by t.vigneswaran agder university college

by

T.Vigneswaran

Agder University college

Contents

Uses of zero The Babylonian Number System The Greek Number System The Mayan number system Indian numerals Abu Rayhan al-Biruni Brahmi numerals Gupta numerals

Nagari numerals Aryabhata (475 A.D. -550 A.D.) Brahmagupta (598 A.D-670 A.D) Bhaskara (1114 A.D-1185 A.D)

One of the common questions

Who discovered zero?

It was India that first domesticated zero, through the Hindu familiarity with the concepts of infinity and the void.

The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different

One use is as an empty place indicator in our place-value number system.

Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different.

The second use of zero is as a number itself in the form we use it as 0.

There are also different aspects of zero within these two uses, namely the concept, the notation, and the name.

(Our name "zero" derived ultimately from the Arabic sifr which also gives us the word "cipher".)

The Babylonian Number System

• Nabu - rimanni and Kidinu

Nabu - rimanni and Kidinu are two of the only known mathematicians from Babylonia.

Historians believe Nabu - rimanni lived around 490 BC and Kidinu lived around 480 BC.

• Pythagorean triples for the equation

• The Babylonians divided the day into twenty-four hours, each hour into sixty minutes, and each minute into sixty seconds. This form of counting has survived for four thousand years.

222 cba

The Babylonians developed a form of writing based on cuneiform. Cuneiform means "wedge shape" in Latin

Example:      47     

Example:     64   

 Example: 79883

• They did not have a symbol for zero, but they did use the idea of zero. When they wanted to express zero, they just left a blank space in the number they were writing.

23)60*11()60*22( 2

The Greek Number System

The Greek alphabet Attic symbols Greeks did not have a symbol for zero.

The first two Greek alphabet -- "alpha" and "beta."

Attic symbols

   = 500, =100, =10, =5, =1

For example:    

represented the number 849

The original Greek alphabet consisted of 27 letters

The Mayan Number System

The Mayan number system dates back to the fourth century.

Mayan's used a vigesimal system, which had a base 20.

The Mayan system used a combination of two symbols. 1. dot (.) was used to represent the units

(one through four). 2. A dash (-) was used to represent five.

The Mayans were also the first to symbolize the concept of nothing (or zero).

The most common symbol was that of a shell ( ) but there were several other symbols (e.g. a head).

They would have been written:

Indian numerals

The two different aspects of the Indian number systems.

First, the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 evolved into the form which we recognise today.

The second aspect of the Indian number system is the place value system.

The Indian number system use almost exclusively base 10

For example: 5864

4106108105 23

Abu Arrayhan Muhammad ibn Ahmad

al-Biruni

Born: 15 Sept 973 in Kath, Khwarazm

(now Kara-Kalpakskaya, Uzbekistan)Died: 13 Dec 1048 in Ghazna (now

Ghazni, Afganistan)

Al-Biruni was born near Kath and the town were he was born is today called Biruni

al-Biruni was engaged in serious scientific work around 990

One of the important sources of information which we have about Indian numerals comes from Al-Biruni

Al-Biruni wrote 27 works on India and on different areas of the Indian sciences.

Brahmi numerals

The Brahmi numerals have been found in inscriptions in caves and on coins in regions near Poona, Bombay, and Uttar Pradesh.

Here is the Brahmi one, two, three.

There were separate Brahmi symbols for 4, 5, 6, 7, 8, 9.

"THE BRAHMI NUMERALS"

Gupta numerals

The Gupta period is that during which the Gupta dynasty ruled over the Magadha state in northeastern India

Nagari numerals

The Gupta numerals evolved into the Nagari numerals.

Aryabhata the Elder

Born: 476 in Kusumapura (now Patna),India

Died: 550 in India

Aryabhata (475 A.D. -550 A.D.) is the first well known Indian mathematician. Born in Kerala, he completed his studies at the university of Nalanda.

Aryabhata gave an accurate approximation for pi(π).

This gives pi = 62832/20000 = 3.1416

•He also gave methods for extracting square roots, summing arithmetic series, solving indeterminate equations of the type

ax -by = c

Aryabhata gives formulae for the areas of a triangle and of a circle.

Aryabhata gives the incorrect formula V = Ah/2 for the volume of a pyramid.

• He gave the circumference of the earth as 24 835 miles.(The currently accepted value of 24 902 miles)

• Aryabhata gives the radius of the planetary orbits.

• His value for the length of the year at 365 days 6 hours 12 minutes 30 seconds.

Aryabhata knew the sign for zero and the numerals of the place value system.

• first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system.

•secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero.

Brahmagupta

Born: 598 in (possibly) Ujjain, India

Died: 670 in India

•Brahmagupta, whose father was Jisnugupta,

wrote important works on mathematics and

astronomy.

•Brahmagupta became the head of the

astronomical observatory at Ujjain

Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century.

Addition: The sum of zero and a negative number is

negative.ie. -2+0 = -2

The sum of a positive number and zero is positive.ie. 2+0 = 2

The sum of zero and zero is zero. ie. 0+0=0

Subtraction: A negative number subtracted from zero is

positive.ie. 0-(-1)=1

A positive number subtracted from zero is negative.ie. 0-(1)=-1

zero subtracted from zero is zero.

ie. 0-0=0.

Multiplication:

Any number when multiplied by zero is zero.

ie. 1x0=0.

Division: Zero divided by a negative or positive

number is zero.

ie. 0/1=0 and 0/-1=0 Zero divided by zero is zero.

ie. 0/0=0?? A positive or negative number when

divided by zero.

ie. 1/0=?? and -1/0=??

Brahmagupta is saying very little when he suggests that n divided by zero is n/0.

Arithmetical rules • A debt minus zero is a debt. • A fortune minus zero is a fortune. • Zero minus zero is a zero.• The product of zero multiplied by a debt or

fortune is zero.

solve indeterminate equations of the form ax + c = by

solves quadratic indeterminate equations of the type

For example:

22 ycax 22 ycax

22 18 yx 22 111 yx

22 161 yx

Rules for summing series

The sum of the squares of the first n natural numbers as n(n+1)(2n+1)/6.

The sum of the cubes of the first n natural numbers as 2)2/)1(( nn

Bhaskara

Born: 1114 in Vijayapura, IndiaDied: 1185 in Ujjain, India.

Bhaskaracharya became head of the astronomical observatory at Ujjain.

The formula

Pell's equation

Trigonometry

sin(a + b) = sin a cos b + cos a sin b

and

sin(a - b) = sin a cos b - cos a sin b.

22 1 YPX

Solve the problem by writing n/0 =∞.

If this were true then 0 times must be equal to every number n, so all numbers are equal.

•Properties of zero.

ie. , and √0 = 0. 002

Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January 2000.

Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21st century begin on 1 January 2001.

Zero is still causing problems!

Thank you.