More Islamic math

Al Karaji (?- 1019) wrote "The Marvelous" and studied algebra of exponents. So
1:x= x:x^2 =x^3: x^2.....

He also gave a limited algorithm for square roots of a polynomials. Al Karaji also proved
1^3 +2^3 +....+10^3 = (1+...+10)^2 (which you may have seen in some "intro to proof" class) by inverse induction.

His proof as follows: Using the diagram above
Area(red square) =  (1 + 2 +  · · · + 9)²
Area (yellow rectangle) =  10(1 + 2 +  · · · + 9)
Area(blue square) = 10² 

Then,
Area(2 yellow rectangle + blue square) = 2* 10(1 + 2 +  · · · + 9) + 10²
                                                                   = 2*10(9*10/2)  + 10² 
                                                                  = 10 * 10²
                                                                  = 10³      (1)

Area (of Entire gnomon) = (1 + 2 +  · · · + 9)² +2*10(1 + 2 +  · · · + 9) +10²

 Plugging (1) into the above equation yields
Area (of Entire gnomon) = (1 + 2 +  · · · + 9)² + 10³
                                           = (1 + 2 +  · · · + 9+ 10)²


By repeating a similar argument for the rest of the number yields Al karaji's results
 1^3 +2^3 +....+10^3 = (1+...+10)^2

QED

Al Samaw al (1112-1174) wrote "Shining Book of Calculation" and performed long division.

Unanswered Question

1. Was there any practical purpose for Al karaji discovery of  1^3 +2^3 +....+10^3 = (1+...+10)^2 ?

Evaluation:
Interesting: 8
Quality: 7
Complexity: 7

Mathematics of Islam

A library in Baghdad was established by the caliph Harun al Rashid (786-809) and books that were collected where translated in Arabic. Rashid successor , his son, Al Mamun (813-833) established a research institute call Bayt al Hikmat or House of wisdom; the institute lased 200 years. Despite being a center of learning, the House of wisdom was restricted by belief. Scholars were invited to the institute to translate work and conduct research. They translated many work by the end of the 9th century such as Euclid, Archimedes and other Greek Mathematicians. At the institute they also learn Babylonian mathematics.

Al Khwarizmi(~825) worte "The condensed book on calculations of al-jabra and al-muqabola.
Some now and then:
"al- jabra"====> "algebra
"Sunya"====>sifr===>Zephirum====> zero
sifr====> cifra===>?(guess)

Al Khwarizmi solved 6 types of equatioins

1. square equal to roots (such as ax^2 = bx)
2. square equal to numbers
3. Roots equal to number
4. squares and roots equal to numbers (ax^2 + bx = c)
5. squares and numbers equal to roots
6. roots and numbers equal to squares

This showed a move from concrete math to abstract math


Unanswered Question:

1. How heavily was the House of wisdom restricted by religion? Did it interfere with any mathematical development?

Evaluation:
Interesting: 7
Quality: 8
Complexity:  5

Chinese remainder theorem

One of the most famous technique that comes from china is know as the Chinese remainder theorem. The theorem is used to solve problems dealing with system of linear congruence and its earliest uses is dated back in the book Mathematical Classic Master sun.

Example:
"We have things of which we do not know the number: if we count them by threes, the remainder is 2; if we count them by fives, the remainder is 3; if we count them by sevens , the remainder is 2. How many things are there?"

This is translated into
N= 3x+2
N=5y+3
N=7z+2

or for those with a background in number theory
N== 2(mod 3)
N== 3(mod 5)
N== 2 (mod 7)

where == stands for congruence. For simpler purpose, we will use the earlier notation. So,
1) 3*7=21
     21 /5 give you a remainder 1
    so 3*21= 63

2) 3*5= 15
    15/ 7 gives you a remainder 1
  so 2* 15= 30

3) 5*7 = 35
35*2= 70   we multiply again because 35/ 3 gives you a remainder 2. We want a remainder of 1
70/3 gives you a remainder 1
70*2 = 140


Then adding 63 + 30 + 140 = 233
so, 233-210 = 23                             210= 2(3*5*7)
thus N= 23

When plunging in our answer we see that it satisfies the system of congruences.
Did you notice any patterns during the steps? 
In number theory terms:

• We multiplied two mods together and then divided by the third and we did that for each of the mod
• When we divide the product by the third mod and got a remainder of 1 we multiplied the product by the remainder of that mod

Unanswered Question:

1. Was the Chinese remainder theorem very applicable? I find it hard to find a situation where the objects are unable to be counted

Evaluation:
Interesting: 7
Complexity: 6
Quality: 8

Millenium prize problem

The Millennium prize problems are seven problems in mathematics where a  prize of $1,000,000 is given to the person who can solve one of them. So far only one problems has been solved: the Poincare conjecture. The Poincare conjecture is one of the most important question in the topology field.

Poincare conjecture:
   Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

 Grigori Perelamn presented a proof of the conjecture in 3 papers. He was offered the prize money and a field medal to which he both declined, stating that it was not only him working on the problem; that people as far back as Hamilton has been working on and contributing to the proof of the conjecture. His proof was honored "Break through of the year".

The remaining Millenium prize problem are

1. P versus NP problem
2. Hodge conjecture
3. Riemann hypothesis
4. Yang–Mills existence and mass gap
5. Navier–Stokes existence and smoothness
6. Birch and Swinnerton-Dyer conjecture

Maybe you'll be the next to solve these famous problems

Unanswered Question:

1. How did the University, that propose these problems, know that these conjectures are true? Are they basing these conjectures on repeated observations?

Evaluation:
Complexity: 5
Interesting: 8
Quality: 7

Final chapters of Greek mathematics

By Ptolemy's time

• the entire eastern Mediterranean was part of the Roman Empire
• the local rulers and romans left the local language and culture intact.

and the great Cicero admitted that the Romans were not that interest in math. 

The last of the Greek mathematians were

1. Nicomachus of Gerasa
2. Diophantus of Alexandria
3. Pappus
4. Hypatia

Not much is know about Nicomachus although it is speculated that he studied in Alexandria. He dealt with perfect primes where he calculated the first four: 6, 28, 496 and 8128. He, however did not present any proof. Besides perfect numbers, Nicomachus also worked with pentagonal, hexagonal and heptagonal numbers.

Pentagonal Numbers

Nicomachus used a proportion in a different sense form Euclid"s. For Nicomachus there was 3 types of proportions:

1. arithmetic proportion- consecutive pair of numbers differ by the same quantity (for example 1,5,9,13 ... etc differ by 4)
2. geometric proportion- "the greatest terms is to the next greatest as that one is to the next" (for example 3,9,27...etc)
3. Harmonic proportion-"greatest term is to the smallest as the difference between the great and mean terms is the the difference between the mean and the smallest terms" (for example: 3,4,6 are harmonic proportion since 6:3 = (6-4): (4-3)

Diophantus worte arithmetica which is divided into 13 books. He introduced symbolic abbreviations and dealt with powers higher then 3, a break frmo the traditional Greek usage.He also presented some intersting problems

1. To divide a give number into two having a given difference
2. To divide a given number into two number such that given fractions (not the same) of each number when added together produced a given number
3. To divide a given square into two squares

Pappus is best know for his Collections,  which are works on various topics in mathematics. Through one of his remarks we learned that women where involved in mathematics in Alexandria.

Hypatia(355-415) was the 1st women mathematician of whom any details are know and her violent death marked the effective end of the Greek mathematician tradition in Alexandria.

Hypatia

Hypatia wa a respected and eminent teach in Alexandria. She was given a thorough education in math and philosophy by her father, Theon of Alexandria. Despite having influential friends, when a rival of her friends spread rumors about her about practicing sorcery a group of citizens gruesomely ended her life.

Heron wrote a detailed work on indirect measurement. For instance, we have two points, A and B, on one inaccessible side of a river (where the observer is on the other side). We want to find the distance between A and B. Heron used similar triangles.

We can measure the distance CA. And the two similar trianlges are DAB and CED.
So we have

CD/AD = CE/AB

So now we can solve for AB

Unanswered questions:

1. How steady was the decline in Greek mathematics? Did it slowly start to fall apart due to political strife and war or was it a long drawn out process?

Evaluations
Interesting: 8
Complexity: 7
Quality:  8

Astronomy and Ptolemy

Ptolemy(100- 178) also made numerous observations of the heavens and wrote several books, one being the Matematical Collections, which consisted of 13 books. The collection was the most influential astronomical work from its time. Islamic scientist bgan calling the book al-magisti or Almagist.

Ptolemy also gave a theorem, which helps derive sum and difference formulas of sine and cosine:


Ptolemy's Theorem:
Given any quadrilateral inscribed in a circle, the product of the diagonals equals the sum of the products of teh opposite sides: AC*BD = AB*DC + AD*BC

The proof of Ptolemy's theorem involves finding 2 pairs of similar triangles,in the diagram above, thus getting 2 different ratios. Then combining the 2 ratios yields the desired equation. With tables of trigonometric ratios the astronomers were able to solve right triangles easily and approach Apollonius' questions.

Unanswered Questions:

1. Did Ptolemy's Theorem have any other practical uses other than find trig identities?

Evaluation:
Interesting: 8
Quality: 7
Complexity:  7

Astronomy

The main reasons that mathematicians studied astronomy was to solve problems dealing with

• the calendar (such as determining the seasons)
• predictions of eclipses
• finding the beginning of the lunar month

In studying astronomy, the mathematics  of Greece created a plane and spherical trigonometry and developed a mathematical model of the universe.

Some terms:
Celestial equator- the axis of which the daily rotation of teh celestial sphere takes place.

vernal and autumnal equinoxes- the points of which the equator and ecliptic intersects one another

The 7 wanderers- They participated in the daily east to west rotation of the celestial sphere but also had their own motions. The 7 wanderers were the sun, the moon, Mercury, Venus, Mars Jupiter, and Saturn.

Great Circle-  A section of a sphere by a lane trough its center. For example the equator is one great circle. The earth has infinitely many great circles.

• The ecliptic, is a great circle that passes through the 12 constellations of the zodiac.

The major contributors to mathematical astronomy were

1. Eudoxus
2. Apollonius
3. Hipparchus
4. Menelaus
5. Ptolemy

Eudoxus was responsible for turning astronomy into mathematical science and is accredited with the invention of the two sphere model. He regarded the models as computational devices.

Hipparchus (190-120) began calculating trigonometric ratios. He also did numerous observations of planetary positions. For his trigonometry, he used chords subtending an arc or central angle in a circle with a fixed radius

Unanswered Question

1. Where there any other attempts to created a spherical model of the Universe? If so, how did they explain the heaven's movement? How did it fail?

Evalueation:
Quality: 8
Complexity: 7
Interesting: 7

Parabolic section

Area(Dark blue triangle) = (1/4)area( light blue triangle)
note: this formula can be applied to any triangle that is drawn in. In general the area of the newer(smaller) triangle will be 1/4th the area of the triangle that was drawn previously.




The formula will be used to find the sum of series in "Quadrature of the parabola" or to approximate the area of a segment of parabola inscribing triangles. Archimedes, using method of exhaustion, constructed more triangles and the area of the triangle, like described above, had 1/4 the area of the triangle drawn in the previous step. The area of the segmet becomes:
                   a +(1/4)a +[(1/4)^2]a+[(1/4)^3]a+...... = k        (any similarities to the geometric series?)

Then k + (1/3)(k-a)=  (4/3)a


Unanswered Questions

1. What were the uses for knowing the area of a segment of paraobla inscribing trianlges  or knowing that a area(one triangle)= (1/4) area(another triangle)?

Evaluation:
Quality: 9
Interesting: 6
complexity: 9

Archimedes and Lemmas

We know that the area of a circle is give by the formula: Area(Circle) = Pi(r^2). However, Archimedes came up with another formula and achieved the same answer: The area of a circle is equal to the area of a right triangle whose legs are the radius and circumference.

One way to see Area(circle) = (1/2)rc is to cut a circle into pieces(or slices) and arrange them as show below. As the slices become thinner (thus making the area more accurate) the rectangle becomes closer to having the sides r and c/2

 

In Archimedes' Measurement of the Circle was two lemmas used to calculate the ratios to the diameter of the perimeters of  regular polygons.

Lemma 1
Suppose OA is the radius of a circle and CA is tangent to the circle at A. Let DO bisect angle(COA) and intersect the tangent at D. Then DA/OA = CA/ (CO+ OA).

The proof of lemma 1 involves extending the line CO to COE where E is a point of the circle and connecting E and A. From there, angle(DOC) is equal to angle(AEO)  which imples that DO is parallel to AE. Using a property involving parallel lines intersecting 2 lines one can derive the desired result.


Lemma 2
Let AB be the diameter of a circle and ACB a right triangle inscribed in the semicircle. Let AD bisect angle(CAB) and meet the circle at D. Connect DB. The (AB/BD)^2 = 1 + [(AB+Ab)/ BC]^2

The proof of Lemma 2 involves manipulating the equation and similar triangles

The first lemma was repeatedly used to develop an algorithm for determining the desired ratio using the circumscribed polygons. Archimedes used the second lemma to get an algorithm for inscribed polygons

Unanswered Question:

1. Is there an advantage to using one formula, for the area of a circle, over the other? Which one appears more in the real world?

Evaluation
Interesting: 8
Quality: 8
complexity: 7 

Levers and Archimdes

Archimedes (287-212 BCE) has more information known about him then any other Greek mathematician. As the son of an astronomer, Phidias, he is credited with the invention, that is used for raising water for irrigation the Archimedian screw. He spent his time solving various problems for Hiero and his successor and it is noted that Archimedes, being very dedicated to his work, would neglect his health and hygiene. For month it was Archimedes military engineering that kept the Roman army  at bay for months during the siege of Syracuse, however eventually the Roman entered the city. During all this chaos Archimedes was working on a math problem and did not obey the Roman solider,  who found him, to come follow him to Marcellus. The solider ended taking his life.

No one before Archimedes had created a mathematical model of the lever by which one could derive a mathematical proof of the law of levers
4 of the 7 Postulate that Archimedes stated in his Planes in Equilibrium

A and B)
Equal weights at equal distances are in equilibrium, and equal weights at unequal distances will have the lever incline towards the weight at a greater distance

C)
When weights are at certain distances are in equilibrium and if more weight is added to one side then the lever will incline towards the side were the weight was added

E)
When weights are at certain distances and weight is taken away from one then the lever will incline towards the side where no weight is taken away.

F)
If magnitudes at certain distances are in equilibrium, other magnitudes equal to them will also be in equilibrium at the same distances

 The Law of the Lever is stated in Proposition 6 and 7:

Two magnitudes, whether commensurable [prop 6] or incommensurable [prop 7], balance at distances inversely proportional to the magnitudes.

Archimedes gave the first proof of the law of the lever and with that law Archimedes found the center of gravity for various figures

Propositions leading to the law of lever

Prop1
Weights which balance at equal distances are equal

Prop2
Unequal weights at equal distances will incline toward the greater weight

Prop3
Suppose A and Bare unequal weights with A> B which balance at point C. Letting AC=a and BC=b then a<b. Conversely, if the weights balance and a<b then A>B

Unanswered Question

1.  What other application does the law of the lever have besides finding the center of gravity?

Evaluation:
complexity: 7
interesting: 7
quality:  8

Construction (with only a ruler and compass)

Surprisingly  there is quite a few things that you can construct with just a ruler and a compass such as

1. A triangle of sides a,b,c
2. A square of side a
3. a perpendicular bisector
4. A square inside a circle
5. A regular hexagon inside a circle
6. An (equilateral) triangle inside a circle

For the first one note that a triangle can not be constructed if a+b>c

•  Starting at one end of the length of a, with the compass stretch out to the lenght of b, mark an arch. do a similar process for the length of a at the other end of a.
• The intersection of the two arcs will be the third point in the abc triangle. 

In drawing a perpendicular bisector to a line AB

• With the compass get the length of AB
• At both ends of AB, using the compass and keeping the length of AB, draw a large enough arc so that there is two intersection
• Align the ruler with the two intersection and draw a line (the resulting line is the perpendicular bisector)

 A square inside a circle
This construction is quite similar to the previous one. In a circle with diameter AB

• draw a perpendicular bisector like before and long enough that the bisector instersects the circle on both end
• Connect the 4 points on the circle which yields a square in the circle

Unanswered Question

1. Was there any practical application, back then,  of constructing with just a ruler and a compass?

Evaluation:
Complexity: 4
Interesting: 5
Quality: 8

Euclid Book 2 propositions

Before diving into some propositions from Euclid's second book some language barriers:

•  The statement "the rectangle contained by 'a' and 'b' " meant a rectangle with a the side length 'a' and ''
• The statement "a together with b" means a+b

Proposition II 4
If a straight line is cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.

In modern terms: This is just squaring a binomial (a+b)(a+b)
note:

• The whole is (a+b) so squaring the whole would yield (a+b)^2
• The segments would be a and b
• rectangle contained by the segments will be ab

Proposition II- 5:
If a straight line is cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.

In modern term: (b-x)x+[(b/2)-x]^2 = (b/2)^2
note:

• (b/2)-x  is CD

• Let b (or AB) be a line and divide it into equal and unequal segments
• The equal segment is b/2 or AC and CB
• The unequal segment is AD and DB

Proposition II 11
To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment

In the above diagram the straight line is 'a' where a= x+a-x
In modern term the proposition states that a(a-x)= x^2
note: 
Unanswered Question

1. In book 1, all the propositions lead to the proof of the Pythagorean theorem so do all the proposition, in book 2, lead to any major theorem?

Evaluation:
Complexity: 7
interesting: 8
quality: 7

Zero

All numbers are represented with 10 digits (0,1,2,....9) using a positional system. A number system in which the value of a digit depends on its position. For example,
 1+1 = 2  ====> one +one =two
11+1 = 12  ===> eleven +two = twelve

Even though those equations are only represented by two different numbers, 1 and 2, depending on the place of the digit changes the value of the characters.

The Babylonians were the first to invent a positional system and in the case of a value was "empty" a blank space was used. Around 300 BC they developed a symbol to denote an empty place value:

For now we will regard a value that is empty as zero.
There are two ways to use zero

1. Placeholder (which is what the Babylonians and the Mayans considered zero as)
2. Number (India)

Any interesting fact is that the Babylonians never "discovered" zero as a number. It wasn't until around 458AD Zero become know as a number.

A funny video about zero that my classmate and i enjoyed:

Unanswered Question:

1. How exactly did zero go from a place value to an actual number?

Evaluation:
Interesting: 8
Complexity: 5
Quality:7

Pythagorean theorem proof and Euclid

Today a group did a presentation on the Pythagoras and the Pythagorean theorem proof. Their proof for the theorem is as followed:
Proof:
We have a right triangle

and with the hypotenuse we can construct a square with side length c. The the Area( big square) = c^2. Within the square, the right hand triangle can be placed in such a way that it can fit into the square 4 times.

The area of the big square can be also be calculated by finding the area of the individual shapes and summing them together.
Area(big square) = 4*Area(triangle)*Area(small square)
                   C^2 = 4[(1/2)ba] *[(b-a)(b-a)]
                          = 2ba*[(b^2)-2ba+(a^2)]
                         = a^2 +b^2

Thus c^2 = a^2 +b^2                                                                                           QED

Little is know about Euclid of Alexandria, a Greek mathematician; he is most famous for his book, Elements, which is broken down into 13 books. Elements is responsible for motivating famous mathematicians to become mathematicians. In book I, Euclid proves various theorems, all which help him prove a proposition at the end (the Pythagorean theorem). His proof of the Pythagorean theorem is different from the one mention above.

Euclid's proof of Pythagorean theorem

Some proposition Euclid also proved include

1. One a given straight line to describe a square
2. If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal sides also equal, then the two triangles are congruent.
3. If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another.

 In one of the proposition, prop I- 41, Euclid uses the phrase "in the same parallels" and in modern terms it would mean that two figures have the same height. The author, Katz, stated that Euclid, on occasion, relied on diagrams more than modern mathematicians would allow.

Unanswered question:

1. What other proofs of the Pythagorean theorem exist?

Evaluation:
Interesting: 4
quality:7
complexity: 6

Plato's famous student

Aristotle(384-322) was Plato's most famous student and rival. He studied at Plato's academy in Athens from the age of 18 until Plato's death in 347 (20 years total). He was the private teacher of Price Alexander, who later became a  successful king conquering the Mediterranean world. Once Alexander took over the throne, Aristotle became his trusted advisor. Later, like Plato, Aristotle founded his own school in Athens, the Lyceum, where he spent the rest of his life writing and teaching. Aristotle introduce written examination. In regards to his studies, Aristotle studied 3 related topics in mathematics: incommensurable, infinity(potential and actual infinity) and continuity.

He believed that logical arguments should be constructed out of syllogisms. Syllogism are "statements that are taken as true and certain other statements that are then necessarily true"(pg 43 Katz). In a modern way, they are the if-then statement students learn in an introduction proof class. An example the author gives in the book is 'if all monkeys are primates, and all primates are mammals, then it follows that all monkeys are mammals." (pg 43)

Aristotle distinguished between a postulates and an axioms. To a modern students they are just facts to memorize but he regarded a postulate as "basic truths that are peculiar to each particular science" and an axiom as truths that are common to all.

Numbers and Magnitude was also clarified by Aristotle. Magnitude(which is continuous) is "divisible into divisibles that are infinitely divisible while a number(which he claimed is discrete) are made up indivisible units.

Two things are in succession if there is none of their kind in between them. Aristotle claimed things are continuous "when they touch and when 'the touching limits of each become one and the same'" (pg 45 Katz)

Although Aristotle emphasized syllogism Greek mathematicians chose to use other forms to build their logical argument. Chrysippus of Soli(280-206) was one of the more notable people that studied the the basic forms of argument used to construct mathematical proof. The basic rules of inference, which is what Chrysippus called the forms of arguments are based on propositions, statements that can be either true or false.

Rules of inference:

1. Modus ponens: If p, then q. [ex: p. Therefore q]
2. Modus tollens: If p then q.   [ex: not q. Therefor no p]
3. Hypothetical syllogism: If p, then q. If q, then r. Therefore if p, then r.
4. Alternative syllogism: p or q [ex: Not p. Therefore q.]

Unanswered question:

1. Is there an advantage of using the rules of inference over syllogisms since many Greek mathematics did not use the later?
2. Did Aristotle use syllogism himself?

Evaluation:
Interesting: 7
quality: 8
complexity: 5

More mathematicans and famous problems

Given the length of a square can you find another square whose area will be twice the area as the first? Once options is guess and checking, but that can take a while. Instead there is an algorithm. Say we have a square, labeled PQRS with side length "p":


 Then area(PQRS) = p^2








The area of the square that we want, Sqaure(ABCD) needs to be twice the area as square(PQRS) so
Area(ABCD) = 2* area(PQRS)
                a^2= 2 *(p^2)
===> thus a= p*sqaure root(2)

Thus you have your new square with twice the area as the first one.

On a related note from last post: another mathematician was Plato(429-347). He was a student of Socrates for 8 years. He traveled about Greece and visited Egypt before journeying to southern Italy and Sicily. Plato perused the truth and wanted to master the 'art of argument'. He was the founder of the Academy in Athens and above the entrance of the academy was the phrase 'Let no one ignorant of geometry enter here'. Plato believed that a "student 'ignorant of geometry' would also be ignorant of logic and hence unable to understand philosophy". (pg 41 Katz)

Plato(left) and Aristotle, his student(right)

Portrait of the academy

 

Academy of Athens today

Plato restricted geometry to a compass and a straight edge since he believed any other instrument would lead you astray.

Plato wrote that philosophers and kings, the ideal rulers of a state, should learn mathematics to exercise their mind and "'to practice calculations, not like merchants or shopkeepers for purposes of buying and selling, but with a view to war and to help in the conversion of the soul itself from the world...."(pg 41 Katz)

Another notable mathematician was Hippocrates of Chios. He was one of the first to try 2 of the 3 famous Greek problems: to square a circle and doubling a cube. Hipporates was able to make progress in squaring a circle by being able to square a lune( figures founded by arcs of two circles). To square a lune is to show that "their areas could be shown equal to certain regions bounded by straight lines." (pg 41 Katz). 

Area(Lune BDC) = Area(triangle ABC)

Unanswered question:

1. Is their a practical purpose for squaring a lune? 
2. How does squaring a lune help towards squaring a circle?

Evaluation:

Interesting: 8

quality: 7

complexity:7

Greece

The people of Greece started to ask "why?" which lead to them developing the idea of logic. One mathematician, Thales of Miletus, proved various theorems that some we have seen in previous math classes such as:

• Two angles between two intersecting straight lines are equal   

•  An angle in a semi-circle is a right angle

Another mathematician, Pythagoras, started a school in Greece and helped distinguished odd and even numbers using pebbles as well as identify square numbers and triangular numbers. Pythagoras and his students believed that "numbers was the basis of the universe, everything could be counted, including lengths" and to the Greeks, a number was a 'multitude composed of units.' However that belief was broken with the discovery of the Pythagorean triple. Using the knowledge that the difference of 2 consecutive square numbers is an odd number, Pythagoras was able to construct the Pythagorean triple.

Pythagorean triple

• n is odd, the triple was : ( n, [(n^2)+1]/2 , [(n^2)-1]/2 )
• n is even, the triple was: (n, ((m/2)^2)-1, ((m/2)^2)+1 )

The numbers that are given from these triples satisfy the Pythagorean theorem( which, interestingly, has be known to other cultures long before Pythagoras lived). From these triples they encounter a triangle that did not up hold their belief:

The Pythagoreans assumed that one could always find a common measure, the unit, between two length; however upon looking closely they could not find a unit between the leg,1, and the hypotenuses (they were incommensurable), the square root of 2 and thus breaking their belief. The idea that their were numbers that could not be measured open up the possibility to new mathematical theories. The Pythagorean theorem would lead to the discovery of irrational numbers.

note: 1 was not a number to the Greeks; it was a unit. 2,3,4,5, etc were consider numbers(composed of unit)

Unanswered question

1. Why did the Greeks did not consider 1 as a number? Did they think fractions as numbers or units since multiple fractions can make whole numbers?
2. If the Pythagorean theorem, as it is know now,  was known to others cultures before Pythagoras birth why was it named after him and his students?

Evaluation
Interesting  7
quality       6
complexity 8

Lecture: More on Egyptian

Since the Egyptians did not like to used complex fractions, except 2/3, it would be convenient to have a simple way to express them as a unit fractions. As an example take 5/6

5/6 = (2+3)/6
      = 2/6 + 3/6
      = 1/3 + 1/2

As a more complicated example try 2/35. One would expect  1/35 +1/35 however the Egyptians did not like to have repeated fractions most likely due to the argument that any fraction can be written as repeating unit fractions and that mathematically it would not get them any where. Instead


2/35 = (2/35) (18/18)
        = 35/ (35*18)  +  1/ (35*18)
        = 1/18 + 1/ 630   [Final answer]

They multiplied by 1 (18/18 = 1) and manipulated the equation to get a sum of units.

Another math problem the Egyptians might run into is that they are presented with a list of unit fractions and they would need to figure out what other unit fractions will be needed to have a sum of 1. They would follow the steps below.

Problem: Complete the list to 1:  2/3, 1/15
Starting off with  2/3     1/15   
                                             Multiply by 15 to each fraction (multiply by the largest denominator)
                          30/3    15/15
                                                   Simplify
                          10       1

                           Add 10 +1 = 11

                           15 -11 = 4         This is the remainder
Then,
           4/15 =  in unit fractions?
                   = 1/15 + 3/15
                   = 1/15 + 1/5

Thus,  2/3 +1/15 + 1/15 + 1/5 = 1.

Unanswered Questions:
1) Was there any practical uses knowing how to add to a list of unit fractions to have a sum of 1? How did they come up with those steps since the modern way seems more intuitive?

Evaluation:
Interesting: 8
Quality: 7
Complexity: 5

Mesopotamia

Reading: 1.2

Summary:
The Mesopotamia civilization contributed a large number of tablets that contained math problems, their solutions and mathematical tables. The Babylonians from Mesopotamia had used various systems of numbers over time, but generally they used based 60. They had placed value of 60, 60^2, 60^3, etc. Being in a different number system their representation of number are different then the present. For instance 90 would be 1,30 since (1x60^1)+(30x60^0) = 90. To represents decimals, the Babylonians used ";" instead of ".' and the place value would work in a similar process as mention above. Many of the tablets that were preserved were multiplication tables and scratch tablets. Since there were no any addition tablet some speculated that the scribes knew their addition well enough.

The Babylonians had an algorithm to find square roots. To find the square root of a number, N, they would:
1) Find the largest number, K, such that K^2 < N^2
2) Find b = N-K^2
3) Find c= (1/2)b(1/a)
4) N ^(1/2) =  K+c

For system of equations, the Babylonians also used false positions.

Q&A
1) Did the Babylonians have another method to find the square root of a number?
      Yes, having found your K, the the square root of your number N would be [ (N/k)+k ] / 2.

2) Using a different number system, did the Babylonians have a different concept of zero?
       Yes, they did not have an assigned symbol for zero in terms of "nothingness" like our present day zero.

3) What type of symbols did the Babylonians used for their numbers?

1. A triangle pointing downward represented 1's
2. A triangle laying on its side represented 10's

Unanswered Question:

1. Why did the Babylonians changed their number system numerous times? Was their an advantage one had over all the others?

Evaluation of reading:
Interesting: 7
Quality: 8
Complexity: 7

Egypt

Reading: 1.1

Summary:
In Section 1.1, the author writes about the Egyptians and gave some back ground information of their number system. The Egyptians preferred to deal with unit fractions and wrote more complicated fractions as sums of unit fractions; the only exception was 2/3. Unlike the present numbers, they had characters to represent the 1's, 10's, 100's etc. For instance to represent 27 they would use two 10's and seven 1's characters; arranging them from least to greatest value.

Egyptians had some helpful techniques to help them with math problems. One technique they was used for multiplying numbers. If they wanted to multiply two numbers, A and B, they would have two columns: the left column started with 1 and the right the smaller of A and B. They would then proceed to double each number until the left columns contain numbers that will add up to the other, A or B. Having chosen the the numbers that would be added they would also add the corresponding number in the right column. As a result, the right column will add to the product of A and B.

 To solve linear equations false position was used. The scribes would guess the answers to the variables and upon finding them incorrect would adjust their educated guess by a proportion, of the corrected answer and the incorrect answer the got. 

Q&A:
1) How do mathematicians know the mathematical procedures and knowledge used in ancient Egyptian?
         The scribes left clay tablets and papyrus that contained problems and the steps that were used to solve them.

2) Any particular text that made significant contribution to understanding how the ancient Egyptians did              math?
     The Rhind Mathematical Papyrus and Moscow Mathematical Papyrus made where two particular texts

3) Did the Egyptians have any other fields of mathematics did they work in?
       They did some geometry. Through a method they were able to find the area of a rectangle, triangle and trapezoid.

Unanswered Question:
1) How did the ancient Egyptians develop their various algorithms?

Evaluation of Reading: 
Interesting: 7
Quality: 5
Complexity: 6