Tuesday, May 27, 2014

Nature of Mathematics

Understanding Our Number System

In order to be confident in our use of numbers, we must understand how our number system works and how numbers are related to each other.  We use a single number structure.  All numbers in our single number structure can be represented using combinations of a finite set of digits--0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
An important understanding of how our number system works comes from understanding "place value." This means that a digit can carry a different value depending on where in a number it is located.  We use a base 10 system.  Depending on where a digit is placed, that digit represents a power of 10.  

When learning about number systems, it is important to remember 3 things.
1.)  Use of numbers in the world around us can take on many different forms.
2.)  The same digit takes on different values depending on where it is placed .
3.)  The number system is finite.  
Below is an interactive chart that helps explain the different ideas behind the number system.

Ideally, what our numeral system does is it
  • Represents a useful set of numbers
  • Give every number represented a unique representation 
  • Reflect the algebraic and arithmetic structure of the numbers. 
Our numeral system can represent a useful set of numbers such as integers, rational numbers, or irrational numbers, for example. Our numeral system gives every number represented a unique representation.  For example, the usual decimal representation of whole numbers gives every non zero whole number a unique representation as a finite sequence of digit, beginning by a non zero digit.  However, when decimal representation is used with rational or real numbers, such numbers have an infinite number of representations (2.31, 2.310, 2.309999999).  The following link gives great insight into how our numeral system reflects the algebraic and arithmetic structure of numbers.  



Wednesday, May 21, 2014

Doing Math


The Islamic Empire has made significant contributions to the study of mathematics.  Islamic mathematics allowed math to be expressed in an art form.  There was an extensive use of geometric patterns that was used to help decorate buildings.  
Around 810 AD, the House of Wisdom was set up in Baghdad and the work done there to translate the major Greek and Indian mathematics and astronomy into Arabic began immediately.  An early director of the House of Wisdom, Muhammad Al-Khwarizmi, made an extremely important contribution to mathematics with his adaption of the Hindu numerical system (numbers 1-9 and 0).  Some 400 years later this was adopted by the entire Islamic world and Europe.

To make sense of Islamic mathematics, I obviously looked online for information.  I came across a website which I found to be extremely easy to follow and shows an example of how algebra and geometry are entwined in a tessellation.  Let a, b, and c each be the color represented in the second triangle.  When you add the color a+b, you are given this new color.  Each different triangle in the second BIG triangle is represented by some algebraic expression. 

Below is the tessellation that I worked on in class.  I had a lot of fun with this activity which is why I decided to research it further. 

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Within this blog, I also touched a little bit on the history of the House of Wisdom, as well as the connection between geometry and algebra.  Tessellations are such an interesting subject.  A tessellation is known as pictures or tiles, which cover the surface of a plane in a symmetrical way without overlapping or leaving gaps.  Symmetry is a big mathematical idea in tessellations.  There are 17 possible ways for a pattern to tile a surface without overlapping or leaving gaps.  There are also four different ways of moving a motif to another position in the pattern: translation, reflection, rotation, and glide reflection.  I absolutely encourage anyone to visit this website (first website listed under source) which was used as a source for the information in this blog post.  It has some awesome examples of tessellations and is filled with even more information.  Below are some tessellations that really appealed to me.

Eglish trinity Church tile

Escher Alhambra Tile

                                                   Islamic Star Pattern tile

Sunday, May 11, 2014

History of Math

Why is Euclid important?
Euclid was a famous Greek mathematician known as the Father of Geometry.  He wrote a textbook, Elements, which was considered one of the most important textbooks in the history of mathematics.  In this textbook the principles of Euclidean Geometry were introduced from a small set of axioms.  Euclid understood that building a logical geometry depends on the foundation, this began with Euclid's book.  It has 23 definitions (some listed below), five unproved assumptions which Euclid called postulates (now known as axioms), and five further unproved assumptions that he called notions.  He then goes on in the book proving elementary theorems about triangles and parallelograms and ends with The Pythagorean Theorem.   Below are the basic definitions brought to us by Euclid that are used as the foundation of much of mathematics:

Definition 1. A point is that which has no part. 

Definition 2. A line is breadthless length. 

Definition 3. The ends of a line are points. 

Definition 4. A straight line is a line which lies evenly with the points on itself. 

Definition 5. A surface is that which has length and breadth only. 

Definition 6. The edges of a surface are lines. 

Establishing these terms was a major step in mathematics.  Not much is known about Euclid's life except that he taught at Alexandria.

Below is a representation of Euclid's Windmill Proof which we saw in class. 
Euclid: Windmill proof


Wednesday, May 7, 2014

What is Math?

So what is math?  Originally when I read this question I was going to respond by saying math is the study of different topics including calculus, algebra, trigonometry, and geometry.  But those are just different subjects within the field of math.
I think math is a language in which people speak using numbers, formulas, critical thinking, and problem solving.  Those are not the only components related to the language of math though, there are many different ones.  I think critical thinking and problem solving are a big chunk of what math is.  Math is interesting because whether you are aware of it or not, you use it every day.  Math is used when you go shopping, when you cook, while playing sports, exercising, and in many other every day activities.  Answering this question is not easy because people use math in many different ways.  

Milestones in Math
1)  The Pythagorean Theorem: I have always found this theorem to be a significant tool used to help us further our knowledge of trigonometry.  It's main purpose to find an unknown in an equilateral triangle is just one of the many uses for this theorem but as we have learned one you can do that you are able to figure out angles and lengths in other shapes as well.

2)  The Quadratic Formula: Since my very first algebra class I was asked to memorize the Quadratic Formula which has proved to be a very useful tool in algebraic problems.  I would consider this one of my favorite tools in math because it is used quite often and once memorized is very helpful.

3) When it comes to calculus which has surprisingly become one of my favorite topics in math, you cannot forget to mention Sir Isaac Newton.  He was largely influential in the history of mathematics for creating calculus.  This was so important because it allowed us to take our calculations a step further.  The study of calculus was such an advancement in the field.

4) Numbers & Counting: I guess none of this would be possible before the discovery of numbers and counting themselves.  This would be considered the very center of all things math.  

5)  Proofs:  As much as I hate to admit it, I think proof writing is a very big aspect of math.  The ability to solve an equation is one thing, writing proofs uses your critical thinking.  I have always been rather good at solving equations but the tough part for me is typically writing down the steps, theorems, and thought processes I used to arrive to an answer.  It makes you question different things we learn in math and prove them to be true rather than just excepting an answer or rule because someone says so.