SCIENCE CAPSULES

    If 5^2 = 25 Then square root of 25 =5 If 3^5 = 243 (?)243 = 5      Inverse function of square is square root.  3 to the power 5 is 243.  What is its inverse function?  Here logarithm appears.  Log  3 (243) =5 3---base 5---power Meaning :  Logarithm of 243 to the base 3 is 5.  Hence, "logarithm of x is the power of a number (base) which yield x.     Usually, logarithm of base 10 is used.  Because it is easy to find the decimal places. Example 10^3 = 1000 Log 10 (1000) = 3.000      Logarithm of base 2 is used to find,  h ow many times a 'thing' has doubled                                                                 Suppose, we play a game in which money is doubled for every win.  Say, total mon...

     First, will understand, why we have numbers.  Say, you are the manager of a restaurant.  People sitting in the table order for 4 burgers.  In the 'numberless world' , you will pass the order to the kitchen by calling out 'burger, burger, burger, burger'.  We know saying 'four burgers' is easy.  Four is a number.  So, numbers are a shortcut for counting by ones'.  Number is abstract form.  Four may mean four butterflies or four stones.  Numbers are concepts like beauty and intelligence.      Let us use pebbles to represent the numbers.  Four can be arranged into 2*2 square form.  Take 8.  It can be represented by 8 pebbles arranged into 2*4 rectangular form.  Many numbers can be represented by a square or a rectangle formed using pebbles.  Examples 6,8,9,10,12,14....      But some numbers cannot be made into rectangle.  Example 1,3,5,7,11.....  These nu...

       Say, you are given  seven cakes  of water colour - as in rainbow.  You can select any 3 colours from the array of 7 in fixed quantity.  mix them.  You will get a new shade of colour.  How many colours can you create by choosing any 3 from 7 colours.  Here, the math branch of 'permutations and combinations' will help as to calculate the number of colour shades you can make.   What exactly permutations.  Suppose you have 3 elements named A, Band C.  How many ways you can arrange them.   For first positions, there are 3 choices. A              B                  C For second position there are 2 choices for each first choice. AB and AC; BC and BA; CA and CB For third position, there are only one choice for each combination. ABC and ACB; BCA and BAC; CAB and CBA; finally, we get six arrangements.  We als...

     Imagine a growing circle.  What is the 'most-least' area you can add to the circle.  or what is the area of the thinnest ring that can be added to the circle.  It is 2*pi*r.  How?  Consider a rectangle.  It's area is length*breadth.  Suppose breadth vanishes, the area is only length.  Similarly length of the thinnest ring is 2*pi*r-- the circumference.  Also we can say 2pi*r is the infinitesimal change in the area of the circle.      From Dot, if you go on add big and bigger rings, a circle will be formed.  If we add the areas of the all the rings, we will get area of the circle.  In calculus adding is called "integrating".  Hence, integrating '2pi*r' from 0 to r we get, We got, total area by integrating small change in area. Calculus works.      Add circular sheets of diminishing area one above the other, you get a solid cone.  correct?  Refer figure.  ...

          One of the aim of statistics is to find pattern in the web of data.      There was a serial killer in Uk by name herald shipman.  He was arrested in 1998.  He killed at least 215 and 45 escaped.  He used his position of authority as a doctor to murder many of his elderly patients.  His Modus operandi was to inject his patients with a lethal dose of morphine and then alter their medical records to make their deaths look natural.      The million dollar question is, ' could we have detected shipman's murders earlier?"     The statistics entered.  People analysed deaths in shipman's area from 1977.  All death certificates were scrutinized.  They came to the following conclusions. 1. Shipman's practice record shows much higher number of deaths than average for his area. 2. Time of death of of patients of other doctors were distributed through out the day.  but shipm...

       Say, you have a habit of studying in the morning of some days; spend time in gym in 'other days or play video games.     We have probabilities for your habits.  Suppose you study one day.  There is chance of 70% or (.7) for going to gym the next day.  But going to video games has only the probability of 0.1.  But studying again next day has the probability 0.6.     Your behavior pattern can be depicted using matrix as given below.               study    gym    video     next day study        0.6     0.3       0.1  gym         0.7    0.1       0.2 video        0.1    0.2       0.7 given day     Consider the last element 0.7 in the matrix.  It says, "if you...