Mathematics
Mathematics is the science of structure and change. Mathematics is important to the other sciences because it provides rigourous methods for developing models of complex phenomena. Such phenomena include the spread of computer viruses on a network, the growth of tumors, the risk associated with certain contracts traded on the stock market, and the formation of turbulence around an aircraft. Mathematics provides a kind of "quality control" for the development of trustworthy theories and equations which are important to people in most modern technical discplines such as engineering and economics.
Overview
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 the base ten number system. |
Modern mathematics is built around a system of axioms, which is a name given to "the rules of the game." Mathematicians then use various methods of formal proof to extend the axioms to come up with surprising and elegant results. Such methods include proof by induction, and proof by contradiction, for example.
Mathematical Subject Classification
There are numerous categories and subcategories of mathematics, as shown by the American Mathematical Society's Mathematics Subject Classification scheme.
A common way of classifying mathematics is into Pure Maths, and Applied Maths. Pure Maths is maths which is studied in order to make mathematics more stable and powerful, and a knowledge of Pure Maths is required to understand the foundations of Applied Maths. Pure Maths is often considered to be divided into the areas of Higher Algebra, Analysis, and Topology.
Applied Maths consists of taking the techniques from Pure Maths and using them to develop models of "the real world." Applied Maths is sometimes considered to be divided into the areas of Dynamical Systems, Approximation Techniques, and Probability & Statistics. There are also various Applied Mathematical disciplines which use a combination of these areas, but focus on a particular type of application. Examples include Mathematical Physics and Mathematical Biology.
History of Mathematics
Mathematics was noted by the earliest humans. Over time, as humans evolved, the complexity of mathematics also evolved. There was an astounding discovery on how the numbers correlated with each other, as well as in nature, so well, as they created the concept of numbers. Many cultures throughout the world contributed to the development of mathematics in historical times, from China and India to the Middle East and Greece.
Modern Mathematics began in Europe during the Renaissance, after various Arabic texts were translated into European languages during the 12th and 13th centuries. Islamic cultures in the Middle East had preserved various ancient Greek and Hindu texts, and had furthermore extended these old results into new areas. The popularity of the printing press combined with the increasing need for navigational accuracy as European powers began colonising other parts of the globe initiated a huge mathematical boom, which has continued to this day.
"God created the integers. All the rest is the work of man."
-Leopold Kronecker
A police station has 25 vehicles consisting of motorcycles and cars. The total number of tyres of both motorcycles and cars equal to 70. Find the number of motorcycles and cars the station has. Answer First strategy : Draw a Diagram Step 1: Understanding the problem Given : 25 vehicle (motorcycles and cars) Total num of tyres = 70 Find : Num of motorcycles Num of cars Step 2: Develop a plan How should we approach this problem? We can make a diagram. First, we must draw 25 vehicles with two tyres. Then, we must add the remaining tyres to the vehicle until the number of tyres equal to 70. After that, we can see how much motorcycles and cars. Step 3: Implement the plan Before we add 2 more tyres to make the number of tyres become 70: OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO Total number of tyres before adding the remaining tyres are 50 The remaining tyres are 20 After we add 2 more tyres to make the number of tyres become 70: OOOO OOOO OOOO OOOO OOOO OOOO OOOO OOOO OOOO OOOO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO
Sign: OOOO – Car OO - Motorcycle
From this sketches, we can see how much number of motorcycles and cars at the police station. There are 15 motorcycles and 10 cars in the police station.
Step 4: Looking Back Did we answer the correct question, and does our answer seem reasonable? Yes. (If you want to know our answer is correct or not, you must count the number of vehicle’s tyres).
Second strategy : Use a table
Step 1: Understanding the problem
What is the question we have to answer? How many motorcycles and cars in the police station.
How many vehicles in the police station? 25 vehicles.
How many number of vehicle’s tyres in police station? 70 tyres.
How many tyres that motorcycles have? 2 tyres.
How many tyres that cars have? 4 tyres.
Step 2: Develop a plan What strategy will help here? We could model this on paper, but accuracy would suffer. We could also use equations. But, let’s make a table.
Step 3: Implement the plan Firstly, we make a table with 5 rows and 3 columns. Then, we choose our target. For example, in the police station have 6 cars and 19 motorcycles. So we can see the total of vehicles in the police station is 62 vehicles. Then, we try and error with the same ratio until we get the answer which are 15 motorcycles and 10 cars: CARS ( 4 TYRES) MOTORCYCLES (2 TYRES) TOTAL OF VEHICLES (70 TYRES) 6 19 62 7 18 64 8 17 66 9 16 68 10 15 70
Calculation: 1 car have 4 tyres. 1 motorcycles have 2 tyres. So, if there has 6 cars and 19 motorcycles... 6 x 4 = 24 and 19 x 2 = 38 And the total of vehicles are 62 (24+38). But, the answer is wrong.
So, we try and error with the same ratio until we get the correct answer which are 10 cars and 15 motocycles: 10 x 4 = 40 and 15 x 2 = 30. The total of vehicles are 70 (40 + 30). The answer is correct.
Step 4: Looking back Did we answer the question asked? Yes. Does our answer seem reasonable? Yes.
ONE STRATEGY THAT IS DEEMED TO BE MOST EFFICIENT AND JUSTIFY After studied these two strategies, this problem can be overcome by using a diagram. This strategy is more efficient because it is easier to solve this kind of problem. In addition, by using table, we have to list the information and do a lot of calculation. But, by using a diagram, we can easily see the situation and understand the problem.
