Transcript Chinese Mathematics

The History of
Mathematics
http://www.math.wichita.edu/~richardson/timeline.html
http://www-groups.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html
http://www-groups.dcs.st-and.ac.uk/~history/BiogIndex.html
Completing a Square
Solving a Quadratic Equation
al-Khwarizmi
Iraq (ca. 780-850)
x2 + 10 x = 39
x2 + 10 x + 4·25/4 = 39+25
(x+5)2 = 64
x + 5 = 8
x = 3
The Bridges of Konigsberg
Topology
Leonhard Euler
Switzerland 1707 - 1783
•In Konigsberg, Germany, a river ran through the city
such that in its centre was an island, and after passing
the island, the river broke into two parts. Seven
bridges were built so that the people of the city could
get from one part to another.
•The people wondered whether or not one could walk
around the city in a way that would involve crossing
each bridge exactly once
Infinite Prime Numbers
Euclid
Greece 325 – 265BC
Theorem: There are infinitely many prime numbers.
Proof:
Suppose the opposite, that is, that there are a finite
number of prime numbers. Call them p1, p2, p3, p4,....,pn.
Now consider the number
•(p1*p2*p3*...*pn)+1
•Every prime number, when divided into this number,
leaves a remainder of one. So this number has no prime
factors (remember, by assumption, it's not prime itself).
•This is a contradiction. Thus there must, in fact, be
infinitely many primes.
The Search for Pi
Person/People
Year
Value
Babylonians
~2000 B.C.
3 1/8
Egyptians
~2000 B.C.
(16/9)^2= 3.1605
Archimedes - Italy
~300 B.C.
proves 3 10/71<Pi<3 1/7
uses 211875/67441=3.14163
Ptolemy - Greece
~200 A.D.
377/120=3.14166...
Tsu Chung-Chi China
~500 A.D.
proves 3.1415926<Pi<3.1415929
Aryabhatta - Indian
~500
3.1416
Fibonacci - Italy
1220
3.141818
Ludolph van Ceulen German
1596
Calculates Pi to 35 decimal places
Machin - England
1706
100 decimal places
CDC 6600
1967
500,000 decimal places
The Search for Pi
•François Viète (1540-1603)
France - determined that:
•John Wallis (1616-1703)
English - showed that:
•While Euler (1707-1783)
Switzerland derived his famous formula:
•Today Pi is known to more than 10 billion decimal places.
Laura T
Ancient Babylonia
The Sumerians had developed an abstract form of
writing based on cuneiform (i.e. wedge-shaped)
symbols. Their symbols were written on wet clay
tablets which were baked in the hot sun and many
thousands of these tablets have survived to this day.
It was the use of a stylus on a clay medium that led
to the use of cuneiform symbols since curved lines
could not be drawn. The later Babylonians adopted
the same style of cuneiform writing on clay tablets.
The Babylonians had an advanced number system, in
some ways more advanced than our present
systems. It was a positional system with a base of
60 rather than the system with base 10 in
widespread use today.
The Four Colour Theorem
The Four Colour Conjecture
was first stated just over
150 years ago, and finally
proved conclusively in 1976.
It is an outstanding example of
how old ideas combine with new
discoveries and techniques in different fields of
mathematics to provide new approaches to a problem. It is
also an example of how an apparently simple
problem was thought to be 'solved'
but then became more
complex, and it is the first
spectacular example where
a computer was
involved in proving a
mathematical theorem.
Laura T
Ancient Babylonia
The Babylonians divided the day into 24 hours, each
hour into 60 minutes, each minute into 60 seconds.
This form of counting has survived for 4000 years. To
write 5h 25' 30", i.e. 5 hours, 25 minutes, 30
seconds, is just to write the sexagesimal fraction, 5
25/60 30/3600. We adopt the notation 5; 25, 30 for
this sexagesimal number, for more details regarding
this notation see our article on Babylonian numerals.
As a base 10 fraction the sexagesimal number 5; 25,
30 is 5 4/10 2/100 5/1000 which is written as 5.425
in decimal notation.
Laura T
Ancient Babylonia
Perhaps the most amazing aspect of the Babylonian's calculating
skills was their construction of tables to aid calculation. Two
tablets found at Senkerah on the Euphrates in 1854 date from
2000 BC. They give squares of the numbers up to 59 and cubes
of the numbers up to 32. The table gives 82 = 1,4 which stands
for
82 = 1, 4 = 1 x 60 + 4 x 1 = 64
and so on up to 592 = 58, 1 = 58 x 60 +1 x 1 = 3481).
The Babylonians used the formula
ab = [(a + b)2 - a2 - b2]/2
to make multiplication easier. Even better is their formula
ab = [(a + b)2 - (a - b)2]/4
which shows that a table of squares is all that is necessary to
multiply numbers, simply taking the difference of the two
squares that were looked up in the table then taking a quarter
of the answer.
Egyptian numerals
Laura T
The following hieroglyphs were used to denote powers of ten:
Value
1
10
Hierogly
ph
Descriptio
n
100
1,000
10,00
0
100,00 1 million,
0
or
infinity
Coil of
rope
Water
lily
(also
called
Lotus)
Finger
Tadpol Man with
e
both
or Frog
hands
raised
or
Single
stroke
Heel
bone
Multiples of these values were expressed by
repeating the symbol as many times as
needed. For instance, a stone carving from
Karnak shows the number 4622 as
Chinese Mathematics
Chinese mathematics has developed greatly since at least 100 BC.
Although the Chinese refer back to their ancient texts, many of which
were written on strips of bamboo, they are constantly coming up with
ways of working out problems. One of the earliest Chinese
mathematicians was a man named Luoxia Hong (130BC – 30BC). He
designed a new calendar for the Emperor, which featured 12 months,
based on a cycle of 12 years. This inspired many people to design
calendars and the one we have today.
The Chinese also came up with a rule called the Gougu Rule. This is
the Chinese version of Pythagoras. Liu Hui (220AD – 280AD) tried to
find pi to the nearest number. He eventually got to 3.14159, which in
those days was thought to be an incredible achievement.
Jeremy
The Moscow Papyrus is located in a
museum hence the name. The
papyrus was copied by a scribe and
was brought to Russia. The papyrus
contains 25 maths problems
involving simple “equations” and
solutions. The problems are not in
modern form. The problem that has
generated the most interest is the
volume of a truncated pyramid (a
square based pyramid with the top
portion removed). The Egyptians
discovered the formula for this even
though it was very hard to derive.
The Actual author of the
equation is unknown.
But this is what he/she
discovered:
The Moscow Papyrus is 15 feet
long and about 3 inches wide.
Alannah
Johannes Widman was a German mathematician
who is best remembered for an early arithmetic
book which contains the first appearance of + and
– (both adding and subtracting, and positive and
negative) signs in 1498.
Alannah
His book was better than anybody else's
because he had more and a wider range
of examples.
The book remained in print until 1526.(28
years after it was first published.)
Moscow Papyrus: Arithmetic
As it name may suggest, the Moscow papyrus is located in the
Museum of Fine Arts in Moscow.
In around 1850BC, the papyrus was copied by an anonymous
scribe and was bought to Russia in the 19th Century.
It contains 25 problems containing simple equations and
solutions.
However, the equations are not in
modern form.
The problem that generates the most
interest is the calculation of the volume
of a truncated pyramid (a square based
pyramid with the top cut off)
The Egyptians seemed to know this
difficult formula.
V= (1/3)(a² + ab + b²)(h)
Alex
Johannes Widman (1462 – 1498)
Widman is best known for a book on arithmetic
which he wrote (in German) in 1489AD.
This contains the first appearance of + and –
signs. This was better than those that had come
before it with a wider range of examples.
The book continued to be published until 1526AD.
Then, Adam Ries – amongst others – produced
superior books.
Alex
Chu Shih-chieh (Zhu
Shijie)
Zhu Shijie was one of the greatest
Chinese mathematicians. He lived
during the Yuan Dynasty.
An early form of
Pascal’s triangle
Yang worked on magic squares and
binomial theorem, and is best known for
his contribution of presenting 'Yang Hui's
Triangle'. This triangle was the same as
Pascal's Triangle, discovered
independently by Yang and his
predecessor Jia Xian .Yang was also a
friend to the other famous
mathematician Qin Jiushao. Calum
Magic squares
An example of a
magic square
In mathematics, a magic square of
order n is an arrangement of n²
numbers, usually distinct integers,
in a square, such that the n
numbers in all rows, all columns,
and both diagonals sum to the
same constant.
A normal magic square contains
the integers from 1 to n².
The
equation
All non trivial magic squares exist
for n≥3.
Calum
Johannes Widmann
+
-
Johannes Widmann (born c. 1460 in
Eger; died after 1498 in Leipzig) was
a German mathematician who was the
first to use the addition (+) and the
subtraction (-) signs. Widmann
attended the University of Leipzig in
the 1480s, and published Behende
und hubsche Rechenung auff allen
Kauffmanschafft, his work making use
of the signs, in Leipzig in 1489.
Calum
Aristarchus
Charis
Aristarchus of Samos was a Greek mathematician and astronomer.
He was born in about 310BC and died at around 230BC.
He is the first person to suggest a universe with the Sun at the centre
instead of the Earth.
He tried to work out the sizes of the Sun and
the Moon and how far away they are. He
worked out that the Sun was 20 times further
away than the moon and 20 times bigger. Both
these estimates were too small but the
reasoning behind it was right.
Blaise Pascal (1623-1662)
Charis
Pascal was a French mathematician and physicist.
His father was a tax official and Pascal made a calculating machine
that did addition and subtraction to make his work easier.
He wrote about Pascal's triangle. Each
number is the sum of the two above it.
There are lots of different patterns in the
triangle. Some are shown on the diagram.
Pascal also
worked with
Fermat on the
theory of
probability, and
he wrote about
projective
geometry when
he was only 16.
Pythagoras and the Mathematikoi
- Pythagoras was the leader
of a Society which consisted
mainly of followers called
the mathematikoi.
- The mathematikoi owned
nothing personal and were
vegetarians.
- Any mathematical
discoveries they made the
credit was given to
Pythagoras.
- Everything we know about
Pythagoras and the
mathematikoi was only
recorded properly 100 years
later as they apparently
wrote none of their
information down.
David
Trigonometry of Hipparchus
-Hipparchus was a Greek mathematician
who invented one of the first trigonometry
tables which he needed to compute the
orbits of the Sun and Moon.
-The table on the right represents the Chord
function. The chord of an angle is the length
between two points on a unit circle
separated by that angle.
-If one the angles is zero it can be easily
related to the sine function. And the used
in the half angle formula:
David
JAPANESE MATHEMATICS
= 16
= 60
The system of Japanese numerals is the system of
number
names used in the Japanese language. The Japanese
numerals in writing are entirely based on the Chinese
numerals and the grouping of large numbers follow
the
Chinese tradition of grouping by 10,000. Like in
Chinese
numerals, there exists in Japanese a separate set of
kanji for numerals called daiji (大字) used in legal and
financial documents to prevent unscrupulous
individuals
from adding a stroke or two, turning a one into a two
or a three.
The formal numbers are identical to the Chinese
formal
numbers except for minor stroke variations George
http://en.wikipedia.org/wiki/Japanese_numerals
John Napier ( 1550-1617)
•
•
•
1.
2.
Napier was a Scottish mathematician
who studied math like a hobby as he
never had time to spend on
calculations between working on
theology.
He is best known, along with Joost
Burgi, for his invention of logarithms
He is also famous for the invention of
two theories:
Napier’s analogy (used in solving
spherical triangles)
And Napier’s bones. (used for
mechanically multiplying, dividing,
taking square roots and cube roots
Hannah
Pierre de Fermat (1601- 1665)
• Fermat was a French mathematician
who is best known for his work on
number and theory
• One of his last theorem’s was proven by
Andrew Wiles in 1994.
• Whilst in Bordeaux, Fermat produced
work on maxima and minima, which was
important. His methods of doing this
were similar to ours , however as he has
not a professional mathematician his
work was very awkward.
• Fermat’s last theorem was that if you
had the equation : xn + yn = zn
n in this equation can be no more that
two.
When n is more than two the equation
does not work
Hannah
Issy
Hipparchus
Hipparchus is most known for Trigonometry.
He did not discover this on his own however.
Menelaus and Ptolomy, helped with this.
“Even if he did not invent it, Hipparchus is the first person
of whose systematic use of trigonometry we have
documentary evidence." some historians say. Some even go
as far as to say that he invented trigonometry.
Not much is known about the life of Hipparchus. But it is
believed that he was born at Nicaea in Bithynia, and lived
from 190 BC to 120 BC
Issy
Algebra
Algebra is a branch of mathematics concerning the study of structure,
relation, and quantity. Together with geometry, analysis,
combinatorics, and number theory, algebra is one of the main branches
of mathematics.
Elementary algebra is provides an introduction to the basic ideas of
algebra, including effects of adding and multiplying numbers, 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.
The history of algebra began in ancient Egypt and Babylon, where
people learned to solve linear (ax = b) and quadratic (ax² + bx = c)
equations, as well as indeterminate equations such as x² + y² = z²,
whereby several unknowns are involved. The ancient Babylonians
solved arbitrary quadratic equations by essentially the same
procedures taught today. They also could solve some indeterminate
equations
The Roman Abacus
John-Jack
The Roman Abacus was devised by Roman traders adapting
ideas that had been picked up in Egypt.
•
•The Abacus is made up of grooves in a slate tile with marbles
that run in them.
•The Abacus was originally made in Babylon using stones and
ditched made in the dry soil in 2700 BC.
•The Abacus was originally made in Babylon using stones and
ditched made in the dry soil in 2700 BC.
•Each Abacus used a different scale depending on the user.
Traders often used ones with fractions up to 1/12.
•This would mean that they could subtract quite
accurately eg. To subtract 1/3 you would take a
bead from the 1/4 column and one from the
1/12 column.
Buffon’s needle problem
Laura W
• Georges-Louis Leclerc, Comte de Buffon lived from in September 7, 1707 to April 16, 178.
• He had many different careers, as a naturalist, mathematician, biologist, cosmologist and
author.
• The Lycée Buffon in Paris is named after him.
• The problem he is famous for is:
Suppose we have a floor made of parallel strips of wood, each the same
width, and we drop a needle onto the floor. What is the probability that the
needle will lie across a line between two strips?
Or in more mathematical terms …
Given a needle of length l dropped on a plane ruled with parallel lines t units
apart, what is the probability that the needle will cross a line?
• For n needles dropped with h of the needles crossing lines, the probability is:
This is useful because it can be rearranged
to get an estimate for pi
Kyle
The Egyptians had a writing system based on hieroglyphs from around 3000 BC.
Hieroglyphs are little pictures representing words. It is easy to see how they would
denote the word "bird" by a little picture of a bird but clearly without further
development this system of writing cannot represent many words. The way round
this problem adopted by the ancient Egyptians was to use the spoken sounds of
words. For example, to illustrate the idea with an English sentence, we can see
how "I hear a barking dog" might be represented by:
"an eye", "an ear", "bark of tree" + "head with crown", "a dog".
Of course the same symbols might mean something different in a different
context, so "an eye" might mean "see" while "an ear" might signify "sound".
The Egyptians had a bases 10 system of hieroglyphs for numerals. By this we
mean that they has separate symbols for one unit, one ten, one hundred, one
thousand, one ten thousand, one hundred thousand, and one million.
Kyle
The Addition & Minus Signs
The plus and minus signs are symbols representing positive
and negative, the meaning of them has been around since the
Egyptian times, but the actual symbols + and – were first
published by Johannes Widmann.
Minus
A Jewish tradition that dated from at least from the 19th
century was to write plus using a symbol like an inverted
T. This practice was then adopted into Israeli schools (this
practice goes back to at least the 1940s) and is still
commonplace today in some elementary schools
(including secular schools) while fewer secondary schools.
It is also used occasionally in books by religious authors,
but most books for adults use the international symbol
"+". The usual explanation for the origins of this practice
is that it avoided the writing of a symbol "+" that looked
like a Christian cross. Unicode has this symbol at position
U+FB29 "Hebrew letter alternative plus sign"
-Addition
Jewish
Addition
Symbol
Kyle
CHAOTIC BEHAVIOR
In mathematics, chaos theory describes the behaviour of
certain dynamical systems – that is, systems whose states
evolve with time – that may exhibit dynamics that are highly
sensitive to initial conditions (popularly referred to as the
butterfly effect). As a result of this sensitivity, which manifests
itself as an exponential growth of perturbations in the initial
conditions, the behaviours of chaotic systems appears to be
random. This happens even though these systems are
deterministic, meaning that their future dynamics are fully
defined by their initial conditions, with no random elements
involved. This behaviour is known as deterministic chaos, or
simply chaos.
Chaotic behaviour is also observed in natural systems,
such as the weather. This may be explained by a
chaos-theoretical analysis of a mathematical model
of such a system, embodying the laws of physics that
are relevant for the natural system.
Hieroglyphic numerals in
Egypt
Hieroglyphs were introduced for
numbers in 3000BCE. Their number
system was based on units of 10. They
used simple grouping to make different
numbers.
The Egyptians
used different
images for
their hieroglyphs.
Nina
Horus was Egyptian God who
fought the forces of darkness
(in the form of a boar - a pig)
and won. His eye is a symbol
for Egyptian Unit Fractions.
Each part of the eye is a part
of the whole. All the parts of
eye, however, don't add up to
the whole. This, some
Egyptologists think, is the
sign that the knowledge can
never be total, and that one
part of the knowledge is not
possible to describe or
measure.
Nina
Pythagoras of Samos
• Pythagoras was an ancient Greek mathematician.
Pythagoras was born about 569 BC in Samos, Ionia and
died about 475 BC.
• Pythagoras invented Pythagoras's theorem which is the idea
that in a right angled triangle the two shorter sides squared
and added equals the longest side (the hypotenuse)
squared.
• It was thought that the Babylonians
1200 years earlier knew this before
but Pythagoras was the one to prove it.
It is said that this is the oldest number
theory document in existence.
This theorem works for every right
angled triangle
Algebra
Jack G
• While the word "algebra" comes from Arabic word
(al-jabr)its origins are from the ancient
Babylonians. With this system they were able to
discover unknown values for a class of problems
typically solved today by using linear equations,
quadratic equations, and indeterminate linear
equations.
• The geometric work of the Greeks, typified in the
Elements, provided the framework for finding the
formulae beyond the solution of particular
problems into more general systems of stating
and solving equations.
• The Greek mathematicians Hero of Alexandria
and Diophantus ("the father of algebra") made
algebra into a much higher level. People argye
that al-Khwarizmi, who founded the discipline of
al-jabr, deserves that title instead.
Algebra
Jack G
• Later, the Indian mathematicians developed
algebraic methods to a high degree of
sophistication. Al-Khwarizmi produced the
"reduction" and "balancing" (the transposition of
subtracted terms),He gave an explanation of
solving quadratic equations supported by
geometric proofs.
• The Indian mathematicians Mahavira and
Bhaskara II, the Persian mathematician Al-Karaji
and the Chinese mathematician Zhu Shijie,
solved various cases of cubic, quadratic, quintic
and higher-order polynomial equations using
numerical methods.
• Gottfried Leibniz discovered the solution to
simultaneous equations
Napier’s Bones
• Napier’s bones are
basically a big
multiplication square.
• It was used before
calculators for
multiplication of !HUGE!
Numbers.
• To do a sum using them
you arrange the bones in
the order of the number
to multiply like in the
example: the sum is
46785399*7.
Paul
• Then, starting from the
left, you just add all the
numbers in the row,
carrying the tens.
http://en.wikipedia.org/wiki/Napier%27s_bones
Nicole
Ahmes
Ahmes was the Egyptian scribe who wrote
the Rhind Papyrus - one of the oldest
known mathematical documents.
Born: about 1680 BC in Egypt
Died: about 1620 BC in Egypt
• The Rhind Papyrus, which came to the British Museum in
1863, is sometimes called the 'Ahmes papyrus' in honour
of Ahmes. Nothing is known of Ahmes other than his own
comments in the papyrus.
• Ahmes claims not to be the author
of the work, being, he claims, only a
scribe. He says that the material
comes from an earlier work of about
2000 BC.
• The papyrus is our main source of
information on Egyptian mathematics.
Nicole
Hieroglyphic Numerals
Hieroglyphics were used
by the Egyptians in
around 3000BC. These
symbols below are what
they would use as
numbers. Although they
only have to write one
symbol for one million and
we have to do seven,
there is a fault. To write
one million take one they
would have to write 54
symbols.
1
10
100
Single
stroke
Heel
bone
Coil of
rope
=
999999
100000 1million or
Infinity
1000 10000
Water
Lily
Finger
Tadpole
or frog
Man with
both hands
raised
Michael
Xenocrates of Chalcedon
Xenorcrates was a Greek
philosopher, mathematician and
leader of the platonic army from
339BC to 314BC. Xenocrates is
known to have written a book On
Numbers, and a Theory of
Numbers, besides books on
geometry. Plutarch writes that
Xenocrates once attempted to
find the total number of syllables
that could be made from the letters of the alphabet. According
to Plutarch, Xenocrates result was 1,002,000,000,000. This
possibly represents the first instance that a combinatorial
problem involving permutations was attempted. Xenocrates
also supported the idea of indivisible lines (and magnitudes)
in order to counter Zeno's paradoxes.
Birth: 396BC,
Chalcedon
Died: 314BC,
Athens
Interests: logic,
physics,
metaphysics,
epistemology,
mathematics, ethics.
Ideas: developed
the philosophy of
Plato.
Michael
Abu'l Hasan Ahmad ibn Ibrahim
Al-Uqlidisi
Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi
was an Arab mathematician who was active
in Damascus and Baghdad. He wrote the
earliest surviving book on the positional use
of the Arabic numerals, around 952. It is
especially notable for its treatment of decimal
fractions, and that it showed how to carry out
calculations without deletions.
While the Persian mathematician Jamshīd alKāshī claimed to have discovered decimal
fractions himself in the 15th century, J.
Lennart Berggrenn notes that he was
mistaken, as decimal fractions were first used
five centuries before him by al-Uqlidisi as
early as the 10th century.
Michael
Ollie
Zhu Shijie of China
Zhu Shijie was born in the 13th
century near Beijing.
Two of his mathematical works
have survived; “Introduction to
Computational Studies” and
“Jade Mirror of Four Unknowns”.
This book brought Chinese
algebra to its highest level and it is his most
important work.
He makes use of the Pascal Triangle centuries
before Blaise Pascal brought it to common
knowledge.
Francesco Pellos 1450 – 1500 AD
• Francesco Pellos, from Nice, is the earliest
example of the use of the decimal point
• He wrote an arithmetic book, called
Compendio de lo Abaco, in 1492.
• In this book he makes use of a dot to
denote the division of a number by a
power of ten. This has evolved to what we
now call a decimal point.
Thales (620-547BC) Jack S
Discoverer of deductive Geometry
•
•
•
•
•
“Father of deductive geometry”
Credited for five theorems
1) A circle is bisected by any diameter.
2) The base angles of an Isosceles Triangle are equal.
3) The angles between two intersecting straight lines
are equal.
• 4) Two triangles are congruent if they have
two angles and one side equal.
• 5) An angle in a semicircle is a right angle.
Jack S
Bhaskara
• Can be called Bhaskaracharya
meaning “Bhaskara the teacher”.
• Lived in India.
• Famous for number systems and solving
equations which was not achieved in
Europe for several centuries.
• More information at
http://www.maths.wichita.edu/~richardson/
Rachel
Hypatia of Alexandria
(AD 355 or 370 – 415)
Hypatia’s father (Theon) was a mathematician in Alexandria in Egypt
and he taught her about mathematics. From about the year 400
onwards she lectured on mathematics and philosophy. She also
studied astronomy and astrology and may have invented astrolabes
(which can be used to study astronomy) . However, there is no proof
that she did this.
Although she did not make any discoveries herself, she helped her
father Theon with some of his works, and was the first woman to
make a significant contribution to the development of mathematics.
She was believed by some people to
practise magic and was also hated for
being a pagan. In AD 415 she was
murdered in the street by a group of
monks.
The Crater Hypatia and Rimae Hypatia
(features of the moon) are both named after
Hypatia.
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Hypatia.html
Julia Hall Bowman
Robinson (1919 – 1985)
Rachel
Julia was born in Missouri in the USA.
When she was nine, she caught scarlet fever,
which was followed by rheumatic fever. In
total she missed two years of school. Over
the next year, she had lessons three
mornings a week and managed to get
through four years of education (fifth to
eighth grades). In her last year at school she
was the only girl in her maths and physics
classes.
In 1948 she started work on Hilbert’s tenth problem (Given a
Diophantine equation with any number of unknown quantities and
with rational integral numerical coefficients: To devise a process
according to which it can be determined in a finite number of
operations whether the equation is solvable in rational integers) and
came up with the Robinson hypothesis. This helped Yuri Matijasevic to
find the final solution to the problem in 1970.
Mary Ann Elizabeth Stephansen
(1872 -1961)
Rachel
She was born in Bergen in Norway on 10 March 1872. She studied at
university in Zurich in Switzerland. She was the only Norwegian to
pass the entrance exam. When she left in 1896 she became a
teacher in Norway, which was unusual for women at that time.
During her time as a teacher she also worked on partial differential
equations.
In 1906 she was appointed to the
Norwegian Agricultural College where she
taught maths and physics. She retired in
1931 and went to live with her sister Gerda
in England.
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Stephansen.html
Hieroglyphic numerals in Egypt
- Brought in +/- 3000BC
Egyptian hieroglyphics were a
symbol for each power of 10.
Rosa
This is the
number 4622
It did not matter if there was a
0 in the number because of the
powers.
1
10
10
101
100
102
1,000
103
10,000
104
100,000
105
1,000,000
106
10,000,000
107
Rosa
Roman Numerals
I
V
Symb
ol
=1
Symb
ol
=5
X
L
C
D
Symbo Symbol Symbol Symb
l
= 50
= 100 ol =
500
= 10
To get a number you put the
symbol’s together in the correct
order unless there is a shorter
way of writing it…
For Example - To get some numbers you
can put a smaller number in front of a
larger number indicating a subtraction.
E.g. 999 can be written like this… IM
which is a lot easier than
M
_________
__
Symbol
= 1000
Symbol =
Number
times by
1,000 to get
the value,
in this case
5,000
V
Roman
Numerals are
mainly used
for writing
dates and on
clocks
John Napier 1550-1617
Sam
He is most well known for his inventions of logarithms
but also invented ‘Napier's bones’ which are a way of
multiplying, dividing, and taking square and cube
roots.
The board consists off 9 rods which have the times
table of 1-9 on each and the number of the
corresponding times table at the top
You turned them so at they top it made your number
then add the numbers in the row if you are multiplying
it by a number with more than one digit you would do it
for both add a zero on they tens 2 zeros on the
hundreds etc then add together
By Sam allum
Arabic/Islamic mathematics
Arabic mathematics : forgotten brilliance?
Recent research has proved the debt that we owe to
Arabic/Islamic mathematics. Certainly many of the ideas
which were previously thought to have been brilliant new
conceptions due to European mathematicians of the
sixteenth, seventeenth and eighteenth centuries, were
actually developed by Arabic/Islamic mathematicians around
four centuries earlier! In many respects the mathematics
studied today is far closer in style to that of the Arabic/Islamic
contribution than to that of the Greeks.
Sameer
Sameer
Arabic Numerals
The Indian numerals were not transmitted directly from
India to Europe but rather came first to the Arabic/Islamic
peoples and from them to Europe. The story of this
transmission is not, however, a simple one. The eastern
and western parts of the Arabic world both saw separate
developments of Indian numerals with relatively little
interaction between the two. By the western part of the
Arabic world we mean the regions comprising mainly
North Africa and Spain. Transmission to Europe came
through this western Arabic route, coming into Europe
first through Spain.
Aristarchus’ Heliocentric Astronomy.
Sophie
(310BC - BC 210BC)
He was a Greek mathematician and an
astronomer. He is widely known for proposing the
theory that the universe was sun-centred
(heliocentric).
He also made some calculations that gave him an
estimation of the sizes and the distance of the
Sun and the Moon, for example, the volume of
Aristarchus's Sun would be almost 300 times
greater than the Earth.
Ptolemy’s Almagest.
Sophie
(85AD- 165AD).
Claudius Ptolemy put a book together called
Mathematical Compilation. It was a book on
everything that people knew about astronomy at the
time.
He thought that the Earth was the centre of the
universe, but surprisingly the calculations he made
were fairly accurate. Up until 1542, Almagest was
still the primary source of astronomical knowledge.
Johannes Kepler (1571 -1630) Germany
Kepler was a mathematician who studied astronomy. In his book, he
gave his first 2 laws of astronomy:
1.
The planets move around the Sun in elliptical orbits.
2.
The radius vector joining a planet to the sun sweeps out equal
areas in equal time.
(1)The orbits are ellipses, with focal
points ƒ1 and ƒ2 for the first planet and
ƒ1 and ƒ3 for the second planet. The
sun is placed in a fixed point ƒ1.
(2) The two shaded sectors A1 and A2
have the same surface area and the
time for planet 1 to cover segment A1 is
equal to the time to cover segment A2.
(3) The total orbit times for planet 1 and
planet 2 have a ratio a13/2 : a23/2.
Sruthi
Isaac Newton later used Kepler’s theory
for his gravitational theory.
Euclid of Alexandria (325BC – 265BC)
Euclid was a Greek mathematician best
known for his treatise on geometry: The
Elements . This influenced the development
of Western mathematics for more than 2000
years.
Sruthi
Pythagorean Arithmetic
and Geometry
• Pythagoras was a Greek philosopher in 500BC. He is
best known for discovering the formula for finding the
hypotenuse on a right angled triangle, using the other
two sides.
• Around 518BC, Pythagoras started a school based on
religion and philosophy. The school had many followers,
and was to be found in Crotone in Southern Italy.
Pythagoras had started another school in Samos, which
he abandoned. Pythagoras named his followers the
Mathematikoi. They were pure mathematicians. They
followed strict rules of respect and humility. They had
very few possessions.
Ben
Ben
Sieve of Eratosthenes
• Eratosthenes was a Greek philosopher, but was also an author,
poet, athlete, geographer, and astronomer. He was the first to
calculate the circumference of the Earth.
• Eratosthenes was to first to conceive a method of finding all the
prime numbers up to a certain integer. This link explains more;
http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes. However, a brief
•
•
•
•
•
explanation is this;
1. Think of a continuous list of numbers from two to some
integer.
2. Cross out all multiples of two.
3. The next lowest, uncrossed number is a prime.
4. Cross off all of this number’s multiples.
5. Repeat step’s 3 and 4 until you have no more multiples.
Buffon’s (1777)
Needle Problem
Buffon’s needle problem
involves probabilities.
A bunch of needles are scattered on
a set of parallel lines and we need to
find the probability of a needle falling
on one of the lines.
The general formula for the
probability is
2l
p( x) 
d
Total Needles= 500 tosses
Vivek
Probability of crossing in this one
is..(107/500)*100= 0.214*100= 21.4%
Red needles/ needles on a line= 107