DO numbers exist independent of Humans?

Plato’s heaven of numbers. Numbers exist irrespective of human existence.

Humans just give them names and connect them with physical world.

What is a number?

• Greeks Created numbers as units of length.

• It is a representation of physical world.

• They exist in an abstract way but are attachable to things for quantifying a set.

Various number systems

• Tally mark system,

• Earliest numeral system. 30000 year old bone in Czeck have 55 tally marks. First ever evidence of counting system.

• Question: What are the difficulties faced by this method?

Various number systems--Sexagesimal

• Ancient Sumerians began it 5000 yrs ago. Based on counting with finger bones

• They used counting of bones of four fingers of one palm with the thumb. One set=12( three bones per finger) .

• Set apart one finger of other hand for this. So five fingers of the other hand mean 60 which is max that can be counted.

• So they based counting based on 60. Higher numbers are expressed with these 60 numerals. They did not know about zero.

We still use this system in measuring time. Hours and minutes are divided into 60 parts

each.This is how the numerals looked

Line and circle

• Counting means moving from one number to another.

• So the earliest mathematicians developed counting and measurement following the two types of movement they knew of:

• Linear movement, and (after discovery of wheel) Circular movement.

• Number line was created mimicking linear movement.

• Angle measurement was developed following circular movement.

Angle counting.

• Why is a circle divided in 360 unit angles?

• Babylonians counted year in 360 days. This they considered as a full circle of seasons. As it was divided into 360 parts (one day each) their geometry also followed the same.

• The entire mathematics of trigonometry is based on this.

Mayan (South America)counting

• These ancients took a simpler means. They made the total number of fingers and toes as basis of counting. So their system was 20 based. All nos >20 were expressed in terms of the first twenty independents.

ROMAN SYSTEM

• They had symbols for one five, ten, fifty, Hundred, five hundred and thousand.(I,V,X,L,C,D,M) Rest of the numbers they created by putting to left to mean less and to right to mean more. They would write Nine as IX (one less than ten) and 51 as LI( One more than fifty). In their numeral 779 will be DCCLXXIX(500+100+100+50+10+10+9).

• It is impossible to do long problems with such system.

Limitations

• Such systems worked well for small societies with few people and fewer needs.

• The numbers needed were small, the calculations needed were simple.

• As long as human societies remained primitive and small they did not need a more powerful method of calculations.

The BIG problem

• As societies grew , population increased, operation with bigger numbers were needed.

• Suppose in the month of January a king has an army of 50000 soldiers. Each soldier is given a pay of 100 coins every month. How much money should the King’s cashier issue on first of every month?

• Try to find the answer in Roman numerals (EXERCISE)

• Try to find the answer in Sumerian calculation (EXERCISE)

Now let us add some other factors

• The soldiers are also given 10 coins on every Sunday for keeping their uniform clean, and 40 soldiers retire every month and 1000 soldiers are recruited every January. Now, how much shall the king spend on the soldiers in next January?

• This is not a big sum. Still, calculations would be next to impossible and time consuming using the Roman, sexagesimal or 20 based methods.

• Can you say why it would be so difficult?(EXERCISE)

WHY are they impossible?

• These systems forces one to keep counting every bit forward and backward to reach the final answer. This becomes impossible with bigger and more complex calculations.

• Its like Walking. Alright if distance is small. Progressively difficult as distance increases.

Advantage of Standard methods

• Solve the problem with the methods you are taught in school. (EXERCISE)

• What’s the difference? (EXERCISE)

You used some tricks unimaginable by Roman method or Sumerian methods • You used multiplication and addition. • Romans could not. They had to take the addition

route only. • (Note: they had a type of multiplication that was

extremely stupid and difficult and not of much use in serious math)

• For travelling long distance you use powerful machines like cars.

• For doing bigger sums you use powerful tools like multiplication and division.

HOW

• HOW did man increase his speed with a car?

• He had to invent something new. A new machine that was totally different from earlier ones. It was wheel.

• In maths too, a new machine was required to cross the barrier.

• Indian mathematicians built that machine.

The Indian system

• The Indians first proposed a unique method of counting bigger numbers by using a peculiar thing called ZERO.

• They asked, what is there before 1? Nothing.

• So they argued, everything begins at nothing

• So their counting method had at the start, NOT 1 BUT “NOTHING”

The Indian system

• For ten fingers in hand they devised ten symbols—0,1,2,3,4,5,6,7,8,9 and called them shunya,ekam,dve,treeni,chatvaari,pancha,shat,sapta,ashta,nava.

• After 9, they proposed to start counting again , this time starting with 1 and again putting 0,1,2,3,4,5,6,7,8,9 after that

• 10 they named dasha and from eleven they just started calling them eka-dasha, Dva-Dasha and so on.

The Indian system

• This resulted into a magical power to the numerals to handle bigger problems.

• How? • Say 51+48 • Roman system: LI + ILVIII = ? The symbols will not give a clue.

You need to count another ILVIII starting from LVI to reach the answer IC

• Hindu system: 5 tens and one 1 + 4 tens and eight 1s = 9 tens and 9 ones =99.You are just jumping from fifty to ninety in one go. Use of 0 makes it possible.

• How?

Let us try adding fifty with fifty

• Roman numeral: • L+L • Start counting from fifty and keep counting next fifty , find the answer and then

recall the symbol for the answer number • Anser is C. • So L+L=C

• Indian system • 50+50 is • Add 5 with 5. It is a small sum. Answer is 10. • put the zero behind the answer. To get 100.

• Which one is simpler and easier? And less time taking? Why?

Now let us try adding fifty thousand with fifty thousand

• With Roman Numeral (EXERCISE)

• With Indian Numeral( EXERCISE)

• Which one is simpler and why?

The cause

• Zero makes recursive counting possible. NO need to remember anything however big the number is. Just follow the rule of 10.

• 10,20,30,40,50,60----------600,700.800----------------8000,9000,10000 • The same series in Roman numerals would become X ,XX, XXX,

XXXX, L, LX----CCCCM, CCCM,CCM-----,MMMMMMM, MMMMMMMM, MMMMMMMMMM

• For smaller numbers it is complicated. For larger numbers it is impossible. Try writing Twelve lakh eighty five thousand six hundred and forty five in Roman numerals and see. (EXERCISE)

• Now do it with Indian numerals. • Notice any difference?

• 10 based Indian system remains equally simple at any size.

Part 2

• Dashamik—The Road to Decimals

The issue of fractions

• Now, in a simple life all you need to count and calculate are basically whole numbers.

• Like, number of humans, numbers of swords, number of horses—no problem with them. Such counts are called Discrete, as they have to take only some fixed values like 1,2,3,4,----100000000 etc. They cannot take value between say 1 and 2.

• For such numbers all types of number systems work well. • With advancement of society when bigger numbers were

needed Indian mathematicians addressed that by inventing the ten based system.

•

Are all counting discrete?

EXERCISE

• Is there any other kind of counting that are not discrete ?

• What about length, area or volume?

• What about weight?

• They can take any value and not only certain fixed values. Such counting are called continuous.

Fun fact

• Time is discrete.

• It cannot take any value less than 10-43 seconds.

• Time is a kind of stream of little time packets (or Time quanta) each of length 10-43 seconds, called plank time.

• Lets go back to main discussion:

• Can you express this number with roman numerals or Sumerian numerals?(mind it, they did not have decimals and you cannot use it) –”10 and one third”.

• But why do we need such numbers a all?

The issue of fractions

• As life progressed, newer factors like, rituals, science, medicine , music etc started coming in. The humans felt the need for more calculations based on continuous numbers.(WHY? ASK AND EXPLAIN)

• Use of continuous numbers: – Measurement of movement of heavenly bodies,

– Measurement of Yajna Vedikas,

– measurement of medicine ingredients,

– Measurement of meters (music, verses).

The initial solution

• When faced with the issue of continuous numbers we tried to solve them by creating smaller units.

• For handling parts of a day (that is less than one day) we created hours. So instead of half a day we can call it 12 hours. Then we went further down to minutes and seconds, pals, anupals, etc.

Drawbacks of this solution

• To express one tenth of a meter we create 1 decimeter, for one hundredth we create 1 centimeter for a thousandth part we create 1 millimeter.

• Now what if we need to measure a millionth part? A trillionth part? It never ends. How many new units should we create?

• Also , what about four sixth part of a KG?

• Or seven twelfth part of one meter?

• Trying to create new, smaller units for each of them will create uncountable number of units. Nobody can remember so much.

Drawbacks of this solution

A new tool is needed.

• So , here also , what was needed was a powerful new tool, that, like zero , shall create a system to mechanically express any such part of a whole.

• Initially it was solved without any new tool by writing the expressions as (4/6),(7/8),10 ⅓

• But it was a clumsy affair to calculate with such expressions, especially when the numbers were real complicated and/or big.

Why clumsy?

• Not easy to compare big or small.

– Take 38/69 and 35/67.

– Which one is smaller?

• Calculation rules are different and more complicated than calculation with whole numbers.

– (you need to do LCMs, multiplications, additions—all for solving one small issue)

The magic solution

• The Indian mathematicians solved this major problem with a beautiful imaginative jump. They created a (.)

• They converted every fraction to exact or approximate parts of 10 or its multiples.

• For example 2/5 (2 parts out of 5 parts) will be 4/10, 7/8 (7 parts out of 8 parts) will be 875/1000. 6/7 (6 parts out of 7 parts) will be approximately 857/1000 (or 857/(10x10x10) and so on.

• This they did by creating a method of dividing the numerator with the denominator by using zeros whenever needed. (we still use it everywhere in fifth standard mathematics)

• Once a fraction is thus expressed as a number on top and a multiple of 10 at bottom (say 10, or 10x0 or 10x10x10and so on), they brought in this dot.

• First they put it at the end of the number at the top. Now, for every 10 at the bottom , they moved it to the left by one digit Har dus ke liYe ek= dasham+ek =dashamik. And thus the almighty “point” got its name.

• Afterwards when the Sahibs learnt about this from the Arabs they gave it the same name, Decimal, in their language, taking the Latin word Deca for ten.

The magic solution

That simplified the matter a lot

• 38/69 and 35/67. Which one is smaller? How to add? COMPLICATED.

• In decimal system they are 0.55 and 0.52. Easy to compare. Easier to add or do any other operation.

What happened with the discovery of decimal

• Mathematics leaped ahead. • Intricate and superfine calculations became

possible. Every branch of knowledge from chemistry to physics to astronomy to music to architecture flourished.

• Human civilization started jumping ahead. • In its first 50 lakhs year of existence we had

achieved only fire and stone tools. With the help of zero and decimal , in the next five thousand years we sent men to moon and our spaceships to beyond solar system.

EXERCISE

• Do you agree that Decimal system of calculation formed the basis of all the Scientific and technological progresses of Human civilization.

• Justify , if you agree.

• Justify if you do not.

Where do we find the first applications Of decimal system

• First Use –INDUS Valley–weights 0.05, 0.1, 0.2,0.5, 1,2,5,10,20,50,100,200,500 ratios.

• Such kind of measurements would not be possible without the powerful tool of decimal method.

• Length measurement 2 mm graduation lothal ivory scale, 35.5 mm Mohen jo daro scale

• Used for planned city making-bldgs, sewarage, canals (gradient calculn) for trade and business, for navigation in sea (astronomy).

But , was the calculations of the Indus people so advanced?

• Stand in the open in a clear night. Take a star in the sky. Draw two imaginary lines, one on the ground from where you stand upto the horizon and the other in air, from where you stand upto the star.

• The angle between this line is the declination of the star. • Aryan book Satapatha Brahmana (1800 BC ) mentions Krittika

Nakshatra to be right at the horizon, that is with 0 Deg Declination. • This measure used to position and shape some yagna vedikas. • Modern scientists have calculated that actually the star was at 0

deg declination during Indus time, i.e. 3000BC. • So Indus people measured them while the Aryans just followed

that information without doing the actual calculations during their times.

• This race of tech savvy people knew what they were doing. Once we know how to read their languages we shall know for sure.

• Look at these perfect measurements.

Part 3 Development of Maths in India—the

Aryan days

Maths of the Aryans

• Started using maths mainly for religious rituals. Construction of Yagna Vedika of various types, gave rise to the branch of Geometry.

• Calculations for Yagnavedikas of various geometric shapes were described in :Shukla Yajurveda and Shatapatha Brahmana. (1800 BC)

• In the same book, calculations for brick making re-appeared long after the Indus civilization. It indicated the second wave of urbanisation in India.

• Knowledge of geometry amplified maths for land distribution Mensuration.

• Need for trade, commerce, astronomy formalisation of decimal system Arithmatic.

• First details of arithmetic in 1500 BC in the book Narada Vishnupurana. (contains details of four basic operations.

• Geometry, Mensuration, Arithmatic all these branches of Mathematics were “applied maths” created for utilitarian purposes.

Maths of the Aryans: evolution of branches

• Non utilitarian math (pure mathematics without direct application in everyday life):

• There was pure math also.

• Starting 800 BC a series of Sulva sutras were written.

• These books brought in the concepts of rational and irrational numbers.

• .

Maths of the Aryans evolution of branches

Pure maths in Sulva Sutra

• rational number—number expressible in a/b form. (Ex: 1,2,3,34/67, 0.25 etc)

• Irrational number—number that cannot be expressed as a/b. (Ex : sq. root of two.)

• The book described a method of expressing an irrational numbers as a sum of a never ending chain of rational numbers.

• Say , Square root of 2 will be 1+4/10+1/100+4/1000+ ---------so on.

Why this is needed?

• Discovery of this method simplified many real life problems. Suppose there are three pieces of land each measuring 1 km by 1 KM and you need to lay ropes along their Diagonals for some work. Each diagonal will measure √(2)KM. How shall you cut a rope of that length? The above method will come to your rescue here.

Another gem from Sulva Sutra

• Pythagorean Theorem

• H(2)=b(2)+h(2) Amount of a square land on hypotenuse is equal to amount of square land on base and height.

• This was first discussed in Sulva Sutra far before Pythagoras mentioned it.

More on Sulva Sutras

• Being able to relate hypotenuse to base and height formed the basis of another powerful branch of mathematics –Trigonometry.

• Pythagoras, the Greek from Samoa had visited India from where he learnt this, went back to Europe and spread it as his discovery. He is called Yavanacharya or Pitar guru in India.

• His Indian visit has been described by Voltaire.

Panini and his maths of Language

• In sixth Century BC Panini wrote a book called AshTadhyayee

• Here he organised Sanskrit language in a scientific method and developed a grammar for writing in this language

• If one wants to explain what he eats, as per Panini’s grammar one should write

• Main khaataa roti. • But what’s the big deal in that? We all use such

sentences when we write something.

Maths of language

• How will you ask your mom to give you food? • Your asking can be:

– Bhukh lagi hain – Mummy khana dO – Mummy khana bana kya? – Kob degi tu khana? – Mummy khana---a—a—a--

• Mom shall have no problem in understanding all these expressions. • 2500 years after Panini, in 1958 when the first computers were

being made, it was seen that a computer was not capable of understanding such variations . It needed a very specially structured language to understand the commands of its Human masters.

Computer language—Normal form

• In 1958 , a man called Baccus started formulating the computer language.

• He created a writing style called normal form

• It is like this:

• Suppose I want to give this instruction to a computer:

• “Any address should contain a name followed by Road name and then by Postal Code.”

• Baccus found that computers would understand this instruction if it is arranged as below:

• <address> ::= <name> <Road name> <post code>

• Computer language developed on this format.

• After this another scientist called INGERMAN found that The language structure formed by Panini was exactly like this NORMAL FORM, a form that is required to do scientific works.

Computer language—Normal form

HOW

• Question: What is my food? • Answer (in normal form) • <Mera khana> ::= <Main (Subject)>+ <Khaataa (Verb)> + <Roti (Object)>

• i.e. Main khaataa roti, which as already seen is as per Panini’s grammar.

• Thus, far before discovery of computers , Panini created a language style that would have such a strong scientific basis that that would be usable in machines to be discovered 2500 years later.

Also:

• Let us take this problem: • 15 people eat 10 rotis each per day and 6 Rotis per

night. What is the total no. of rotis they would eat in a week?

• The mathematical solution to this problem is: • <total> ::=15X(10+6)X7 • In words this will be : • <total> ::=<Persons> <multiply><roti in

day<sum>Roti in night> <multiply><Days> • Can you see the similarity of this with the Panini

language structure?

So, Not only for computers---

• Panini’s language arrangement methods gave us the tool of arranging long mathematical problems in proper order so that they could be easily solved.

• This proved immensely helpful in development of higher mathematics with large and complicated calculation.

• So, while writing a simple grammar book, Panini actually created the arrangement methodology for solving difficult mathematical problems and also a writing method that could be used in computers, to be discovered 2500 years after writing of his book.

Fun Fact

• Earlier there was no Sanskrit Language in India

• The Aryans here spoke “Vedic Language.”

• Panini gave this language a structure ,i.e. he made “Sanskaar”(Renovation) of the Language.

• From then the Vedic language changed and its name was changed to “Sanskrit” (meaning “Jiska sanskaar kiya gaya hain” or, “the ranovated one”.

Part 4

• JAIN MATHEMATICS—ROAD TO INFINITY

Jain maths

• In 6th Century BC when Panini was restructuring the Vedic language another religion was becoming very important in India side by side with Vedic religion.

• This was Jainism.

• Have you seen a Jain Sadhu?

• They are very religious persons.

Not only Religious

• Maybe today they are only religious, but from sixth century BC to 500 AD, there were many Jain hermits who also studied mathematics deeply and made fantastic contribution to mathematical sciences.

• Strange, isn’t it? What does maths have to do with Jain religion?

Maths and the Jains

• The Jain religion believed that time has no beginning or end.

• The religion also held that the universe is endless, its life is also endless.

• Now, The Jain sage mathematicians wanted to prove to people that what their religion says is correct.

• But how to show that?

The road to the proof

• To prove the infinite nature of time , life and universe the Jain sages started studying mathematics.

• They wanted to prove these by mathematical logic.

• But mathematics at that time played with only smaller numbers needed for real life trade, astronomy and religion. There was no method of catching the infinite in the existing maths.

The road

• The Jain mathematicians made a very simple assumption before going into the work. They said, to reach the highest place you first need to go higher and then even higher. Only then you can reach infinite.

• So they started their work for finding huge numbers using known units and things.

• How?

Measuring huge distances

• RAJJU

• If a god travels 1 lakh Yojana (i.e.10 lakh kilometers as we call it today) in a second, then the distance he will travel in six months is called a RAJJU. This is equal to 15552000000000 kilometers (this is equal to distance travelled by light in 60 days)

Measuring huge numbers

• Pallya • Take a pot of length breadth and depth 1 Yojana each.

Now, take one hair from a sheep every century and put in the pot.

• The time it will take to fill the pot is called one Pallya. • An average 5 cm long hair has a volume of 0.0001 cubic

cm. • The volume of the pot will be 1000000000000000000

cubic centimeter. So pallya will be equal to 1,000,000,000,000,000,000,000,000 years.

• That’s a pretty long time , isn’t it?

Easy to say but how to write such big numbers?

• If you start following the advice of the Jain mathematicians you shall face a problem.

• DO you know the biggest number that has a name? • It is Googoloplexian. How big is it? • 1 followed by 100 zeros is called a googol. • 1 followed by one googol number of zeros is a googoloplex. • 1 followed by one googoloplex number of zeros is called a Googolplexian. • It is so big that if you want to write it down you will need to cut all the

trees in the world and shall still be short of paper. • You saw the result of applying the Jain advice on Pallya. • Now, if you want to apply the Jain advice on this number, how shall you

write this down? • Also, how to do calculations with such big numbers? A lifetime may be

required to just write down and do one sum only!!!

The Jain Solution

• The Jains solved this problem by creating a new number writing method called index numbers.

• They started representing big numbers with help of small numbers raised to powers.

• The number of times a number can be divided by 2 without leaving any remainder was given the name ‘Ardhacched.” of that number.

• Let us take 524288. • You can divide it repeatedly with 2 for 19 times. So 19 will be called

the ardhacched of this number. In other words, 219=524288, that is TWO RAISED TO THE POWER 19 is 524288.

• Similarly 1162261467 can be written as 319 . Here 19 will be called the Trikacched of this number.

• An so on.

• For 1000000 they would write 106 .Nice and short, isn’t?

• In this method you can write googoloplexian in a small chit of paper as

• How powerful is this method?? A number that cannot be written with all the papers made by all plants on earth can be compressed in just half a inch place!!!!

• This discovery drastically reduced the size of large calculations and helped development of complex mathematics immensely.

• These techniques were written down in the Jain book Shatakhandagam of Second Century AD.

The power of “To the power”

Arriving at the logical idea of an infinite number

After experimenting with measurement of huge time and huge length, and solving the problem of writing huge numbers in short form, the Jain sages now devised a method for guessing an infinite “NUMBER”.

They proposed:

Take a pot as wide as the Earth.

Start putting and counting mustard seeds in that pot.

When the pot is filled, still you shall not be able to reach the highest number.

If you take a pot one thousand time larger than this , fill it with mustard seeds and count each seed, still this will be smaller than the highest number that can be.

And then they concluded:

The number that is bigger than any big number that human mind can imagine or calculate is called ASEEM or INFINITY.

Different types of infinite

• The jains proposed the following types of infinite:

• Unidirectional infinite (The destination number if you start counting all positive numbers starting at 1)

• Bidirectional Infinite (The two opposite destinations if you count all positive and all negative numbers)

• Two dimensional infinite: Place having infinite length and infinite breadth

• Three dimensional infinite: Place having infinite length breadth and depth

• Four Dimensional infinite: Three dimensional infinite spread over infinite time.

Different types of infinite

Different types of Jain counting

• Sankhyat (Countable)

• Asankhyat (Countable but infinite)

• Anant (Endless and not countable

• Exercise: Can you find some examples??

Reaching infinite through calculation

• Besides theory Jains also made an actual calculation model to reach infinite

• They proposed:

– Take the biggest number you can imagine. Let it be A.

– Now multiply A with A. You get A square.

– Take this to be B.

– Now do the same to B and get B square.

– Keep repeating this and you shall be on your way to infinity.

And thus--

• The Indian mathematicians worked for almost three thousand years and created a fantastic basis of mathematical calculations that has created computers, sent men to space, solved the problem of many diseases, created our cities, aeroplanes and everything we need in modern lives.

• All with a 0 , a dot and Infinity.

Before we end

• Side by side with this main flow of mathematical progress, there were many interesting branches. Here is one such interesting work--

Mathematics of music and verses

• Side by side with religion, Jain mathematicians tried to catch hold of music and verses with the laws of mathematics.

• Take this example

• Water water everywhere / not a drop to drink

• Drink water Everywhere/drop water not

• EXERCISE

• Read them aloud. Is there any difference in the Rhythm?

Mathematics of music and verses

• Rearrangement of the position of the words completely changes the rhythm.

• Now, let us make it a bit more interesting.

• Lets sing this line:

• Ga ma pa sa’ ni pa ga ma ga

• Now lets sing this:

• Ga ma ga pa ga ma pa ni sa’

• Any difference?

• Rearranging the music notes create completely different music, right?

• That’s how different Ragas and songs are made. They all are based on 8 basic notes but just arranging and rearranging them differently we get such variety of music!!

Mathematics of music and verses

Mathematics of music and verses

• So the Jain sages started examining in how many possible ways the musical notes or syllables in a verse can be rearranged to create different meters or Ragas.

• Their findings , written in a book called Bhagabatisutra of three hundred BC gave the world the mathematics of permutation, combination.

But, what for?

• Permutation and combinations , or the science of arranging things in different orders and selecting a few from many things , is a very useful thing in scientific experiments, Tests, Market analysis and many other modern day processes. Let’s see how?

Permutation

• A scientist is testing the effect of three drugs A,B,C on people.

• They have to be administered one each on three consecutive days.

• The drugs have different effect depending on the order in which they are administered.

• Only one order will give the right result.

• How many people will be required for the experiment?

Combination

• Your Teacher decides to select 6 questions from a chapter having 20 questions for a test.

• In how many ways can the questions be chosen?

A game of chances

• A scientist is making a machine that has 200 ball bearings arranged in a big ring that rotates once in every 10 minutes.

• It has been seen that when the ring rotates, it heats up at three different points heating up three ball bearings.

• 60 ball bearings among the 200 are made of plastic and melt when they heat up.

• If three ball bearings melt at a time the machine breaks down.

• Now, What is the chance that the machine will run successfully for 10 minutes?

So--

• Solving these problems is the only way for the Doctor, for the teacher or for the scientist to find appropriate way of working with their experiments, their tests and their machines.

• Permutation and combination mathematics devised by the Jain saints help us solve our modern day problems in this fashion.