Math - Page 10

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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Medieval mathematician and businessman Fibonacci (Leonardo Pisano) posed the following problem in his treatise Liber Abaci (pub. 1202):

How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?

This problem can be illustrated in such way:

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