HISTORY OF TRIGONOMETRY

 Ansua XI C

The history of trigonometry goes back to the earliest recorded mathematics in Egypt and Babylon. The Babylonians established the measurement of angles in degrees, minutes, and seconds. Not until the time of the Greeks, however, did any considerable amount of trigonometry exist. In the 2nd century bc the astronomer Hipparchus compiled a trigonometric table for solving triangles. Starting with 7½° and going up to 180° by steps of 7½°, the table gave for each angle the length of the chord subtending that angle in a circle of a fixed radius r. Such a table is equivalent to a sine table. The value that Hipparchus used for r is not certain, but 300 years later the astronomer Ptolemy used r = 60 because the Hellenistic Greeks had adopted the Babylonian base-60 (sexagesimal) numeration system.

In his great astronomical handbook, The Almagest, Ptolemy provided a table of chords for steps of y°, from 0° to 180°,that is accurate to 1/3600 of a unit. He explained his method for constructing his table of chords, and in the course of the book he gave many examples of how to use the table to find unknown parts of triangles from known parts.

                                Indian astronomers had developed a trigonometric system based on the sine function rather than the chord function of the Greeks. This sine function, unlike the modern one, was not a ratio but simply the length of the side opposite the angle in a right triangle of fixed hypotenuse. The Indians used various values for the hypotenuse.

Late in the 8th century, Muslim astronomers inherited both the Greek and the Indian traditions, but they seem to have preferred the sine function. Trigonometric calculations were greatly aided by the Scottish mathematician John Napier.

                                     Finally, in the 18th century the Swiss mathematician Leonhard Euler defined the trigonometric functions in terms of complex numbers. This made the whole subject of trigonometry just one of the many applications of complex numbers, and showed that the basic laws of trigonometry were simply consequences of the arithmetic of these numbers.

AryabhatTa

Bijeesh A B - VI C -

Aryabhata was the great Hindu Mathematician. He lived from 475 AD to 550 AD. He wrote on arithmetic including undo this heading arithmetic and geometric progressions, quadratic equations, and indeterminate equations. His work often called Aryabhatiya consists of a collection of astronomical tables and the Aryastasata which include the Ganita, a note on arithmetic, the Kalakriya on time and its measure and the Gola on the sphere. Aryabhata was one of those ancient scholars of India who could stand with pride among the greatest scholars even of the modern age.

Puzzle

   1)There is a number which is very peculiar. This number is 3times the sum of its digits. Can you find the number?

 2) Write the biggest number that can be written with four one’s?

3) Can you write 1789 in roman numbers?

 4) What is the sum of first 70 odd numbers?

5) Find the value of:

                                   CCLXV+CXVI+XIII+VI=?

Beginning of algebra

 It is said that algebra as a branch of mathematics began about 1550 BC, that means more than 3500 years ago, when people in Egypt started using symbols to denote unknown numbers .Around 300 BC, use of letters to denote unknown and forming expressions from them was quite common in India. Many great Indian mathematicians Aryabhata, Brahmagupta, Mahavira and Bhaskara II and others, contributed a lot to the study of Algebra.

Branches of mathematics

The branches of mathematics in which we study about numbers is called arithmetic. The branch in which we learn about figures in two and three dimensions and their properties is called geometry. The branch in which we use letters along with numbers to write rules and formulas in general is called algebra.    

Answers for the puzzleS

1Ans) 27

            2+7=9

            9x3=27

2Ans) 285311670611

3Ans) MDCCLXXXIX

4Ans) 4,900 (70x70)

5Ans) CD  
  
                        CLXV              265

                      C XVI               166

                                 XIII                 13                        

                  VI            +       6    
  

           Ans          CD               400  

                                                          

Math’s Interesting Facts

                                                                                                                  ANILA

                                                                                                                      IX-B

·         The word "mathematics" comes from the Greek máthēma, which means learning, study, science.

·         What comes after a million, billion and trillion? A quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion and undecillion.

·         Different names for the number 0 include zero, nought, naught, nil, zilch and zip.

·         The = sign ("equals sign") was invented by 16th Century Welsh mathematician Robert Recorde, who was fed up with writing "is equal to" in his equations.

·         Googol (meaning & origin of Google brand) is the term used for a number 1 followed by 100 zeros and that it was used by a nine-year old, Milton Sirotta, in 1940.

·         The name of the popular search engine ‘Google’ came from a misspelling of the word ‘googol’.

·         Abacus is considered the origin of the calculator.

·         12,345,678,987,654,321 is the product of 111,111,111 x 111,111,111. Notice the sequence of the numbers 1 to 9 and back to 1.

·         Plus (+) and Minus (-) sign symbols were used as early as 1489 A.D.

·         An icosagon is a shape with 20 sides.

·         Trigonometry is the study of the relationship between the angles of triangles and their sides.

·         If you add up the numbers 1-100 consecutively (1+2+3+4+5...) the total is 5050.

·         2 and 5 are the only primes that end in 2 or 5.

·         From 0 to 1,000, the letter "A" only appears in 1,000 ("one thousand").

·         'FOUR' is the only number in the English language that is spelt with the same number of letters as the number itself

·         40 when written "forty" is the only number with letters in alphabetical order, while "one" is the only one with letters in reverse order.

·         Among all shapes with the same perimeter a circle has the largest area.

·         Among all shapes with the same area circle has the shortest perimeter .

·         In 1995 in Taipei, citizens were allowed to remove ‘4’ from street numbers because it sounded like ‘death’ in Chinese. Many Chinese hospitals do not have a 4th floor.

·         In many cultures no 13 is considered unlucky, well,there are many myths around it .One is that In some ancient European religions, there were 12 good gods and one evil god; the evil god was called the 13th god.Other is superstition goes back to the Last Supper. There were 13 people at the meal, including Jesus Christ, and Judas was thought to be the 13th guest.

Some beautiful examples with multiplication

12345679 x 9 = 111111111;

12345679 x 8 = 98765432

An interesting fact about primes
Mathematicians of XVIII^th century proved that numbers 31; 331; 3331; 33331; 333331; 3333331; 33333331 are primes. It was a big temptation to think that all numbers of such kind are primes. But the next number is not a prime.
333333331 = 17 x 19607843

NUMBER FACTS

                                           - ANUNANDA.C -

                                                  VIII-C

·         The numerical digits we use today such as 1, 2 and 3 are based on the Hindu-Arabic numeral system developed over 1000 years ago.

·         Different names for the number 0 include zero, nought, naught, nil, zilch and zip.

·         The smallest ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

·         2 and 5 are the only prime numbers that end with a 2 or a 5.

·         The golden ratio of approximately 1.618 between two quantities such as lengths often appears in nature (tree branching, uncurling ferns, pine cone arrangements etc) and has been used throughout history to create aesthetically pleasing designs and art works such as Leonardo da Vinci’s Mona Lisa.

·         Fibonacci numbers are named after Italian mathematician Leonardo of Pisa (better known as Fibonacci) who introduced them to Western Europe after they had earlier been described by Indian mathematicians. They are related to the golden ratio and proceed in the following order: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, .... Can you see the pattern?

·         The number Pi (the ratio of the circumference to the diameter of a circle) can’t be expressed as a fraction, making it an irrational number. It never repeats and never ends when written as a decimal.

·         Here is Pi written to 100 decimal places:
3.1415926535897932384626433832795028841971693993751
058209749445923078164062862089986280348253421170679

·         What comes after a million, billion and trillion? A quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion and undecillion.

·         The name of the popular search engine ‘Google’ came from a misspelling of the word ‘googol’, which is a very large number (the number one followed by one hundred zeros to be exact).

·         A ‘googolplex’ is the number 1 followed by a googol zeros, a number so ridiculously big that it can’t be written because there literally isn't enough room in the entire universe to fit it in!

·         Check out some more big numbers.

·         You might have heard the word ‘infinity’ before or seen its symbol that looks like the number 8 placed on its side. Infinity means a limitless quantity or something that goes on forever. While it’s not really a number like 1, 2 or 3, infinity is often used in math as part of equations and formulas.

·         111111111 x 111111111 = 12345678987654321

·         12 + 3 - 4 + 5 + 67 + 8 + 9 = 100

                                              

                                            

                              puzzles

                                                  Vishnu R - VI A     

1: Tricky Questions

à The following equation is wrong:   101-102=1 Move one numeral to make it correct…

à What mathematical symbol can be put between 5&9 to get a number bigger than 5 and smaller than 9?

à You are making an opaque cube,(ie.you cannot see through the sides.)It can be any size you want it to be. Where do you place the cube so that you can see as many sides as possible?

2: Logic questions

à You have a basket having 10 apples. You have 10 friends, who each desire an apple .You give each of them an apple. After some time, each of your friends have an apple each, still there is one apple remaining in the basket. HOW?

à There are five gears connected in a row, the first one is connected to the second one, the second one is connected to the third one and so on. If the first gear is rotating clockwise, what direction is the fifth gear turning?

à A boy and a girl are talking. `I AM A BOY’ said the child with black hair. `I AM A GIRL’ said the child with white hair. At least one of them lied. Now who is the boy and who is the girl

NOW CHECK THE SOLUTIONS.


       SOLUTIONS

1:

a.  Move the numeral 2 half a line up to achieve 101-102=1

b.  A decimal point.5.9 works nicelyJ

c.  You place it so you are standing inside it, in a corner. Then you can see all the sides. ‘IT IS A VERY BIG CUBE’J.

2:

a.  You give an apple to each of your nine friends, and a basket with an apple to your tenth friend. Each friend has an apple, and one of them has it in the basket.J

b.   Clockwise.J

c.    They both lied. The child with the black hair is the girl, the child with the white hair is the boy.(if only one of them lied, they would both be boys or  both will be girls.JJJ

Tease Your Brains!!

                                           KAVYA MOHAN - VII - B

Q: Fifty minutes ago if it was four times as many minutes past three ‘O’ clock, how many minutes is it to six ‘O’ clock?

ANS: Twenty six minutes.

Q: A clock takes six seconds to

Strike 4 ‘O’ clock, Then how much will it take to strike 8 ‘O’ clock?

ANS: 14 seconds!

Q: One person has 4 sons , if each brother had one sister, what is the total number of students?

ANS: Five

Q: Vivek and Prasobh  have  the same amount of money with them. How much amount should Vivek give Prasobh in order that the amount with Prashob be 10 more than that with Vivek?

ANS: Rs 5/-

Q: Find the number which is three times the sum of its digits.

ANS: 27.

      

   

    

     The evolution of mathematics

GIRISH V GOPINATH - X A -

The evolution of mathematics might be seen as an ever-increasing series of abstractions , or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals, was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.

In addition to recognizing how to count  physical objects, prehistoric peoples also recognized how to count abstract quantities, like time – days, seasons , years. elementaryarithmetic (addition, subtraction, multiplication & division)   naturally followed.

Since numeracy pre-dated writing ,further steps were needed for recording numbers such as tallies or  the knotted strings called Quipu  used by theinca  to store numerical data. Numeral systems have been many and diverse, with the first known written numerals created byEgyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.

Mayan numerals

The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when theBabylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.                         

                                                                     G

BRAIN TEASERS - Thulasi XI B

1)   Team A wins 19 out of 20 games. Team B wins 7 out of 8 games. Which team has a higher percentage of wins?

2)   How can you get 26 using 5 fives?

3)   A two digit number equals five times the sum of its digits. If you add 9 to the number, the order of its number is reversed. What is the number?

4)   What number gives the same result when it is added to 1.5 as when it is multiplied by 1.5?

5)   Replace the asterisks by the correct mathematical signs to make the expression   equal to 64: 18*12*3*8 = 64?

6)   On the first day of summer camp, every child met and shook hands with every other child, if there were 12 children present, how many total handshakes were there?

7)   A box has 16 candies: 12 red and 4 yellow. If you were picking candies from the box without looking, how many candies would you pick up to be certain to find two of the same color?

8)   In the parking lot of your building, there only are two wheelers and four wheelers. If there are a total of 40 vehicles and 104 wheels in the parking lot, how many four wheelers are there?

9)   Here’s a variation on a famous puzzle by Lewis Carroll,    A group of 100 soldiers suffered the following injuries in a battle: 70 soldiers lost       an eye, 75 lost an ear, 85 lost a leg, and 80 lost an arm. What is the minimum number of soldiers who must have lost all 4?

10)          Mr.Seibold has 6 daughters. Each daughter is 4 years older than her next younger sister. The oldest daughter is 3 times as older than her youngest sister. How old is each of the daughters?

  

ANSWERS

1)   Team A

2)   5^5-5 +5(5)

3)   45

4)   3

5)   18 x 12/3- 8 = 64

6)   66[n X (n-1)/2]

7)   03

8)   12- four wheelers, 28- two wheelers

9)   100 soldiers suffered a total of 310 injuries which means, at a minimum, 100 soldiers lost 3body parts, and 10(the remainder when dividing 310 by 100)must have lost all 4 body parts.(In reality, as many as 70 may have lost all 4 body parts)

10)          From youngest to oldest, the 6 daughters are 10, 14, 18, 22, 26, and 30.

                                                  *****

INTERESTING FACTS AND TRICKS      

                                   - Thulasi XI B -

1. Multiply Up to 20X20 In Your Head

In just FIVE minutes you should learn to quickly multiply up to 20x20 in your head.  With this trick, you will be able to multiply any two numbers from 11 to 19 in your head quickly, without the use of a calculator.

Try this:

• Take 15 x 13 for an example.
• Always place the larger number of the two on top in your mind.
• Then draw the shape of Africa mentally so it covers the 15 and the 3 from the 13 below. Those covered numbers are all you need.
• First add 15 + 3 = 18
• Add a zero behind it (multiply by 10) to get 180.
• Multiply the covered lower 3 x the single digit above it the "5" (3x5= 15)
• Add 180 + 15 = 195.

 2. The 11 Rule

You likely all know the 10 rule (to multiply by 10, just add a 0 behind the number) but do you know the 11 rule? It is as easy! You should be able to do this one in you head for any two digit number.

To multiply any two digit number by 11:

• For this example we will use 54.
• Separate the two digits in you mind (5__4).
• Notice the hole between them!
• Add the 5 and the 4 together (5+4=9)
• Put the resulting 9 in the hole 594. That's it! 11 x 54=594

The only thing tricky to remember is that if the result of the addition is greater than 9, you only put the "ones" digit in the hole and carry the "tens" digit from the addition. For example 11 x 57 ... 5__7 ... 5+7=12 ... put the 2 in the hole and add the 1 from the 12 to the 5 in to get 6 for a result of 627 ... 11 x 57 = 627

 3. Square a 2 Digit Number Ending in 5

For this example we will use 25

• Take the "tens" part of the number (the 2 and add 1)=3
• Multiply the original "tens" part of the number by the new number (2x3)
• Take the result (2x3=6) and put 25 behind it. Result the answer 625.

Try a few more 75 squared ... = 7x8=56 ... put 25 behind it is 5625.
55 squared = 5x6=30 ... put 25 behind it ... is 3025.

4. Square 2 Digit Number: UP-DOWN Method 

Square a 2 Digit Number, for this example 37:

• Look for the nearest 10 boundary
• In this case up 3 from 37 to 40.
• Since you went UP 3 to 40 go DOWN 3 from 37 to 34.
• Now mentally multiply 34x40
• The way I do it is 34x10=340;
• Double it mentally to 680
• Double it again mentally to 1360
• This 1360 is the FIRST interim answer.
• 37 is "3" away from the 10 boundary 40.
• Square this "3" distance from 10 boundary.
• 3x3=9 which is the SECOND interim answer.
• Add the two interim answers to get the final answer.
• Answer: 1360 + 9 = 1369

5. The 11 Rule Expanded

You can directly write down the answer to any number multiplied by 11.

• Take for example the number 51236 X 11.
• First, write down the number with a zero in front of it.

051236

The zero is necessary so that the rules are simpler.

• Draw a line under the number.
• It is simple if you work through it slowly. To do this, all you have to do this is "Add the neighbor". Look at the 6 in the "units" position of the number. Since there is no number to the right of it, you can't add to its "neighbor" so just write down 6 below the 6 in the units col.
• For the "tens" place, add the 3 to the its "neighbor" (the 6). Write the answer: 9 below the 3.
• For the "hundreds" place, add the 2 to the its "neighbor" (the 3). Write the answer: 5 below the 2.
• For the "thousands" place, add the 1 to the its "neighbor" (the 2). Write the answer: 3 below the 1.
• For the "ten-thousands" place, add the 5 to the its "neighbor" (the 1). Write the answer: 6 below the 5.
• For the "hundred-thousands" place, add the 0 to the its "neighbor" (the 5). Write the answer: 5 below the 0.
  That's it ... 11 X 051236 = 563596

6. Guess Your Birthday!

Here's a fun trick to show a friend, a group, or an entire class of people.
Step 1) Have them take the month number from their birthday: January = 1, Feb = 2 etc.
Step 2) Multiply that by 5
Step 3) Then add 6
Step 4) Then multiply that total by 4
Step 5) Then add 9
Step 6) Then multiply this total by 5 once again
Step 7) Finally, have them add to that total the day they were born on. If they were born on the 18th, they add 18, etc.
Have them give you the total. In your head, subtract 165, and you will have the month and day they were born on!
How It Works: Let M be the month number and D will be the day number. After the seven steps the expression for their calculation is:
5 (4 (5 M + 6 ) + 9 ) + D = 100 M + D + 165
Thus, if you subtract off the 165, what will remain will be the month in hundreds plus the day!

7. Strange Properties of 666, "The Number of the Beast"
The last book of the Bible, Revelation, brings up the number 666 as being the number of the beast connected with the end of this age and the coming of the Messiah. You will find the direct reference in Chapter 13, verse 18 of Revelation. Besides that cataclysmic reference, the number 666 has quite a few very interesting properties.
666 = 3^6 - 2^6 + 1^6
666 = 6^3 + 6^3 + 6^3 + 6 + 6 + 6

(Mike Keith mentions that there are only five other positive integers that exhibit this property...
666 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2
666 = 1 + 2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9 = 9 + 87 + 6 + 543 + 21
Moreover, 666 is equal to the sum of the cubes of the digits in its square (666^2 = 443556, and the sum of the cubes of these digits is 4^3 + 4^3 + 3^3 + 5^3 + 5^3 + 6^3 = 621) plus the sum of the digits in its cube (666^3 = 295408296, and 2+9+5+4+0+8+2+9+6 = 45, and 621+45 = 666).
Incredibly, the number 666 is equal to the sum of the digits of its 47th power, and is also equal to the sum of the digits of its 51st power. That is,
666^47 = 5049969684420796753173148798405564772941516295265
4081881176326689365404466160330686530288898927188
59670297563286219594665904733945856
666^51 = 9935407575913859403342635113412959807238586374694
3100899712069131346071328296758253023455821491848
0960748972838900637634215694097683599029436416

and the sum of the digits on the right hand side is, in both cases, 666. In fact, 666 is the only integer greater than one with this property. (Also, note that from the two powers, 47 and 51, we get (4+7) (5+1) = 66.)
Mr. Keith also points out that if we assign numerical values for the letters of the alphabet starting with A = 36, B = 37, and so on, we find that the letters in the word
SUPERSTITIOUS = 666!!!

8.  What Day of the Week Were You Born On?
Even though you were there at the moment of your birth, you may not remember exactly what day of the week it was. In fact, not only will this method help you find that out, you can find out the day of the week for any date you want in the 1900's. Here is a little trick to help you figure what day that was:
Step 1) Write the last two digits of the year you were born. Call it A.
Step 2) Divide that number, that is, divide A by four. Drop the remainder if there is one.
Call this answer, without the remainder, B.
Step 3) Find the month number corresponding to the month you were born in from the table below. Call it C.
Step 4) Oh, the date you were born on, call it D. (If you were born on the 12th, call D 12.)
Step 5) Now add A + B + C + D. Divide this sum by 7. The remainder you get is the key to the day of the week.
Step 6) In the table of days below, match the remainder with the day of the week you were born on.
NOTE: This trick will work for any date in the 20th century.

TABLE OF MONTHS			TABLE OF DAYS
			Sunday = 1
January = 1 (0 in leap yr)	July = 0		Monday = 2
February = 4 (3 in leap yr)	August = 3		Tuesday = 3
March = 4	September = 6		Wednesday = 4
April = 0	October = 1		Thursday = 5
May = 2	November = 4		Friday = 6
June = 5	December = 6		Saturday = 0

9. A Strange Prime Number –
The prime number 73,939,133 has a very strange property. If you keep removing a digit from the right hand end of the number, each of the remaining numbers is also prime. It's the largest number known with this property. Take a look: 73,939,133 and 73, 939, 13 and 73, 939, 1 and 73,939 and 7,393 and 739 and 73 and 7 are all prime!

10. Multiply by 5
Most people memorize the 5 times tables very easily, but when you get in to larger numbers it gets more complex – or does it? This trick is super easy.
Take any number, then divide it by 2 (in other words, halve the number). If the result is whole, add a 0 at the end. If it is not, ignore the remainder and add a 5 at the end. It works every time:
2682 x 5 = (2682 / 2) & 5 or 0
2682 / 2 = 1341 (whole number so add 0)
13410
Let’s try another:
5887 x 5
2943.5 (fractional number (ignore remainder, add 5)
29435
.
11. Tough Multiplication
If you have a large number to multiply and one of the numbers is even, you can easily subdivide to get to the answer:
32 x 125, is the same as:
16 x 250 is the same as:
8 x 500 is the same as:
4 x 1000 = 4,000