Month: June 2011

Typically, the conic sections, and that includes the ellipse, parabola, and the hyperbola all have their definitions connected to the intersection a plane makes with a cone. However, more technically useful definitions of the conic sections are those provided by plane geometry. The equivalence of the definitions of the conic sections have been proved by Germinal Pierre Dandelin, a Belgian mathematician who discovered the concept now known as the Dandelin spheres in 1822.

The Dandelin spheres which are sometimes called focal spheres can be used to prove some important theorems; at least two. Though the theorems proved by Dandelin have been known for centuries yet Dandelin made it easier to prove them.
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Cantor set, seen on the number line as the interval between 0 and 1 is an example of a fractal on the real number system as shown on the real number line. The Georg Cantor set is very easy and simple to construct just with the aid of a line that represent numbers where if one remove a section; it amounts to dealing with that part of the set.

Construction of Cantor set involves three steps which are outlined below:

1. Draw a horizontal number line that signifies the interval of real number system with the left and right endpoints labelled 0 and 1 respectively.

2.Cut off or wipe out or simply erase a section of this line that represents middle-thirds, that is between 1/3  and  2/3  of the drawn line segment. Once you have erased this middle-thirds section, you will be left with two thirds of the originally drawn number line.
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Once a while in mathematics, we really need to find the midpoint between two other points, that is, the point that is exactly in the middle of the two other points. A good example is if you have to find the point at which a line bisects or divides a given line segment into two equal parts. The midpoint formula is quite simple and you should endeavour to know how to derive it for future use.

The midpoint formula can be conceived in terms of finding the middle number that exists between two given numbers such as 10 and 15. By adding the two numbers and dividing by 2, we obtain the exact middle number as follows:

Exactly the same way, the midpoint formula works. Let us consider the following question:
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In general, conic sections represent the curves obtained from intersecting by a plane some “double-napped” cone (it’s like two cones were put “nose to nose”, perfectly balancing). The term “sections” is used with the same meaning as in science or medicine. In these areas, a little sample (obtained from a biopsy for example) is frozen and then very thin and little slices (they are also called “sections”) are taken to have a view of them under a microscope. When we’re talking about conic sections, we mean what follows when the cones are sectioned at different angles.

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Logarithms – that is how usually the “inverse” of exponentials are called (just as   division is the opposite of multiplication and subtraction is the opposite of addition). Logarithms “undo” exponentials.

y = b^x is equivalent to log_by =x

Above we can see the exponential statement “y = b^x“.  Then goes the equivalent logarithmic statement “log_b(y) = x”. It is pronounced “log-base-b of y equals x”. The “the base of the logarithm” is the value of the subscripted “b”. In an exponential the base b is not equal to 1 and is always positive just like the base b for a logarithm is not equal to 1 and is always positive. No matter what may be aside, the logarithm is usually called the log “argument”. You should make a note of the fact that the base in the log equation and the exponential equation is “b”. Moreover, when one switches between the two equations, the y and x switch sides.
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