Loving heavenly Father we come to you this hour asking for your blessing and help as we are gathered together.

We pray for guidance in the matters at hand and ask that you would clearly show us how to conduct our work with a spirit of joy and enthusiasm.

Give us the desire to find ways to excel in our work. Help us to work together and encourage each other to excellence.

We ask that we would challenge each other to reach higher and farther to be the best we can be.

We ask this in the name of the Lord Jesus Christ

Amen.

Mathematics inEarly Civilization

Prepared by:Hailin K Bennet XI A

History of Mathematics

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 and12345678987654321 12345679 x 8 = 98765432

Levels of Math

Kindergarten – whole numbers and counting1st – addition, subtraction, measurements2nd – place value, base – 10 system, addition, subtraction, measurement3rd – addition, subtraction, place value, multiplication, division, fractions, geometry4th – multi-digit multiplication, fractions, decimals, mixed numbers, area5th – 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 figures7th – rational numbers, linear equations, proportionality and similarity, surface area and volume, probability and data8th – linear functions and equations, geometry, analysis of data sets9th - 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

MayansMayans are from ____________.

Base 20 system

One of the 1st cultures to invent _________.

Their calendar had 18 months a year; 20 days a month

MEXICO

ZERO

According to the Mayans:

The WORLD WOULD

21 DECEMBER 2012

HAVE ENDED on…

RomansRoman 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 stroke10= hobble for cattle100= coiled rope1000= lotus flower10000= finger100000= frog1000000= a god raising his adoration

Egyptian Fraction

The Eye of Horus

Egyptian

Arithmetic

Addition and Subtraction in

Egyptian Numerals365

+ 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

2346

92

184

123

8

Divide 299/23=?

Dividend: 1+4+8 = 13

Result: 23+92+ 184= 299

2346

92

184

123

8

31

Numbers that cannot divide evenly e.g.:

35 divide by 88 1

16 2

√ 32 4

4 1/2

√ 2 1/4

√ 1 1/8

35 4 + 1/4 + 1/8

doubling

half

EgyptianGeometr

y

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 25 problems with solution.• Older than the Rhind Mathematical Papryus.

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.• Larger than the Moscow

Mathematical Papyrus

AREAObject 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 17 in MMP

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 1/34. Reckon with these 12, 1 1/3 times. Result 165. Calculate thou its angle (square root). Result 4

for the length.6. 1/2 1/4 is 3 for the breadth.

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 RMPThe 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 MMPThe 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 CircleComputing 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 CircleComputing 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

VolumeObject 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

Arithmetic

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

Positional number system

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

Positional number system

(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.

Lack of 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.

FRACTION!!!

• Similarly the Babylonian Sexagesimal fraction 0;7,30 represented

7/60 + 30/3600

which again written in our notation is 1/8.

FRACTION!!!

• 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".

FRACTION!!!

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

Geometry

Theorems

Undefined Terms

Euclidean Geometry

Euclid (c 300 BC), Alexandria

Undefined Terms can be described but cannot be

given precise definitions using simpler known

terms.

3 Main Undefined Terms:

Undefined Terms

Point is thought to be a circular dot that is shrunk until it has no size.

Line is thought to be a wire stretched as tightly as possible of infinite length

having no thickness.

Plane is thought to be a sheet of paper that

has no thickness, stretched tightly , and

extending infinitely in all directions.

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

Undefined Terms

Euclidean Geometry

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 Problem2. Devise a Plan3. Carry Out the Plan4. 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.

POINT

A

LINE

A

B

l

ABLine l

plane

A B

C Plane ABC or Plane P

P

GREAT MATHEMATICIANS

Ptolemy (AD 83–c.168), Roman Egypt

• Almagest: comprehensive treatise on geocentric astronomy

• Link from Greek to Islamic to European science

Al-Khwārizmī (780-850), Persia

• 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

Leonardo of Pisa (c. 1170 – c. 1250) aka Fibonacci

• 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.

Cardano, 1501 —1576)• 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 Ferrari.

John Napier (1550 –1617)• 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.

• “MirificiLogarithmorumCanonisDescriptio,” contained 57 pages of explanatory matter and 90 of tables,

• facilitated advances in astronomy and physics

Galileo Galilei (1564-1642)

• “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.

René Descartes (1596 –1650)

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.

Blaise Pascal (1623 –1662) 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

Pierre de Fermat (1601–1665)• If n>2, then

a^n + b^n = c^n has no solutions in non-zero integers a, b, and c.

Sir Isaac Newton (1643 – 1727)

• 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....

Euler (1707 –1783) • 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

David Hilbert (1862 –1943)

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 mathematicians.”

John von Neumann ) (1903 –1957)

Norbert Wiener (1894-1964)

• 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 of society.

Claude Shannon (1916 –2001)]

• 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 master's thesis of all time

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