The document discusses the history and development of numeral systems. It notes that the most commonly used system today, using the digits 0-9, was developed in India by mathematicians like Aryabhata and Brahmagupta. This Hindu-Arabic numeral system later spread to the Middle East via Arab traders and was modified before being adopted in Europe. Key aspects included the development of place-value notation and the introduction of the zero symbol. This system is now used globally due to its simplicity compared to earlier additive systems.

1. A numeral system is a writing system for expressing
numbers, that is a mathematical notation for representing
numbers of a given set, using graphemes or symbols in a
consistent manner. It can be seen as the context that allows
the symbols "11" to be interpreted as the binary symbol
for three, the decimal symbol for eleven, or a symbol for other
numbers in different bases.

3.  The most commonly used system of
numerals is known as Arabic numerals
or HinduTwo Indian mathematicians are
credited with developing
them. Aryabhata of Kusumapura develope
d the place-value notation in the 5th
century and a century
later Brahmagupta introduced the symbol
for zero.[1] The numeral system and the
zero concept, developed by the Hindus in
India slowly spread to other surrounding
countries due to their commercial and
military activities with India. The Arabs
adopted it and modified them. Even
today, the Arabs called the numerals they
use 'Rakam Al-Hind' or the Hindu
numeral system. The Arabs translated
Hindu texts on numerology and spread it
to the western world due to their trade
links with them.

4.  positional system, also known as place-value
notation. Again working in base 10, ten
different digits 0, ..., 9 are used and the position
of a digit is used to signify the power of ten
that the digit is to be multiplied with, as in 304
= 3×100 + 0×10 + 4×1. Note that zero, which is
not needed in the other systems, is of crucial
importance here, in order to be able to "skip" a
power. The Hindu-Arabic numeral system,
which originated in India and is now used
throughout the world, is a positional base 10
system.
 Arithmetic is much easier in positional systems
than in the earlier additive ones; furthermore,
additive systems need a large number of
different symbols for the different powers of
10; a positional system needs only ten different
symbols (assuming that it uses base 10).
 The numerals used when writing numbers
with digits or symbols can be divided into two
types that might be called
the arithmetic numerals 0,1,2,3,4,5,6,7,8,9 and
the geometricnumerals 1,10,100,1000,10000...
respectively. The sign-value systems use only
the geometric numerals and the positional
systems use only the arithmetic numerals. The
sign-value system does not need arithmetic
numerals because they are made by repetition
(except for the Ionic system), and the positional
system does not need geometric numerals
because they are made by position. However,
the spoken language uses both arithmetic and
geometric numerals.

5.  p/q where p and q are integers and
q ! 0 are known
as rational numbers. The collection
of numbers of the form
p/q , where q > 0 is denoted by Q.
Rational numbers include natural
numbers, whole numbers, integers
and all negative and positive
fractions. Here we can visualize
how the girl collected all
the rational numbers in a bag.
Rational numbers can also be
represented on the number line
and here we can see a picture of a
girl walking on the number line.
To express rational numbers
appropriately on the number line,
divide each unit length into as
many number of equal parts as the
denominator of the rational
number and then mark the given
number on the number line.

6. The process of visualization of
representation of numbers on the
number line through a magnifying
glass is known as the process of
successive magnification.

7. This module is from Elementary
Algebra by Denny Burzynski and
Wade Ellis, Jr. The symbols,
notations, and properties of
numbers that form the basis of
algebra, as well as exponents and
the rules of exponents, are
introduced in this chapter. Each
property of real numbers and the
rules of exponents are expressed
both symbolically and literally.
Literal explanations are included
because symbolic explanations
alone may be difficult for a student
to interpret. Objectives of this
module: understand exponential
notation, be able to read
exponential notation, understand
how to use exponential notation
with the order of operations.

8. In elementary
algebra, root
rationalisation is a
process by
which surds in
the denominator of
an irrational fraction are
eliminated.
These surds may
be monomials or binomi
als involving square
roots, in simple
examples. There are
wide extensions to the
technique.

9. It is difficult to deal
with the expression
having square root
in the denominator.
This raises a need of
removing square
root from the
denominator. It can
be done by
rationlising the
denominator.

10.  1. Numbers 1, 2, 3…….∞, which are used for counting are called
Natural  16. Real numbers satisfy the commutative, associate and distributive
 numbers and are denoted by N.  law of addition and multiplication.
 2. 0 when included with the natural numbers form a new set of  17. Commutative law of addition: If a and b are two real numbers then,
numbers
 a+b=b+a
 called Whole number denoted by W
 18. Commutative law of multiplication: If a and b are two real
 3. -1,-2,-3……………..-∞ are the negative of natural numbers.
 numbers then, a. b = b. a
 4. The negative of natural numbers, 0 and the natural number together
 19. Associative law of addition: If a, b and c are real numbers then,
 constitutes integers denoted by Z.
 a + (b + c) = (a + b) + c
 5. The numbers which can be represented in the form of p/q where
 20. Associative law of multiplication: If a, b and c are real numbers
 q 0 ≠ and p and q are integers are called Rational numbers. Rational
 then, a. (b. c) = (a. b). c
 numbers are denoted by Q. If p and q are co prime then the rational
 21. Distributive of multiplication with respect to addition: If a, b and
 number is in its simplest form.
 c are real numbers then, a. (b+ c) = a. b + a. c
 6. Irrational numbers are the numbers which are non-terminating and
 22. Removing the radical sign from the denominator is called
 non-repeating.
 rationalisation of denominator.
 7. Rational and irrational numbers together constitute Real numbers
 23. The multiplication factor used for rationalizing the denominator is
 and it is denoted by R.
 called the rationalizing factor.
 8. Equivalent rational numbers (or fractions) have same (equal)
 24. The exponent is the number of times the base is multiplied by
 values when written in the simplest form. itself.
 9. Terminating fractions are the fractions which leaves remainder 0 on  25. In the exponential representation
 division.  m
 10. Recurring fractions are the fractions which never leave a remainder  a , a is called the base and m is
 0 on division.  called the exponent or power.
 11. There are infinitely many rational numbers between any two  26. If a number is to the left of the number on the number line, it is less
rational
 than the other number. If it is to the right, then it is greater than the
 numbers.
 number.
 12. If Prime factors of the denominator are 2 or 5 or both only. Then the
 27. There is one to one correspondence between the set of real
 number is terminating else repeating/recurring.
 numbers and the set of point on the number line.
 13. Two numbers p & q are said to be co-prime if, numbers p & q have
no
 common factors other than 1.

11. The origins and history of number system.
We call them Arabic
Numerals, but our
numbers actually find
their origins in the history
of the Hindus of India.
They have changed greatly
over the centuries, passing
first to the Arabs of the
Middle East and finally to
Europe in the Middle
Ages, and are now the
most commonly used
numbers throughout the
world.

12. Thanks
Name - Prajjwal Kushwaha
Class - flyers-1-b
Roll no -28
Special thanks to Mr.
pradeep Kumar lodha