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