Tuesday, May 27, 2014

History of Mathematics: Zero and Brahmagupta

" Zero is both a number and the numerical digit used to represent that number in numerals."

The number zero first appeared in Brahmagupta's book Brahmasputha Siddhanta (The Opening of the Universe) in 628 AD.  Brahmagupta was an Indian mathematician and astronomer who never wrote any proofs for his work so it is unknown how his mathematics were derived.  Most of his works were composed in elliptic verse so it has a poetic ring to them which was common at that time in Indian mathematics. Brahmagupta did a number of important works including algorithms for square roots, the solution of quadratic equations, and the area of a cyclic quadrilateral, known as Brahmagupta's Formula. Brahmagupta established a set of rules for numbers and the modern version includes:
  • The sum of zero and a negative number is negative.
  • The sum of zero and a positive number is positive.
  • The sum of zero and zero is zero.
  • The sum of a positive and a negative is their difference; or, if their absolute values are equal, zero.
  • A positive or negative number when divided by zero is a fraction with the zero as denominator.
  • Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. 
  • Zero divided by zero is zero.
As you can see, Brahmagupta not only mentions zero but also basic mathematical rules for computing zero as well.  According to Brahmagupta, zero divided by zero is zero which was then accepted for centuries later.  In the 12th Century, almost 500 years after Brahmagupta, Indian mathematician Bhaskara II showed that zero divided by zero should be infinity because 1 can be divided into an infinite number of pieces of size zero.  This answer was considered correct for centuries as well, until the modern view, that any number divided by zero is undefined, was established.  Zero divided by zero is still a controversial topic discussed today by many mathematicians. What do you think the answer should be?

My personal opinion is that zero divided by zero is undefined so I agree with the modern view.  I believe it is undefined because zero divided by zero it is like asking "how many zeros are in zero"; are there 1, 0, infinitely many? Since this question doesn't really make since, I believe zero divided by zero is undefined.


Tuesday, May 20, 2014

House of Wisdom

The House of Wisdom was a library translation institute and school established in Baghdad, Iraq in 810; or otherwise known as the Islamic Golden Age.  It was originally created to translated Greek and India mathematics and astronomy into the Arabic language and preserved and modified the translations.  By the ninth century it was the largest repository of books in the world.  The House of Wisdom was founded by a man named Caliph Harun al-Rashid and was later taken over by his son al-Ma'mun.  During the reign of al-Ma'mun the House of Wisdom became an unrivaled center expanding not just for the study of mathematics and astronomy but also for the study of humanities and for the sciences in medieval Islam, including medicine, alchemy and chemistry, zoology and geography and cartography. 

The House of Wisdom wasn't just a place for translations but also a place where many well known scholars from all over the world came to share information and ideas.  These discussions and further research lead to many important original contributions to the mathematics field which were persevered and researched all possible because of the House of Wisdom.  Without these scholars coming together when and how they did, we do not know when the work these scholars did would have come about.  Their work has shaped the way we see and do mathematics today. 
For example, Persian mathematician Muhammad Al-Khwarizmi most important contribution was his strong advocacy of the Hindu numerical system (0-9) which he realized would revolutionize Islamic mathematics and later western mathematics.  Muhammad Al-Khwarizmi also made great contributions to algebra which created "a powerful abstract mathematical language still used across the world today."  Without his work we wouldn't be using the number system we know today or solving algebraic expressions how we are today. 
Al-Karaji, who was a 10th century Persian mathematician, "was the first to use the method of proof by mathematical induction to prove his results, by proving that the first statement in an infinite sequence of statements is true, and then proving that, if any one statement in the sequence is true, then so it the next one."  He introduced and used the method of proof by mathematical induction which is still being used everyday by mathematicians and educators to prove mathematical concepts. 
Nasir Al-Din Al-Tusi was a 13th Century Persian astonomer, scientist, and mathematician who was the first to treat trigonometry as a separate mathematical discipline, different from others.  Without his contribution we might not teach trigonometry separate and people would probably hate mathematics even more if trigonometry was lumped in with algebra.


Brilliant minds came together in the House of Wisdom which further expanded and preserved the mathematics field.  Without these scholars working together when they did, mathematics wouldn't consist of how we know and see it today.  The House of Wisdom was a place for researching, developing, and preserving mathematics in books which still has great influence over us today.  Imagine how different math could be? Would we be using roman numerals instead of numbers? Would trigonometry be taught with algebra? Unfortunately we will never know.  We just have to accept that the world of mathematics would not be as we know it today without the great works from the House of Wisdom. 


Monday, May 12, 2014

Nature of Mathematics

What is an axiom?

An axiom is defined as "a proposition that is assumed without proof for the sake of studying the consequences that follow from it."  Axioms are self-evident truths or universally accepted principles or rules that require no proof. Axioms are used as a starting point by mathematicians and other scholars for proving propositions. 

What is the role of axioms in mathematics?

"Mathematics is the science of proving theorems, and a theorem is a statement that, given the premises laid down by the axioms and certain agreed-upon rules of inference, is apodictically true." The mathematical community has agreed upon a set of axioms used for doing mathematics.  Axioms cannot be proven absolutely true but in turn, are used as a premise or starting point in mathematics.   Most proofs are built on axioms because mathematicians need some basic knowledge when proving conjectures.  It would be extremely tedious and time consuming to prove propositions if mathematicians had to prove everything from the basics upward every time. 

As can be seen, axioms play a huge role in mathematics. Axioms allow mathematicians to be able to progress in the mathematical community without having to prove the basics every time a conjecture is to be proven.    



Thursday, May 8, 2014

495: What is math?

Mathematics gives people a way to examine, understand, and prove the physical world around us.  It is the abstract study of many topics including algebra, trigonometry, calculus, and geometry, just to name a few.  Mathematics is figuring out patterns to hypothesize conjectures about  abstract ideas to explain and understand our physical world; mathematics also gives people a way to prove or disprove those hypothesized conjectures. 

To me, mathematics is the most important subject we learn in school because it is not only a way to exercise our brain but it allows people to improve and/or develop their problem solving skills.  Mathematics shows people how to reach a wanted result by following a succession of steps in a specific order without error.  This process is used in the real world by everyone and can teach people how to optimize their daily lives. Mathematics is useful in every field from art to business.   

I believe the top 5 biggest moments/discoveries in the history of mathematics are:

1.) Numbers.  Being able to quantify things is essential for doing any form of mathematics.  Without numbers there would be no calculus, algebra, statistics, or any other subcategory of mathematics.

2.) Pi.  Pi is important because it allows us to use a symbol as the representation for the ratio of a circle's circumference to its diameter.  This makes for easier calculations especially in long or difficult problems.  

3.) Calculus.  Calculus is the mathematical study of change which allows us to have a better understanding of the nature of space, time, and motion.  We use calculus to study and understand the world we live in.

4.) Pythagorean Theorem.  The Pythagorean theorem allows people to find the third side of a right triangle when the other two sides are given which is useful in a lot of areas of mathematics.  This theorem was developed around 2000 and 1786 BC which is such an advanced concept for that time.

5.) Quadratic formula.  The quadratic formula is the most useful and important method for solving a quadratic equation, which is done in almost every area of mathematics.  The best thing about the quadratic formula is that it works with all equations whether our answer turns out to be real or complex.