sciencecapsules

    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 money earned is 4096 dollars.  How many times the game was won?  Or how many times the money has doubled?  We can just use log. Log 2 4096 = 12 times.     Base 2 logs are used in radioactivity and musical theory.      We know   e = 2.71.... a math c

     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 numbers which cannot subdivided into other numbers are called prime numbers.  The prime numbers ar

       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 also get 6 by multiplying 3*2*1, the number of choices for each position. 3*2*1 is called 'factorial 3' or by symbol 3!.      Hence, if there are n

     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.  Here, the least decrease in volume (by the same argument) is pi *r^2.  By integrating pi*

          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 shipman's victims tended to die between 01.00 pm and 5.00 p.m- precisely when shipma

       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  play video game one day, returning to the video games next day has a high probability of 0.7.     Consider the middle element in the last row 0.2, it tells us that if you play video one day, you may go to gym the next day, the chances are only 20% or 0.2.      This is how, we have to interpret the m

     Let us understand the fractional multiplication visually.  You may feel it is elementary.  But it gives new perspective to fractional multiplication.     First, let us take 4*3.  Let us take a vertical line equally divided into 4 parts and horizontal line into 3 parts.     The area enclosed is the result of 4*3.  There are 12 unit squares.  Hence the answer is 12 units.     Now, how to visualize (1/2)*(1/2).  Take a vertical line, divide into 2 equal parts and name it 1/2 and 1.  Repeat for horizontal line.      Here 1/2 and 1/2 bounds one square which is one fourth (1/4) area of the whole unit square.  So 1/2*1/2=1/4.    In multiplication, the answer need not increase but diminish also.   Now let us prove 1/3*6/2=1.    Draw vertical line and graduate it in 1/3.  Mark horizontal line in terms of 1/2.     1/3 and 6/2 bounds 6 squares.  The area of each square 1/6 (since unit square   contains 6 squares).     Hence the total area of six squares is 6 *1/6 =1.  Th

     Just try to create a five letter word randomly, tidkl,cewkx,dmwol, vuptg,hvwjk,naqid     The words are not only meaningless but also difficult to pronounce.     Actually, how many five letter word you can make in English language. 26*26*26*26*26 =11881376 words. because, there are 26 choices for each position.      English has nearly 200000 words.  There may be 40000 real meaningful 5 letter words. 40000/11881376 = 0.003 or 0.3%     If you or a computer generates five letter words randomly, there is only 0.3% chance for getting real words.     You see, coining a real word or name is very difficult.  People want new names for new born child, medicine, fictional characters etc.  Let us try to get real new names.     1. All English words has vowels.  Hence, add a vowel or two deliberately in each words. Now, actot, verrs,aglgo ...     Some words can be pronounced.  A little improvement.     2. The frequency of each letter in English language is known and given be

     We are always afraid of negative sign. The history books tells us that Julius Ceaser was born in 100 BC, not - 100. Some times, we say it is -5 degrees outside, but many prefer to say 5 below zero.  Sub ground floors in a mall are sometimes called B1,B2 instead of -1,-2.  There is something unpleasant about minus sign.      Why minus multiplied minus is plus.  Let us try to answer this enigma.  We know 1*3 is 1+1+1. ok.  Similarly -1*3 is (-1) +(-1) +(-1) = -3, also ok.  With the same logic, we will do the following    -1*3 = -3     -1*2= -2     -1*1= -1     -1*0=0     -1*-1=? what     At every step, we are adding 1 to the result.  Hence logically (-1)*(-1) =1, must be.   But, how to give a meaning from real life.  Let us try, My friend's friend is also my friend My friend's enemy is also my enemy. But my enemy's enemy is my friend.     (-1)*(-1) =(+1)     Consider toggle switch.  One press changes the state of the switch.  On to OFF, or OFF to ON.  He