History Of Mathematics

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We ask this in the name of the
Lord Jesus Christ
Amen.

Prepared by:
Hailin K Bennet XI A

In 50,000 B.C. there was the first evidence of counting, which was the
same time of the Neanderthal man. Here’s irony; in 180 BC the 360
degree circle was defined. Just thirty years before the 360 degree
circle, the Great Wall of China began construction. Then in 140 BC,
the first Trigonometry was discovered followed by two forms of
Algebra in 830 AD by Al-Khowarizmi and in 1572 AD by Bombelli (the
more popular choice in text books). A few centuries later, we run out
of things to discover, so we improvise; which is what Matiyasevich
does in 1970 by proving that Hilbert’s Tenth Problem is unsolvable.
Six years later the Four Color Conjecture is verified by computer.
The key codes that we use every day to clock in at work, at our ATM,
to secure our house, etc. was introduced in 1977 by three
Mathematicians; Adelman, Rivest and Shamir. In 1994 Wiles proves
Fermat’s Last Theorem. And lastly, in 2000, mathematical challenges
of the 21st century here was announced.

Mathematics have come a great length. What people do not realize is
that mathematics are used everywhere; GPS, cellular phones,
computers, the weather, our television cable guide, sound waves (use
a sinusoidal wave or sine wave), clocks, speed limits, shopping,
balancing our checkbooks, etc. Math is used everywhere. Another
thought; if trigonometry was discovered first, why is it we take
Algebra before Trigonometry? Or how did they do Trigonometry
without knowing Algebra first? Hmm.
A couple of fun facts:
111111111 12345679 x 9 = 111111111
x111111111 and
12345678987654321 12345679 x 8 = 98765432

Levels of Math
Kindergarten – whole numbers and counting
1st – addition, subtraction, measurements
2nd – place value, base – 10 system, addition, subtraction,
measurement
3rd – addition, subtraction, place value, multiplication,
division, fractions, geometry
4th – multi-digit multiplication, fractions, decimals, mixed
numbers, area
5th – multi-digit division, adding and subtracting of fractions
and decimals, triangles, quadrilaterals, and algebra

6th – multiplication and division of fractions and decimals,
ratios, rates and percents, 2-3 dimensional figures
7th – rational numbers, linear equations, proportionality and
similarity, surface area and volume, probability and data
8th – linear functions and equations, geometry, analysis of
data sets
9th - 12th – Algebra 1, Geometry, Algebra 2, Trigonometry,
Statistics, and Calculus

Before the Ancient Greeks:
 Egyptians and Babylonians (c. 2000 BC):
 Knowledge comes from “papyri”
 Rhind Papyrus

Mayans
 Mayans are from ____________.
 Base 20 system
 One of the 1st cultures to invent
_________.
 Their calendar had 18 months a year; 20
days a month

Romans
 Roman Numeral System
 I, II, III, IV, V,…,X, L, C, D, M,…
 The system is based on Subtractive
Pairs
 The Line above Roman Numerals
means multiply by a thousand

Egyptian
Mathematics
(Introduction
)

Egyptians
• 1st to have fractions!
• Geometry invented by Egyptians
– Geo means earth; meter comes from measures
• Used geometry to measure land to assess
taxes

• Came closest to developing pi
– pi is the ratio of diameter of a circle to the
circumference of the circle

Egyptian Numeric Symb ols
The Egyptian zero symbolize beauty,
complete and abstraction.
The Egyptian zero’s consonant
sounds are “nfr” and the vowel
sounds of it are unknown.
The “nfr” symbol is used to
expressed zero remainders in an
account sheet from the Middle
Kingdom dynasty 13.

Numeric Symbols
1= simple stroke
10= hobble for cattle
100= coiled rope
1000= lotus flower
10000= finger
100000= frog
1000000= a god raising his adoration

Egyptian Fraction
The Eye of Horus

Addition and Subtraction in Egyptian
Numerals
365
+ 257
= 622

Egyptian Multiplication• Doubling the number to be multiplied
(multiplicand) and adding of the doublings to
add together.
• Starting with a doubling of numbers from
1,2,4,8,16,32,64 and so on.
• Doubling of numbers appears only once.
Examples:
a. 11= 1+2+8
b. 23= 1+2+4+16
c. 44= 4+8+32

 Applying the distribution law:
a x (b+c)=(a x b) + (a x c)
Example:
23 x 13= 23x (1+4+8)
= 23 +92 +184
= 299

Multiply 23 x 13
1+4+8 = 13
Result: 23+92+ 184= 299
23
46
92
184
1
2
3
8

Divide 299/23=?
Dividend: 1+4+8 = 13Result: 23+92+ 184= 299
23
46
92
184
1
2
3
8

31
Numbers that cannot divide
evenly e.g.:
35 divide by 8
8 1
16 2
√ 32 4
4 1/2
√ 2 1/4
√ 1 1/8
35 4 + 1/4 + 1/8
doubling
half

Egyptian Geometry
• Discusses a spans of time period ranging from
ca 3000 BC to ca. 300 BC.
• Geometric problems appear both the Moscow
Mathematical Papyrus( MMP) and Rhind
Mathematical Papyrus (RMP).
• Used many sacred geometric shapes like
squares, triangles and obelisks.

Moscow
Mathematical
Papyrus
• Golenishchev
Mathematical
Papyrus
• Written down 13th
century based on the
older material dating
Twelfth dynasty of
Egypt.
• 18 feet long, 1 ½ and 3
inches
wide and divided into

Rhind
Mathematical
Papyrus• Named after Alexander
Henry
Rhind
• Dates back during
Second
Intermediate Period of
Egypt
• 33 cm tall and 5 m long
• Transliterated and
mathematically
translated in the late 19th
century.

AREA
Object Source Formula (using
modern notation)
Triangle Problem 51 in
RMP and problem
4,7 and 17 in
MMP.
A= ½ bh
Rectangle Problem 49 in
RMP and problem
6 in MMP and
Lahin
LV.4.,problem1
A= bh
Circle Problem 51 in
RMP and
problems 4,7 and
A= ¼( 256/81)d^2

37
Area of Rectangle
Problem: 6 of MMP
Calculation of the area of a rectangle is used in a
problem of simultaneous equations.
The following text accompanied the drawn
rectangle.
1. Method of calculating area of rectangle.
2. If it is said to thee a rectangle in 12 in the area is 1/2 1/4 of the
length.
3. For the breadth. Calculate 1/2 1/4 until you get 1.
Result 1 1/3
4. Reckon with these 12, 1 1/3 times. Result 16

38
Area of Rectangle
Problem: 49 of RMP
• The area of a rectangle of length 10 khet
(1000 cubits) and breadth 1 khet (100 cubits)
is to be found 1000x100= 100,000 square
cubits.
• The area was given by the scribe as 1000
cubits strips, which are rectangles of land, 1
khet by 1 cubit.

39
Area of triangle
Problem: 51 of RMP
The scribe shows how to find the area of a triangle of land of
side 10 khet and of base 4 khet.
The scribe took the half of 4, then multiplied 10 by 2 obtaining
the area as 20 setats of land.
Problem: 4 of MMP
The same problem was stated as finding the area of a triangle
of height (meret) 10 and base (teper) 4.
No units such as khets or setats were mentioned.

40
Area of Circle
Computing π
Archimedes of Syracuse (250BC) was known as the first
person to calculate π to some accuracy; however, the Egyptians
already knew Archimedes value of
π = 256/81 = 3 + 1/9 + 1/27 + 1/81

41
Area of Circle
Computing π
Problem: 50 of RMP
A circular field has diameter 9 khet. What is its area? The
written solution says, subtract 1/9 of the diameter which leaves
8 khet. The area is 8 multiplied by 8 or 64 khet.
This will lead us to the value of
π = 256/81 = 3 + 1/9 + 1/27 + 1/81 = 3.1605
But the suggestion that the Egyptian used is
π = 3 = 1/13 + 1/17 + 1/160 = 3.1415

Volume
Object Source Formula
Cylindrical granaries RMP 41
Cylindrical granaries RMP 42, Lahun IV.3
Rectangular granaries RMP 44-46 and MMP 14
Truncated granaries MMP 14

Babylonian Math
• Main source: Plimpton 322
• Sexagesimal (base-sixty) originated with ancient
Sumerians (2000s BC), transmitted to
Babylonians … still used —for measuring time,
angles, and geographic coordinates

Babylonian
• The Babylonian number system is old. (1900 BC to 1800 BC)
• But it was developed from a number system belonging to a
much older civilization called the Sumerians.
• It is quite a complicated system, but it was used by other
cultures, such as the Greeks, as it had advantages over their
own systems.
• Eventually it was replaced by Arabic Number.

After 3000 B.C, Babylonians
developed a system of writing.
Pictograph-a kind of picture
writing
Cuneiform - Latin word “cuneus” which means “wedge”
Sharp edge of a stylus made a vertical stroke (ǀ)
and the base made a more or less deep impression (∆).
The combined effect was a head-and-tail figure resembling a wedge .

• Like the Egyptians, the Babylonians used to
ones to represent two, and so on, up to nine.
• However, they tended to arrange the symbols
in to neat piles. Once they got to ten, there
were too many symbols, so they turned the
stylus on its side to make a different symbol.
• This is a unary system.

• The symbol for sixty seems to be exactly
the same as that for one.
• However, the Babylonians were working
their way towards a positional system

• The Babylonians had a very advanced
number system even for today's standards.
• It was a base 60 system (sexigesimal)
rather than a base 10 (decimal).

• When they wrote "60", they would put a single
wedge mark in the second place of the numeral.
• When they wrote "120", they would put two
wedge marks in the second place.

• A positional number system is one where the
numbers are arranged in columns. We use a
positional system, and our columns represent
powers of ten. So the right hand column is units, the
next is tens, the next is hundreds, and so on.
(7 ∙ 100) + (4 ∙ 10) + 5 = 745
10^2 = 100 10^1 = 10 10^0 = 1
7 4 5
123

• The Babylonians used powers of sixty rather than
ten. So the left-hand column were units, the second,
multiples of 60, the third, multiplies of 3,600, and so
on.
(2*602) + (1*60) + (10 + 1) = 7271
(2*3600)+
(1*60) +
(10 + 1)
=7271

x 3600 x 60 Units Value
1
1 + 1 = 2
10
10 + 1 = 11
10 + 10 = 20
60
60 + 1 = 61
60 + 1 + 1 = 62
60 + 10 = 70
60 + 10 + 1 = 71
2 x 60 = 120
2 x 60 + 1 = 121
10 x 60 = 600
10 x 60 + 1 = 601
10 x 60 + 10 = 660
3600 (60 x 60)
2 x 3600 = 7200

 They had no symbol for zero. We
use zero to distinguish between
10 (one ten and no units) and 1
(one unit).
 The number 3601 is not too
different from 3660, and they are
both written as two ones.
 The strange slanting symbol is the
zero.
123

• The Babylonians used a system of
Sexagesimal fractions similar to our
decimal fractions.
For example:
if we write 0.125 then this is
1/10 + 2/100 + 5/1000 = 1/8.

• Similarly the Babylonian Sexagesimal fraction
0;7,30 represented
7/60 + 30/3600
which again written in our notation is 1/8.

• We have introduced the notation of the
semicolon to show where the integer part
ends and the fractional part begins.
• It is the “Sexagesimal point".

Greek Mathematics
• Thales (624-548)
• Pythagoras of Samos (ca. 580 - 500 BC)
• Zeno: paradoxes of the infinite
• 410- 355 BC- Eudoxus of Cnidus (theory
of proportion)
• Appolonius (262-190): conics/astronomy
• Archimedes (c. 287-212 BC)

 Three Types of Geometry:
 Euclidean (what we will study)
 Non- Euclidean
 Elliptic Geometry (Spherical Geometry)
 Hyperbolic Geometry

Theorems
Postulates
and Axioms
Definitions
Undefined
Terms

Undefined Terms can be
described but cannot be
given precise definitions using
simpler known terms.
3 Main Undefined Terms:
Undefined
Terms

Using the undefined terms (point, line, plane)
allows us to define other terms in geometry,
e.g. space: a set of all points
Definitions

Postulates and
Axioms
Postulates and axioms are one in
the same.
They are accepted as
statements of fact.

Theorems
Theorems are results that are deduced
from undefined terms, definitions,
postulates,
and/ or
results that follow from them.

Theorems
Postulates
and
Axioms
Definition
s
Undefined
Terms

HOW TO SOLVE GEOMETRY?
Scientific Method?
 Ask a Question
 Do Background Research
 Construct a Hypothesis
 Test Your Hypothesis by Doing an Experiment
 Analyze Your Data and Draw a Conclusion
 Communicate Your Results

HOW TO SOLVE GEOMETRY?
Four Steps:
1. Understand the Problem
2. Devise a Plan
3. Carry Out the Plan
4. Look Back

1. UNDERSTAND THE PROBLEM
 Is it clear to you what is to be found?
 Do you understand the terminology?
 Is there enough information?
 Is there irrelevant information?
 Are there any restrictions or special
conditions to be considered?

2. DEVISE A PLAN.
 How should the problem be
approached?
 Does the problem appear similar to any
others you have solved?
 What strategy might you use to solve
the problem?

3. CARRY OUT THE PLAN
 Apply the strategy or course of action
chosen in Step 2 until a solution is
found or you decide to try another
strategy

4. LOOK BACK
 Is your solution correct?
 Do you see another way to solve the
problem?
 Can your results be extended to a more
general case?

SOME STRATEGIES…
 Draw a picture!
 Guess and check.
 Use a variable.
 Look for a pattern.
 Make a table.
 Solve a simpler problem.

WHAT TO DO WHEN YOU START WORKING A
GEOMETRY PROBLEM?
1. Understand the problem.
2. Devise a plan.
3. Carry out the plan.
4. Look Back.

• Almagest: comprehensive treatise on
geocentric astronomy
• Link from Greek to Islamic to European
science

• Algebra, (c. 820): first book
on the systematic solution
of linear and quadratic
equations.
• he is considered as the
father of algebra:
• Algorithm: westernized
version of his name

• Brought Hindu-Arabic
numeral system to Europe
through the publication of his
Book of Calculation, the Liber
Abaci.
• Fibonacci numbers,
constructed as an example in
the Liber Abaci.

• illegitimate child of Fazio
Cardano, a friend of Leonardo
da Vinci.
• He published the solutions to the
cubic and quartic equations in
his 1545 book Ars Magna.
• The solution to one particular
case of the cubic, x3 + ax = b (in
modern notation), was
communicated to him by
Niccolò Fontana Tartaglia (who
later claimed that Cardano had
sworn not to reveal it, and
engaged Cardano in a
decade-long fight), and the
quartic was solved by
Cardano's student Lodovico

• Popularized use of the (Stevin’s)
decimal point.
• Logarithms: opposite of powers
• made calculations by hand much
easier and quicker, opened the
way to many later scientific
advances.
• “MirificiLogarithmorumCanonisDes
criptio,” contained 57 pages of
explanatory matter and 90 of
tables,
• facilitated advances in astronomy

• “Father of Modern Science”
• Proposed a falling body in a
vacuum would fall with uniform
acceleration
• Was found "vehemently suspect of
heresy", in supporting Copernican
heliocentric theory … and that one
may hold and defend an opinion
as probable after it has been
declared contrary to Holy
Scripture.

 Developed “Cartesian
geometry” : uses algebra
to describe geometry.
 Invented the notation using
superscripts to show the
powers or exponents, for
example the 2 used in x2 to
indicate squaring.

 important contributions to
the construction of
mechanical calculators, the
study of fluids, clarified
concepts of pressure and.
 wrote in defense of the
scientific method.
 Helped create two new
areas of mathematical
research: projective
geometry (at 16) and
probability theory

• If n>2, then
a^n + b^n = c^n
has no solutions
in non-zero
integers a, b,
and c.

• conservation of momentum
• built the first "practical" reflecting
telescope
• developed a theory of color based on
observation that a prism decomposes
white light into a visible spectrum.
• formulated an empirical law of
cooling and studied the speed of
sound.
• And what else?
• In mathematics:
• development of the calculus.
• demonstrated the generalised
binomial theorem, developed the so-
called "Newton's method" for
approximating the zeroes of a
function....

• important discoveries in
calculus…graph theory.
• introduced much of modern
mathematical terminology and
notation, particularly for
mathematical analysis,
• renowned for his work in
mechanics, optics, and
astronomy.
• Euler is considered to be the
preeminent mathematician of the
18th century and one of the
greatest of all time

 Invented or developed a
broad range of
fundamental ideas, in
invariant theory, the
axiomatization of geometry,
and with the notion of
Hilbert space

• major contributions set theory,
functional analysis, quantum
mechanics, ergodic theory,
continuous geometry, economics
and game theory, computer
science, numerical analysis,
hydrodynamics and statistics, as well
as many other mathematical fields.
• Regarded as one of the foremost
mathematicians of the 20th century
• Jean Dieudonné called von
Neumann "the last of the great

• American theoretical and
applied mathematician.
• pioneer in the study of stochastic
and noise processes, contributing
work relevant to electronic
engineering, electronic
communication, and control
systems.
• founded “cybernetics,” a field
that formalizes the notion of
feedback and has implications
for engineering, systems control,
computer science, biology,
philosophy, and the organization

• famous for having founded
“information theory” in 1948.
• digital computer and digital
circuit design theory in 1937
• demonstratedthat electrical
application of Boolean
algebra could construct and
resolve any logical, numerical
relationship.
• It has been claimed that this
was the most important

ReferencesFrom online
• http://www.math.wichita.edu/history/topics/num-sys.html#babylonian
• http://www.math.wichita.edu/history/topics/num-sys.html#babylonian
• http://www.slideshare.net/Mabdulhady/egyptian-mathematics?from_search=1
• http://en.wikipedia.org/wiki/Egyptian_mathematics
• http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_arith.html

History Of Mathematics

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