Transcript document

by
T.Vigneswaran
Agder University college
Contents
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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
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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
a2  b2  c2
• 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.
The Babylonians developed a form of
writing based on cuneiform. Cuneiform
means "wedge shape" in Latin
 Example:
47
 Example:
64
 Example: 79883
(22 * 60 2 )  (11 * 60)  23
• 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.
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,
For example:
represented the number 849
=5,
=1
 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
5  103  8  10 2  6  10  4
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
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 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.
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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
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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.
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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
ax  c  y
2
2
For example:
ax  c  y
2
8x 2  1  y 2
11x  1  y
2
2
61x 2  1  y 2
2
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 (n(n  1) / 2) 2
Bhaskara
Born: 1114 in Vijayapura, India
Died: 1185 in Ujjain, India.
 Bhaskaracharya became head of the
astronomical observatory at Ujjain.
 The formula
Pell's equation
PX 2  1  Y 2
Trigonometry
sin(a + b) = sin a cos b + cos a sin b
and
sin(a - b) = sin a cos b - cos a sin b.
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. 0  0 , and √0 = 0.
2
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.