THE INFINITY
OT I D LA E IR
What you hold in your hands right now is a lot more than just an assemblage of paper and ink. It is the fruit of hard work, motivation, ideas and talent merged together and molded into the form of this special issue of The Ed Infinity. Battling odds, some criticism and ito r-in -C the frequently posed question “What do hie Ar f : jun you write about in a math magazine?” The Se Se nio t h Infinity has evolved from an eight page blacki rE dit Ar vin ors and-white newsletter to a fully fledged magazine Ab d S : his ha at par with the most prestigious ones in school. he rm kJ
a ain Ed an For this, I’d like to thank our former Chief Editor, d ito rs: Vin an A ay ga Devashish for making the publication what it is today and Re ak raw va Ba raising the standards of this magazine. nt n al Co Na sal rre y sp ar on One important aspect I’d like to point out in my editorial is that de Sh nts : Ujj ivam you don’t have to be mathematician or even good at math to wa Ma l D Goy An s a read The Infinity and enjoy it. I have often seen that many people a ter jan hu l -in ja Ku Ch think that The Infinity will have articles that need one to be good at ma arg rC e: ha math to understand them and don’t read it on this presumption. Math is ud ha a lot more than just what we learn in school. Limiting mathematics to sums ry Sp
is like limiting an entire language to just it’s grammar. In this special issue, we have a large variety of articles, from thought-provoking ones like ‘The Right Number’ to lighter ones like ‘Number Tricks’. There is also an articles focusing on the School’s Platinum Jubilee. I hope you enjoy the issue. Signing off,
ISSUE NO. 7
Misconceptions We often come across many abstract concepts in mathematics in our daily studies, but do not give them much attention, focusing more of what is asked by the examiners. However, this is sometimes used by those making test papers to test the basic concepts which even the smartest of people may not be familiar with, and which indirectly find their way into the syllabus. Let us take one such common misconception in mathematics. Most people, including many of the secondary and high school students and even college students do not know the way to differentiate between equations that result to ‘undefined’ and those resulting to ‘infinite’, or even the way of classifying them under the two aforementioned categories. Suppose we have the equation (x/0) =a, Without giving it a second thought, most of us will conclude that the resultant value is ∞ (infinite). Let us now consider two different values in place of ‘x’ which are 1 and 0. 1/0=? The answer to this equation can be called ∞ (infinite), because on cross multiplication of this equation, the result to (1/∞) will come as ‘0’ since infinite initial decimal places on the right side of ‘0’ will be ‘0’ itself. The probability of some number other than ‘0’coming after the infinite zeroes will be (1/∞), which is a very negligible value. Therefore, the value (1/∞) is proved to be insignificant. So we can take the value, rounding it off to a very approximate value of ‘0’ itself. By another step of cross multiplication, we come back, and now proving the equation corrects as: 1/0 = ∞ (highly approximate).
THE INFINITY | FOUNDERS’ 2010
Now we move onto the second equation (0/0): Let 0/0 = a By cross multiplying: 0/a=0 And now, because the numerator is ‘0’ for the left side of the above equation, the result will be ‘0’ as well. Hence, 0 = 0 In this case, the resultant ‘a’ can be any number from –(∞) to +(∞), including the number ‘0’, we have no single specific result. Thus the result of this equation is called ‘undefined’ for the simple reason that it is nothing definite. The common misconception is now cleared. Now, moving on to a related concept. We have another school formula a0 = 1. Most of us find it irrelevant to know why the result of any number that has the exponential power ‘0’ has the result as ‘1.’Actually, it is rather simple and can be done in just two steps. Take a4/a4 = 1 [As any number on getting divided by itself gives the result ‘1’] a(4-4) = 1 [Exponential powers get subtracted on division of their co-efficient values] Hence, a0 = 1. Thus, this case, we can have any number as ‘a’, except the number ‘0’, because (0/0) will not give us ‘1’, but an undefined value as shown earlier. Accordingly, 00 is ‘undefined’.
On a random note ... Approximately 12,000 square feet of the school area is underbajri .
236 Y.C.’s issued since 2005. On an average that is over 21 Y.C.’s per term!
29 steps to the Headmaster’s office from the ground floor of the main building
The school pumps in 1,75,000 liters of water each day! In the houses, the average usage of water is about 135 liters per boy everyday. This includes bathing, brushing, flushing and drinking.
ISSUE NO. 7
The constant of
6174
The number '6174' is known as Kaprekar's constant, named after the Indian Mathematician, D. R. Kaprekar. Kaprekar was born on January 17, 1905 in Dahanu, near Mumbai. As a child, he was very interested in recreational math. In 1946, he discovered the Kaprekar's Constant.
This is how it works : 1. Take any four-digit number, consisting of at least two different digits. 2. Arrange the digits of the number in descending order, and then in ascending order so that you have two four-digit numbers. 3. From the larger number, subtract the smaller number. 4. Continue step 2 and 3 with the answers you get. This whole procedure is known as Kaprekar's Routine. The end answer will always come out to be 6174 in a maximum of 7 subtractions. This is true for ALL four-digit numbers, except numbers like 1111, 4444 etc. which consist of only 1 digit. For example- Taking the number as '7823' 8732-2378 6543-3456 8730-0378 8532-2358 7641-1467 7641-1467
= = = = = =
6534 3087 8532 6174 6174 6174
If the result comes to three digits, we take 0 as the first digit to make it a four-digit number. For example, taking the number '5666' 6665-5666 9990-0999 9981-1899 8820-0288 8532-2358 7641-1467
= = = = = =
0999 8991 8082 8532 6174 6174
The number which acts like the Kaprekar's Constant for three-digit numbers is '495'
THE INFINITY | FOUNDERS’ 2010
LXXV It’s our school’s 75th year and it’s going to be a great event in the already rich history of the school. Chandbagh has gone through 75 tough years and maintained its status as a leading institution in the country. The celebrations are on, and the board of the Infinity also decided to catch the action. Since the number 75 is on everyone’s lips we decided to look a bit deeper into this number. To begin with, 75 is one of the six two-digit numbers that are ‘Repfigits’. Here is why: 7+5 5 + 12 12 + 17 17 + 29 29 + 46
=12 =17 =29 =46 =75
It is also a Keith number, because it recurs in a Fibonacci-like sequence starting from its base 10 digits: 7, 5, 12, 17, 29, 46, 75… For the musician- ‘75 Numbers See Another Day’ is a lovely song by Don Reno and Red Smiley & the Tennessee Cutups. For the social networking addict-5% of users account for 75% of all activity on twitter! For the ice-cream lover- 75% of a softies is made up of air. For the religious one- The name ‘Lebanon’ appears 75 times in the Old Testament. The names of the 75 Pharaohs are listed on the Tablet of Abydos. Also, Abraham lived for 75 years For the nature lover- The unique Saguaro Cactus, found in South-western United States does not grow branches until it is 75 years old. For the doctor- bones are about 75% water. For the dirty one- 75% of human excreta is water! For the geek- 75 is 1001011 in binary form, 4b in hexadecimal form and 0113 in octal form. For the environmentalist- Since the closing of the UN Climate Change conference in Copenhagen, 75 countries submitted to the United Nations Framework Convention on Climate Change (U.N.F.C.C.C.) their national pledges to cut or limit the emissions of greenhouse gases by 2020. For the journalist- Over 75% winners in Rajya Sabha poll are millionaires. For the biker- Over 50,000 Pulsars are sold in 75 days. For the publishing world- The Paperback Book also turns 75 this year.
Lastly, for the celebrations-75 is the year of the platinum jubilee.
ISSUE NO. 7
The Right Numbers When space and time were born in the mightiest explosion that ever was, so were six numbers; eternal, inherent, but not arbitrary. These numbers can rightly be claimed to be the first numbers that ever existed, and from which all others perceived today can be inherently derived. Yet, to call them ‘random’ somehow does not seem appropriate. For as Richard Feynman rightly remarked that these numbers are ‘fine-tuned’; especially, it seems, to support the existence of complex entitiesstars, planets, heavy elements and ultimately, life. Almost as the universe itself was desirous of housing much more than a mere soup of elementary particles.
The first of these numbers is the ratio of number of protons to number of electrons. If it had been larger or smaller by 1 part in 10 raised to the power -37, then electromagnetism would have dominated gravity. There would have been no stars, galaxies or planets; nothing except elementary particles such as electrons.
Secondly, consider the fact that the universe has always expanded at a fixed rate. If the expansion rate would have been larger by 1 part in 10 raised to the -55th power, then all matter would have dispersed too fast for gravity to bind it together and transform it into a star. If it would have been larger by the same, the universe would eventually have turned on itself; coming rushing back and crushing everything that exists into a point of infinite density-the Big Crunch.
Another ‘fine-tuned’ constant is the strong nuclear force. When two hydrogen atoms fuse together, then matter-energy conversion is 0.7%. Had it been 0.6%, there would have been nothing but hydrogen in the universe; had it been 0.8%, there would have been no lighter elements or hydrogen; again no stars.
THE INFINITY | FOUNDERS’ 2010
Now there are also those constants that would have allowed for star, but not planet or life formation. The universe also has a fixed amount of mass density. If the mass density of the universe had been larger by a grain of sand (1 part in 10 rose to the -59th power), stars would have been too short-lived to sustain life. If it would have been smaller by the same, insufficient helium from big bang would result in a shortage of heavy elements.
Then there is the cosmological constant; the ‘missing matter’ or dark energy. Had it been different by 10 raised to the -120th power, then no star formation would have ever taken place.
If the ratio of electromagnetic to gravitational force would have differed by 1 part in 10 raised to the -40th power, it would have either formed no heavy elements (had it been smaller), or stars would have been too large and too short-lived to sustain life (had it been larger). No life again.
Had the Big Bang and the constants created then simply been random, the universe would probably have just been a soup of elementary particles or gas. The point I am trying to make is that the universe has a tendency for complex entities to exist within itself. Had these constants been random, left to chance, them the maximum probability of complex bodies to have formed, taking nothing else into consideration, would have been 1 in 10 raised to 311th power. An analogy to this bizarre probability; make a pile of coins covering the whole of Asia, reaching up till the moon. Then do so on 1090 other virtual continents of the same size. Then paint one blue and ask a person to pick it from the pile. The probability that he will pick the right one out of the pile is the probability of even star formation having occurred in the universe. We have truly been endowed with just the right numbers. As Hawking said, "The laws of science, as we know them at present, contain many fundamental numbers, like the size of the electric charge of the electron and the ratio of the masses of the proton and the electron. ... The remarkable fact is that the values of these numbers seem to have been very finely adjusted to make possible the development of life."
A DOON SCHOOL PRODUCTION