SCIENCE CAPSULES

    In any artificial and natural growth at some rate, the quantity will double at regular intervals of time- doubling time .  In any decline or decay process at a fixed rate, in the quantity will halve itself at constant intervals of time- half life .      For example: at a particular rate of interest, the amount of money will double in constant period of time.There is a very simple formula to calculate doubling time or half life. Doubling time or half life = natural log [2] / rate of decay  or growth. simply                         t = 0.6931/r Note:  Rate should be expressed in decimal numbers. example: 5% = 5/100 = 0.05 the answer will be in the same unit as r considering real life problems: 1. If the inflation rate is 6% per annum, when the prices will double?    t = 0.6931/0.06 = 11.5 years Every 11.5 years, the prices will be double. 2. If a financi...

Take this 4 letters O,P,S,T.  How many meaningful words one can make with them?  I can make 6 words. Post, stop, tops, opts pots, spot. Actually how many words one can make with 4 letters.  It is mathematically given by factorial 4 =4! 4*3*2*1=24. Hence 24 words are possible.  Some may be meaningless.  How many 4 letter words one can coin using 26 alphabets.  There is a permutation formula to calculate it.    26! /[26-4]! = 26!/22! = 358800  Hence we can make 358800 words. { many words will be meaningless.}  In addition to this, we can coin many 2 letter, 3 letter, 5 letter and so on words out of 26 alphabets.  But the 20 volume oxford dictionary contains only 171476 entries.  So we can understand that there is a room for many more new words.      Similarly using the available words, we can coin many many new sentences.   For example:     On reaching my house, I found the keys w...

    We know the ratio of circumference to the diameter of a circle is pi= 3.14...  This ratio pi is applicable to a electron circling around a nucleus as well as to the planets revolving around the sun.  This mathematical number 3.14.. is a pure constant which always remains true.      If there are minimum 58 persons in a room, two persons will definitely have same birthday(not birth date) according to the probability theory.  This is also a proven truth.      The tangent of angle 45 degree is 1.  That is, in a right angled triangle of angle 45 degree, two sides will be equal. Tan 45 = 1= opposite side /adjacent side or opposite side = adjacent side. This can be applied to a triangle drawn on a paper or to the plane taking off at 45 degree angle. The mathematical facts like given above are always holds good for any persons,  anywhere and anytime.  Now let us consider some physics facts.  The ac...

    There was a chess competition in a king's regime.  The king promised gift of winner's choice.  After the game, the winner just asked some grains of rice  as his price.  He added one condition. He wanted to put one grain of rice on the first square of chess board;  2 grains on the 2nd square; 4 grains on the 3rd; 8 grains on the 4th one.  That is, doubling the number of grains each time till reaching 64th square.  The king was surprised by the humble request and ordered to execute the prize.  The servants brought sack full of rice and begum to arrange the grains on the board as instructed.  Halfway through the work, the king and the servants realized that all rice grains in his kingdom would not be enough to fill all the 64 squares.  The king submitted to the wisdom of the winner.  Then the winner said, "I do not want the price.  Just give rice food to the needy daily".  And he went away.  This i...

Take the function, y = x^2 x        y=x^2 -3        9 -2        4 -1        1 0         0 1         1 2         4 3         9   If we draw the graph, we get a parabola.  Now we know how the function behaves.      I want to know if 'x' changes by 1[unity], What is a change in 'y' at any value of x.  That is, what is the rate of change of function at any point in the curve .  Is there any standard formula?  The question is easy to answer, if the function is linear or the graph is straight line.  The answer would be slope [gradient] of line and the slope will be the same at every point in the line. but Here we have parabolic relationship (refer figures).     Velocity is the rate of change of distance.Inflation is the rate of change of pric...

    Take the point C[ -2+i] in the complex plane.  Let Z =0 +0i and square it. Z^2 = 0+0i..  Now add C = Z  to Z^2 and get Z1. Z1 = Z^2+C Z1 = [0+0i]^2+[-2+i] Z1 = -2+i now let Z=Z1 and calculate Z1 = Z^2+C  again     = [-2+i]^2+[-2+i]     = 1+5i [complex addition,multiplication used] Magnitude of Z1 [1+5i] = root of [1^2+5^2] = 5.09     Now do this repetitive feed back calculation say 100  times according to this formula.  Z[n+1] = Z[n]^2+C Each time calculate the magnitude of Z.  If the magnitude stay stable within 2 , paint the point C as black.  If the magnitude goes beyond 2 and heads towards infinity, either paint the point as white or the point may be given colours as follows.  If the magnitude goes beyond 2 after 10  iterations, the point C may be given red.  For 20 iterations, it may be a orange dot.  For 30  repetitions, C may be given yellow. ...

      At present the world record for 100 m dash remains at 9.578 sec by USAIN BOLT.  Will it further come down?  Will ever a man make it under 9 sec.  We will try to answer these questions via mathematics.   Harmonic series:      Take a deck of 52 playing cards.  Move first top card to half of its length.  That is the maximum distance you can move before gravity pulls it down.  Now consider top first and second card as one unit and move both of them to 1/4 of the card length.  Again both will stay without falling.  Now consider first 3 cards as one unit and move all of them to 1/6 of the card length.  All the 3 cards stay put and we get some length of "overhang". Repeat the process for all the 51 cards.(last card cannot be moved).  The overhang accumulates as given by the following series. = 1/2+1/4+1/6+1/8+.........+1/102 = 2.2594........card length        ...

    What is the probability of getting an odd number in a dice throw?  There are three odd numbers 1,3 and 5.  Hence the probability is 3/6=50%. Draw a circle within a square   Drop pins randomly in the square.  The pin should fall on its tip.  What is the chance of pin falling within the circle? Area of the square = 2r*2r= 4r^2. Area of the circle = pi r^2. Hence the chance of a pin falling in the circle = area of the circle/area of the square.                                                                          = pi*r^2/4*r^2                                                                 ...

    Toss a coin 100 times.  You are likely to get head around 50 times.  So there is a 50 % chance of getting a head or a tail. In other words, the probability of getting a head or a tail is 1/2.  The range of probability is 0 (no chance)   to 1(100% chance).        Suppose you toss a coin 3 times, what is the chance of getting a head in all the 3 throws.  Here the multiplication rule comes to our help.  This rule can be applied when the events are independent of each other.  Just multiply the probability of the event 3 times.  total probability = 1/2*1/2*1/2=(1/2)^3 = 1/8 = 0.125  The probability of getting only heads 3 times consecutively is 1/8 or 1 in 8.  That is , if 8 persons throw the coins 3 times, 'one'  may get the head all the three times.        Let us assume that in a multiple choice question, there are 4 choices and one among them is correct.  Guessi...

    If there are 60  people in a gathering, will any two persons have the same birth day?  We are doubtful. we think if there are more than 365 persons, there may be a possibility of two persons sharing same birthday.  But the statistics' answer is stunning.  Here we go.      In a coin toss, chance of getting head or tail is 50 % or 1/2.  Chance of head is 1/2 and the tail is 1/2.  Hence the total probability is 1.        In a dice throw, chance of getting '1' is 1/6 and the chance of 'not' getting '1' is 5/6 = 1-1/6 .       In a gathering, the chance for a person to have any date in a year as a birthday is 365/365=1=100%.  For another person, the chance of not having a birthday same as first person is 364/365.  For a third person, the chance of not having a birthday same as previous two persons is 363/365.  Hence the probability for not having a match (two persons ha...

  We know, if the switches A and B are closed in the above circuit, the bulb will light up.Even if one is open, the bulb is off.  Logically we can say,  if A and B are true, the output is true else the output is false.  In digital logic it can be written, "if A AND B are 1 the output is 1 else the output is 0".      The AND logic can be wired electronically.  Instead of switches we will be using automatic switches like diode and transistor.       Then we have OR  logic.  That is written as; "if either A OR B is 1 then output is 1" . ( If both are 0 then output is 0).    Next NOT  logic; here the input is reversed and given as output(0 to 1: 1 to 0 ).     NOT  and  AND  can be combined and it is called NAND logic.  NOT + OR  is called NOR  logic.      All this logic statements can be hard wired using electronic components and they are us...

Consider this, 6= 2 * 3 9 = 3* 3 12 = 2* 2* 3 The kind of numbers like 6, 9, 12 are called composite numbers because they can be broken down into smaller numbers.  But numbers like 2, 3, 7, 11 cannot be split up into whole numbers. They are divisible by 1 and by themselves only.  These numbers are called prime numbers.  They behave like "atoms" in the number system.  We can build other composite numbers using them in an unique way.  For example, 28 can be expressed as 7 * 2* 2.  This is the only way to write 28 using primes. Another example:6545448 = 2^3*3^5*7*13*37 The list of prime numbers below 100 is given below. 2, 3, 5, 7, 11, 13, 17, 19, 23 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. There are 25 primes between 1 and 100.  But only 14 primes between 900 and  1000.  There is no formula to find the prime numbers.  There is no pattern in the appearance of primes. Yet, the umber of prime numbe...

    To add;   3/2+5/3, first we have to calculate the least common divisor which is   2*3 = 6.   then,         3/2+5/3 =  (9+10)/6 = 19/6.  Why do we adopt this procedure?         We cannot add apples and oranges. Similarly 3/2 and 5/3 belongs to different categories .  3/2 has 3 times 1/2 and 5/3 has 5 times 1/3.  Hence we cannot directly add them.  First we should bring them to a common platform .  Here LCD 6 comes to our help.   1/6 happens to be the common factor for both the fractions. 9 times 1/6 gives  3/2               9*(1/6) = 3/2 10 times 1/6 gives 5/3             10*(1/6) = 5/3      So instead of adding 3/2 and 5/3; we can easily add 9/6 and 10/6 and get the answer.                             ...

  If a dog bites man, it is not a news.  If a man bites dog, it is a news.  Anything surprise is the news and an information.  If an event certainly is not going to occur then its probability of happening is 0 and it is not an information.  If an event is certainly going to happen then its probability is 1 and it is also not an information.      An event which is having probability between 0 and 1 carries some information.  For example:  In tossing of a coin getting a head or tail is having a 50% chance or 0.5 probability.  Out come of the toss is an information.  In the case of rolling of dice, each outcome has 1/6 chance or 0.167 probability.  Certainly the outcome of a  dice is  the surprise and an information.      Shannon gave a formula to calculate information content in an event based on probability.      I want to calculate the information content of coin ...