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Tuesday, April 29, 2014

Triangles

Let us consider the persons who developed mathematics as one person. Let us call this person as Cofu. Then we could easily understand about the thoughts which was going in the minds of the people who developed mathematics. Human categorizes or differentiate things. They differentiated seasons in different ways and wrote them as tally marks on the walls of caves. Now considering all as cofu. Cofu looked at his surroundings and found many shapes among which was a shape with three sides. He called this shape as triangle. Then he wanted to rigorously study this shape. He found that the triangles may be classified according to the length of triangle's sides. If all the three sides are different then the triangle is called scalene triangle. If two sides are equal then the triangle is called isosceles. If all the sides are equal then the triangles is called equilateral.

As the time passed he developed the method to measure angles. He found that the angles are related to the sides or in other words they depend on each other. If we increase one angle then the side opposite to it also increases and if we increase the sides then the angles opposite to it also increases. If two angles are equal then the two sides are equal in a triangle.


He also differentiated triangles according to the angles. A triangle with one angle greater than 90° is called an obtuse angled triangle and those with all the angles less than a right angle is called an acute angled triangle. If any of the sides is equal to right angle it is called a right angle triangle.

As the time passed he wanted to measure angles. He faced a problem that there are many sizes of triangles and each shape has different relations of sides. After thinking a while he found that every triangle can be decomposed into two triangles with one angle of each triangle right angle. then he thought to develop all the mathematics of triangles according to right angle. We will study about the right angled triangles in the next post.

Monday, April 28, 2014

Bisection Method

There are many methods to find the roots by approximation and one of it is bisection method. This method uses the intermediate value theorem which states that if a continuous curve changes sign then it must have taken zero at some point.

We take the given function f(x) and two values a and b such that f(a)·f(b) < 0. Then we bisect the values a and b. Let it be c. c = (b-a)/2. If f(c)=0 then c is the root else if f(a)·f(c)<0 then b=c else a=c. We continue the same procedure again. This method converges very slowly.

The property of this method is that it does not use the value of f(x) as the formula to compute the root.