Chaos Theory

Chaos Theory is a study of the mathematics associated with a complex system, in order to detect minute changes in the conditions in which the system must operate. The theory is that small changes will upset a delicate balance, and the upset will cascade into an unacceptable (and if the math is not correct, an unpredictable) level of risk to the primary function of the system. You might have heard chaos theory described as “the butterfly effect.”

The butterfly effect, which purports that if enough butterflies in South America flap their tiny wings simultaneously, they will shift the air current traveling north, to change the course of a hurricane as it moves through the Caribbean Sea. Or, you may have seen the 2004 film of the same name, the plot of which was chaos theory. Bear in mind that the operative word here is “theory.”

However, I can tell you that I have had experience with chaos theory in the workplace, and it had little to do with math. Technically smart people with intricate jobs tend to take pride in, or somewhat own their work. They see a correlation between the results of their work and their status among peers. Throw into this mix a new manager, who decides to institute changes in the workplace that he or she manages, to include decisions that affect the technical regimen. Now, you have a risk of chaos. Likely, the technical people will not see the manager as either their superior or their peer.

They will test the manger by instituting a technical discussion with him or her in order to measure the manager’s knowledge about the effect that the manager intends to cause or has caused to their work. If the manager does not pass this test, look out!

The best and brightest technical people will activate their resumes in order to disappear from that now failed workplace (their opinion) swiftly and unexpectedly. It is the technical people who remain that will do harm. They either cannot or will not see a way to leave, so they will institute technical chaos in order to cause the manager to fail at one or more benchmarks. Then, they will enjoy watching that manager fail again as he or she tries to defend what happened even though the manager cannot technically explain how the system works when it works correctly.

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10 Great Uses for Maths

Many people think of maths as a subject, but really it is an exciting world of possibilities and ideas. Like a tree, it has many different branches. Here are ten great uses for maths.

1) Maths is a language. Just like learning German or French can help you communicate with others in the world, studying a new area of maths can help you to be part of a universal conversation.

2) Money. Understanding more about the subject helps a person know more about currency, interest rates, loans and assets. It also allows you to quickly figure out the percentage of a sale or how to invest wisely.

3) Measurements and Cooking. Need to slice a cake into eight equal pieces? You’ll need fractions for that. How about converting between metric and imperial measurements? Yep, numbers are going to come in handy.

4) Programming. Computer coding is based on numbers. Algorithms often involve calculations.

5) Sports. Tallying goals, keeping track of how many points are needed to win, and predicting who will win or lose are a few mathematical tricks that can be performed.

6) Science. Temperature, measurements, conversions… the list goes on and on. Whether you’re studying biology, chemistry, physics or earth science, you’re bound to need mathematics.

7) Music. Musical scales are composed of eight notes and the distance between the notes goes into important things such as harmonies and chords. It’s no coincidence that being good in maths often means a person has a musical ability as well.

8) Puzzles. Being able to think in a new way is one of the strengths of maths. Geometry especially assists in spacial thinking. Children who complete puzzles when young have been shown to have better mathematical aptitude later in life.

9) Problem solving. Need to build a fence? How about deciding how much paint to buy to turn your white walls blue? Algebra is a great tool to be able to do this.

10) Navigation. For centuries ships have used compasses and sextants to measure precise distances. Today, GPS and other digital systems use the power of maths to steer us in the right direction. Without maths, we’re literally lost.

There are many other uses for maths. Numbers surround us wherever we go. Engineering in bridges, code to build the internet, currency exchange rates, combinations of pincodes and locks, weather forecasts, shopping sales and restaurants all rely on mathematics. By learning new areas of maths, you are opening yourself to more possibilities in the world.

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Deriving Values for Trignometric Ratios

Let us first derive the value for Sin(45), Cos(45) and Tan(45).

Let us take an isosceles right angle triangle with base = height. Here the angle made by the hypotenuse with the base is 45 degrees. By the pythogoreas theorm the square of the hypotenuse is equal to the sum of the square of the base and the height. The square of the hypotenuse is thus sqrt(2) * base or sqrt(2) * height.

Sin(45) is hence height/length of hypotenuse = height / sqrt(2) * height = 1/ sqrt(2)

Cos (45) is defined as length of base / length of height and hence it is base / sqrt(2) * base which is equal to 1/sqrt(2).

Tan(45) is hence Sin(45)/Cos(45) which is equal to 1.

Let us derive the expression for Sin(60), Cosine(60) and Tan(60). Let us consider an equilateral triangle. In the equilateral triangle the three angles are equal to 60 degrees. Let us draw a perpendicular between one of the vertex to the opposite side. This will bisect the opposite side by exactly half as the perpendicular line will also be a perpendicular bisector. Let us consider any one of the two triangles created with the perpendicular bisector as the height. So the length of the perpendicular bisector is nothing but sqrt( l ** l – l * * l /4) = l * sqrt(3)/2. By definition Sin(60) is hence height of the triangle / hypotenuse, so Sin(60) can be calculated as l * sqrt(3/2) /l = sqrt(3)/2. Hence Cos(60) can be calculated as sqrt(1 – Sin(60) * Sin(60)) = sqrt(1 – 3/4) = 1/2.

In the same triangle the opposite angle is equal to 30 degrees. So Sin(30) = l/2 / l = 1/2 or 0.5. Using this Cos(30) can be calculated as sqrt(1 – 1/4) = sqrt(3)/2.

Let us go one step further and derive values for Cos(15). Cosine(A + B) is defined as CosineACosineB – SinASinB so when A = B then Cos(A + B) = Cos2A or in other words is equal to Cos (A) * Cos(A) – Sin(A) * Sin(A). Cos2A is equal to sqrt(3)/2 is equal to CosA * Cos A – Sin A * Sin A. Sin A * Sin A can be written as 1 – Cosine A * Cos A. So the expression becomes 2 Cosine A * Cosine A – 1 = sqrt(3)/2. So 2 Cos A * Cos A = (2 + sqrt(3))/2. Cos A * Cos A = (2 + sqrt(3))/2. So Cos 15 = Sqrt(2 + Sqrt(3))/2). Using this values for Sin 15, Sin 75, Cos 75, Sin 7.5. Sin 3,75, Cos 3.75 can be determined

The author is a dual master of science by research in Information Technology and Industrial Engineering. He has worked for many years in leading IT Services firms worldwide. He writes on academic theory, IT services, cricket and current affairs.

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