MATHS AND MUSIC

Math and music have always been considered closely connected in many ways. 

• It is widely believed that students who do well in music also excel in math. 

• Let’s take a look at some of the basic components of music and see what math has to do with them. 

Rhythm is to Music as Numbers are to Math

Rhythm measures time 

• Measure is the space between two bar lines on the staff that 

   represents the division of time by which air and movement of 

   music are regulated 

• When you play a few different notes together or even repeat the 

   same note on an instrument, you create something called rhythm.

Pythagoras 

• The Greek octave had a mere five notes. 

• Pythagoras pointed out that each note was a fraction of a string. 

• Example: Lets say you had a string that played an A.

  The next note is 4/5 the length (or 5/4 the frequency) which is 

   approximately a C. The rest of the octave has the fractions 3/4 

   (approximately D), 2/3 (approximately E), and 3/5 

   (approximately F), before you run into 1/2 which is the octave A


Ratios 


• Pythagoras was excited by the idea that these ratios were made up

  of the numbers 1,2,3,4, and 5. 

• Why ? 

• Pythagoras imagined a   "music of the spheres" that was 

   created by the universe.

 • The 18th century music of J. S. Bach, has mathematical 

    undertones, so does the 20th century music of Philip Glass. 


Golden Ratio and Fibonacci 


• It is believed that some composers wrote their music using the 

   golden ratio and the Fibonacci numbers to assist them

 • Golden Ratio: 1.6180339887 

• Fibonacci Numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21


From Then to Now 


• So, how did we get the 12 notes scale out of these six notes ? 

• Some unknown follower of Pythagoras tried applying these ratios 

   to the other notes on the scale. 

• For example, B is the result of the 2/3 ratio note (E) applied to 

   itself. 2/3 * 2/3 = 4/9 which lies between octave A (1/2) and 

   octave C (4/10). To put B in the same octave we multiply 4/9 by 

   two to arrive at 8/9. G is produced backward from A. As B is a 

   full tone above A at a string ratio of 8/9, we can create a missing

   tone below   A by lengthening the string to a ratio of 9/8. To add 

   G to the same octave we apply 9/8 to 1/2 (octave A) and by 

    multiplication we get 9/16 as the ratio to G. 

• BUT! There was a problem, however, if you performed this 

   transformation a third time. The 12 tone octave created by 

   starting with an A was different than the 12 tone octave created 

   when you started with an A #. 

• Which means that two harps (or pianos, or any other instrument) 

    tuned to different keys would sound out of tune with one 

    another. Also, music written in one scale could not be 

    transposed easily into   another because it would sound quite 

    different.

 • The solution was created around the time of Bach. A "well 

    tempered" scale was created by using the 2 to the 1/12th power 

    ratio mentioned above. Using an irrational number to fix music 

    based on ratios, Pythagoras probably rolled over in his grave.

MATHS AND DANCE

MATHS IN SPORTS

MATHS CROSSWORD

Across 

2. This type of number is the most important to factor. 

3. Twelve inches. 

4. Letters representing numbers. 

6. Three squared. 

7. Nothing with a belt.

 9. An angle less than right. 

11. A regular polygon with interior and exterior angles equal.

Down 

1. An all round perimeter. 

2. An American government polygon. 

5. Powerful numbers found in fishy places. 

8. This sum is mixed up alott. 

10. A selfish average.

ANSWER

AMAZING MATH TRICKS

The 7 - 11 -13 trick!

This trick makes you look like your brain is a mega fast calculator! 

 Ask a friend to write down ANY three digit number such as 231 

    or 884. 

Ask them to multiply the number by 

x 7 

x 11

 x 13 

 ...but even if your friend has used a calculator, you will have 

    written down the answer ages ago! 

 THE SECRET: all you do is write out the starting number 

    twice!

    So 231 will become 231231 and 884 will become 884884. 

You don't believe it? Well try it on this calculator and see for 

yourself! You work this calculator by clicking the mouse on the 

buttons. 

Go on, put in ANY three digit number then x 7 x 11 x 13 and see 

what you get!


The 3367 trick:


This trick is similar to the 7-11-13 trick. It's harder to do, but it 

looks far more miraculous! 

 Get a friend to pick any 2 digit number e.g. 74 

 x 3367

 To work out the final answer you have to imagine the original 

      number written out three times e.g. 747474 then divide it by

      three. 249158 

 This one takes practice, but unlike the others, it's very hard to see

     how it's done! 



THE MISSING DIGIT TRICK!

Here it is!

  Cover your eyes in some way so that you can't see what your 

      friend is writing. 

 Ask a friend to secretly write down ANY number (at least four 

    digits long). e.g. 78341 

 Ask the friend to add up the digits... e.g. 7+8+3+4+1 = 23

  ... and then subtract the answer from the first number. 

     e.g. 78341 - 23 = 78318 

 Your friend then crosses out ONE digit from the answer. (It can 

     be any digit except a zero) e.g. 7 x 318

  Your friend then reads out what digits are left .e.g. 7-3-1-8 

 Even though you haven't seen any numbers, you can say what 

     the missing digit is! EIGHT

  THE SECRET  

This great trick relies on the power of 9. 

 After your friend has added up the digits and subtracted them, 

     the answer will ALWAYS divide by 9. If a number divides by 

     nine, then when you add the digits up, they will also divide by 9.

     If you check our example 7+8+3+1+8 = 27 which does divide 

     by nine.

  When your friend crosses a digit out, he then reads out the 

      digits that are left. You add them up. In the example we had 

      7+3+1+8 = 19 

 All you do now is see what you have to add on to your answer 

     to get the next number that divides by nine! The next number to

     divide by 9 after 19 is 27. So you need to add on EIGHT. This 

     is the number that was crossed out!

RAMANUJAN MAGIC SQUARE

This square looks like any other normal magic square. But this is formed by a great mathematician of our country  –  Srinivasa Ramanujan. 

What is so great in it?

SRINIVAS RAMANUJAN

An equation for me has no meaning unless if expresses a thought of God.                                                  

                                                                     -- Srinivas Ramanujan

Srinivasa Ramanujan Iyengar, known to most simply as Ramanujan as the other components of his name are surnames borne of various traditions, was born on 22 December, 1887 in the small village of Erode, India. Legend tells Ramanujan’s mother became pregnant several years after marriages only after her father prayed to the Goddess Namagiri to bless his daughter with offspring. The son of an accountant, Ramanujan was born into the Brahmin caste - the highest, and most orthodox Hindu, level of the Indian caste system. While he was deeply influenced by the traditions of this caste, his family, like most others in Southern India around this time, survived in relative poverty. Ramanujan was one of six children, three of whom died before their first birthdays. At age two Ramanujan contracted smallpox. Ramanujan survived. He would go on to live a short, often sickly, life. But Ramanujan's life was a life of extraordinary genius which would see him became India's foremost mathematician and a legend for the ages.

Ramanujan started school at age five. While “quiet and meditative”, his mathematical abilities were recognized quite early. He had a great ability to repeat all of the formulas and theorems he had been taught. He could recite the digits of pi and the square root of two to as many places as listeners could bear to hear. The apparently critical moment in Ramanujan’s mathematical development was at age 15 when he finally came into possession of a significant book of mathematics - A Synopsis of Elementary Results in Pure and Applied Mathematics by George S. Carr. This was a lengthy, two volume compendium that contained many of the important mathematical results through the middle part of the nineteenth century.

It was hardly a text from which one could learn - it was very terse, almost dictionary like. Yet Ramanujan set out to understand and establish all of the results in this text on his own. His obsession with mathematics kept him from fulfilling the scholarships that he had won to government colleges. He was inseparable from the two large notebooks that he filled with his mathematical ideas. Ramanujan would later claim that his inspirations came in the form of dreams from the Goddess Namakkal.

In the early part of 1909 Ramanjuan became quite ill. Later that year Ramanujan was married in an arranged marriage. He would not live with his wife, age nine, until she turned twelve. Throughout this time Ramanujan unwillingly accepted financial support from local benefactors so he could pursue his mathematics ruminations. One such benefactor described him as “Miserably poor... A short uncouth figure, stout, unshaved, not overclean, with one conspicuous feature - shining eyes... He never craved for any distinction. He wanted ... that simple food should be provided for him without exertion on his part and that he should be allowed to dream on.” When Ramanujan tired of this support he became an office clerk. But at each opportunity he sought to share his mathematical work with others who might be in a position to judge or appreciate it. 

On the heals of several fortuitous introductions, Ramanujan was encouraged enough to write to G.H. Hardy - then a Fellow at Trinity College, Cambridge and who would become one of the twentieth century’s most famous mathematicians. 

Ramanujan’s letter, dated 16 January, 1913, is a picture of modesty:

I beg to introduce myself to you as a clerk in the Accounts Department... I have no university education... but I am striking out a new path for myself... The results I get are termed by the local mathematicians as “startling”... [Yet] the local mathematicians are not able to understand me in my higher flights... I would request

you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published... Requesting to be excused for the trouble I give you, I remain, Dear Sir, Yours truly, S. Ramanujan.

The letter included more than 100 theorems that Ramanujan had discovered. They are fabulously intricate wonders such as:

Prof Hardy: [These formulas ] defeated me completely. I had never seen anything in the least like this before. A single look at them is enough to show they could only be written down by a mathematician of the highest class. They must be true because no one would have the imagination to invent them

Hardy immediately arranged for Ramanujan to come to England. Yet the prejudices of India’s caste system made Ramanujan feel that he could not accept Hardy’s invitation to come to Cambridge. Moreover, Ramanujan’s mother would not give her consent. Hardy

made every effort to encourage him to come to Cambridge, even enlisting his friends as allies.

While he was able to sway Ramanujan, it was not until Ramanujan’s mother announced that “she had a dream on the previous night, in which she saw her son seated in a big hall amidst a group of Europeans, and that the goddess Namagiri had commanded her not to stand in the way of her son fulfilling his life’s purpose.” On 17 March, 1914 Ramanujan sailed to England. In April he was admitted to Trinity College - a remarkable honor

Yet outside of mathematics Ramanujan did not prosper as well. He had a very difficult time adjusting to life in England. He was a strict vegetarian. He cooked all of his food himself and had difficulty obtaining food that was compatible with his usual diet. The climate of England was totally different from his native India. The stone buildings of Trinity College were cold and damp. Never having encountered such cold, Ramanujan slept in his overcoat, wrapped in a shawl. It was not until a fellow Indian student at Trinity realized that Ramanujan did not understand the purpose of the many blankets spread neatly on his bed that Ramanujan learned to lift up the blankets and slide under them to keep warm. England joined World War I. Trinity College housed open air hospitals for wounded soldiers and adequate vegetarian food became harder to come by. Ramanujan was often sick. In the spring of 1917 he became particularly ill. He was diagnosed with tuberculosis and placed in a sanatorium.

Despite his illness, his impact spread and his reputation grew. In May of 1918 he was elected as a Fellow to the Royal Society - one of the highest academic honors of the time. His election was all the more remarkable as it came on a first ballot, was the first time an Indian had been so honored, and it came at the remarkably young age of 30. Several other major honors were also bestowed on him during this year. These honors seemed to have buoyed his health for a short time and he continued to develop beautiful and important mathematical discoveries. 

Early in 1919 his health worsened again. He returned to India where he died on 26 April, 1920 at the age of 32. He had no children. He was survived by his wife and his parents.

The richness of Ramanujan’s mathematical legacy is in sharp contrast to trials of his brief and difficult life. Ramanujan’s own published works are of sufficient importance to consider him one of the elite mathematicians of the twentieth century. But his impact did not stop there. 

He left many notebooks full of unpublished theorems, results, and ideas. These notebooks have been intensely studied by mathematicians and have resulted in hundreds of papers whose contributions are direct results of the work laid out by Ramanujan. 

Indeed, almost eighty years later, the notebooks of Ramanujan served as the impetus of the major new discovery by Ken Ono that you investigated in Topic 5. Ono says that while he "was familiar with a lot of what he had done through the writings of more modern mathematicians, I didn't suspect that I would learn anything from studying Ramanujan's notes." 

However, one mathematical identity, written in a particularly obtuse fashion, even for Ramanujan, struck Ono. "This can't be right." Yet it was. This one identity helped Ono establish "spectacular" results that are "the most important work on partition congruences since the epic work of Ramanujan" and among the most notable of the past decade. Ono "learned a valuable lesson.

It sometimes really pays to read the original." Who knows how many other mathematical gems are still unearthed in Ramanujan's notebooks.

THE MANDELBROT SET - FRACTAL MATHEMATICS

The Mandelbrot set is one of the most famous sets in mathematics. 

It is an important example of a fractal - a mathematical object that is approximately self-similar across an infinity of scales. 

This set was named after Benoit Mandelbrot (1924- ) who, as an IBM researcher in the 1970’s, was the first to use computers to visually explore the complex mathematical objects that had been first investigated by the French mathematicians Pierre Fatou (1878-1929) and Gaston Julia (1893-1978). 

Fractals play a critical role in many natural and physical processes. 

MAGIC SQUARE OF JUPITER

Another famous magic square appears in a woodcut by the German artist Albrecht Dürer, who lived from 1471 to 1528. It is called the Magic square of Jupiter.

a Find the 4-digit numeral contained within the square that 
   identifies a year that occurred during Dürer’s lifetime.

b What is the magic sum for this 4 × 4 square ?

c Find five 2 × 2 squares within the magic square for which the 
   numbers have the same total as the magic sum.

d Apart from the two diagonals, find four numbers each from a 
   different row and  column that add to the magic sum.


There are more than two solutions.

MAGIC SQUARES

Magic squares have every row, column, and diagonal adding to the same magic sum.

The Lo-Shu magic square dates back to about 2200 BC. It appeared on an ancient Chinese tablet and was first drawn on a tortoise shell given to the Emperor Yu.

GOOGOL-PLEXING

The number ,  the googol, is 1 followed by one hundred zeros. 

The name ‘googol’ was created by the 9-year-old nephew of American mathematician Dr Edward Kasner.

The number, that is 1 followed by a googol zeros, is called the googolplex. 

The googol is a very big number but it is rarely used for practical purposes. 

Even the number of particles in the observable universe, estimated at being between  and  , is less than a googol!

The Internet search engine Google was named after the googol, to reflect the huge size of the world wide web. 

It was invented in 1996 by two Stanford University students, Larry Page and Sergey Brin. 

Google is a powerful search engine because it can find information from at least two billion web pages in less than one second.

How many googols are there in a googolplex?

ARITHMAGONS

Arithmagons are number puzzles made from triangles. The circled numbers are added together to give the number on the line joining the circles. The challenge is to find the correct numbers to go inside the circles.

Find the numbers missing from the circles in this arithmagon:

Solution

Think about the pairs of numbers that add together to give the numbers on the lines.

11 = 10 + 1 = 9 + 2 = 8 + 3 = 7 + 4 = 6 + 5

10 = 9 + 1 = 8 + 2 = 7 + 3 = 6 + 4 = 5 + 5

5= 4 + 1 = 3 + 2

Now, by trial and error, write numbers in the circles to reach a solution:

11 = 10 + 1? 11 = 9 + 2 ? 11 = 8 + 3 ?

HISTORY OF NUMBERS

Mathematics began with the invention of numbers. Early tribes used notches on sticks, pebbles, and knots in ropes to represent numbers.

We use the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This is called the

Hindu–Arabic system, but there have been many other number systems before it. 

It is important to be able to read and write numbers in our own number system and to understand the rules our numbers follow.

The ancient Egyptian number system

The ancient Egyptians used one of the earliest number systems about 5000 years ago. Pictures called hieroglyphs represented

words or sounds. They were written on papyrus (a type of paper made from reeds) or painted on walls.

The hieroglyphic symbols used by the Egyptians were:

The Australian Aboriginal number system

The Australian Aboriginal way of life had no need for a complicated number system. Their society relied on story-telling and did not have symbols for numbers. Different tribes had their own names for numbers. Here are two examples:

Belgando River Aborigines

1 = Wogin 

2 = Booleroo

3 = Booleroo Wogin 

4 = Booleroo Booleroo

Kamilaroi Aborigines

1 = Mal 

2 = Bulan

3 = Guliba 

4 = Bulan Bulan

5 = Bulan Guliba 

6 = Guliba Guliba

The Babylonian number system

The ancient kingdom of Babylon existed from about 3000 to 200 BC where Iraq is today. Babylonian writing used wedge shapes called cuneiform. The wedges were stamped into clay tablets which were then baked. Babylonian numerals also used cuneiform.

While our number system is based on 10 and 100, the Babylonian number system was based on 10 and 60. 

The Roman number system

The Roman empire was one of the greatest empires. Roman numerals were invented about 2000 years ago. They were used until the end of the 16th century. Today they are used mainly in

clocks and for some page numbers in books.

The modern Chinese number system

Chinese people today use the numerals below.

• The Chinese write from top to bottom.

• The symbols in a number are grouped in pairs and the numbers in 

   each pair are multiplied together.

• The products are added to give the number.

The Hindu–Arabic number system

Our number system goes back to the Hindus (who lived in India) and came to Europe through the Middle East/Arabia. 

Our system needs only 10 symbols called digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It is easier to use because it has a zero and the position of each numeral determines its value. This is called place value.

The numerals first appeared in Europe in the 10th century, but were different to the ten numerals we use today.

The following table shows how our numerals have changed over time.

The Hindus called the zero ‘sunya’ meaning a void. Other names used were ‘cipher’, ‘nought’ and the Arabic ‘sifr’.