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