though he is best known for the so-called Fibonacci
sequence of numbers.
Nicole Oresme
A French mathematician and scholar were largely underappreciated and
underestimated despite his significance. Before his compatriot Rene
Descartes made the concept public, he likely drew the first time-speed-
distance graph using a rectangular coordinate system. As a result of his
musicology research, he was the first to apply fractional exponents and work
on infinite series, as well as the first to prove that the harmonic series
1/1+1/2+1/3+1/4+ 1/5… is a divergent infinite series (i.e., not tending to a
limit, other that infinity).
Regiomontanus
A German scholar whose greatest contribution to mathematics was the study
of trigonometry. Trigonometry was mainly regarded as a distinct area of
mathematics, thanks to his contributions to separating it from astronomy.
The first notable work on trigonometry to be published was De Triangulis, in
which he offered most of the essential trigonometric information taught in
high schools and colleges today.
Nicolaus Cusanus
He is also known as Nicholas of Cusa, was a German philosopher,
mathematician, and astronomer who lived during the fifteenth century. He is
also deserving of being included herein the development of mathematics. His
thoughts on the infinite and the infinitesimal had a direct impact on later
mathematicians such as Gottfried Leibniz and George Cantor.
The Golden Ratio
The Greek letter phi (or rarely Phi) is used to indicate this ratio, which is also
known as the Golden Ratio, Golden Mean, Golden Section, Divine Proportion,
and so on. Two quantities are in the Golden Ratio if the ratio of the sum of
the smaller and larger quantities is equal to the ratio of the larger quantity to
the smaller quantity.
Lattice Multiplication
The book unquestionably had an impact on medieval mathematics, but it
also covered a variety of other topics, such as the Chinese remainder
theorem, perfect numbers, prime numbers, formulae for numeric series and
square pyramidal numbers, Euclid’s geometric proofs, and research of
simultaneous linear equations in the style of Diophantus and Al-Karaji. He
also explained how multiple huge numbers use the lattice (or sieve) method,
which was invented by Islamic mathematicians like Al-Kwarizmi and is
algorithmically equal to long multiplication.
