In ancient times, early man use to keep animals…

He faced many problems regarding the taking care of them…

As GOD has given him Brain to utilize it, so He found a way of keeping an eye on his animals…

Every night, he used to keep a stone

with him , correspondence to one animal…

2 stones for 2 animals…

3 stones for three animals…

AND SO ON…

This one to one correspondence gave

birth to many mathematical symbols.Some of these symbols

are now called as NUMBERS.

These numbers/symbols, are

collectively called counting numbers. And

mathematically they are called

Natural Numbers.

The collection of numbers 1,2,3,4,5… is

called as the set of NATURAL

NUMBERS, as these numbers are used

naturally for counting…

The symbol used for natural numbers is

People from different localitiesand different countries used different symbols to denote the numbers…

MORE TO KNOW

MORE NUMBER SYSTEMS

Babylonian numerals were written in CUNEIFORM, using a wedge-tipped REED STYLUS to make a mark on a soft CLAY tablet which would be exposed in the SUN to harden to create a permanent record.

This system first appeared around 3100 BC. It is also credited as being the first known POSITIONAL NUMERAL SYSTEM,

in which the value of a particular digit depends both on the digit itself and its position within the number

Human’s need for numbers seems to be fulfilled here…

BUT its just the beginning…

His Need doesn't end.

NEED is the

MOTHER

of

INVENTIONS

and

DISCOVERIES

One morning, he found

no animalin the shed…

He was amazed to see the

empty shed…

Suddenly a question came into his mind…

No animal = no stone

No stone = what symbol?

Here he realized the need for

“symbolizing NOTHING” for the

first time…

NO ANIMALS

Many mathematicians of different

era has suggested for symbolizing

nothing but ARYABHATTA,

The INDIAN Mathematician was

the first who

Introduced the Symbol “0”

for symbolizing nothing, to the

world and made complicated

things easier.

By the 9th century

AD, The number

system consists of

one more member

namely “0”…

The set of numbers

0,1,2,3,4,5…

is called as the set of

WHOLE NUMBERS.

Is used for

representing

the set of

whole

numbers

But HUMAN NEEDS

DOESN’T FULFILLED HERE.

AS FAR AS THE HE

DEVELOPED THE SENSE

OF TRADE,

HE FOUND SOME OR THE

OTHER WAYS OF

EXCHANGE OF GOODS

AND SERVICES.ONE OF

THE WAY HE ADOPTED IS

THE BARTER SYSTEM

The history of bartering dates all the way back to 6000 BC.

Introduced by Mesopotamia tribes, bartering was adopted by

Phoenicians. Phoenicians bartered goods to those located in

various other cities across oceans. Babylonian's also

developed an improved bartering system. Goods were

exchanged for food, tea, weapons, and spices. At times, human

skulls were used as well. Salt was another popular item

exchanged. Salt was so valuable that Roman soldiers' salaries

were paid with it. In the Middle Ages, Europeans traveled

around the globe to barter crafts and furs in exchange for silks

and perfumes. Colonial Americans exchanged musket balls,

deer skins, and wheat. When money was invented, bartering

did not end, it become more organized.

Due to lack of money, bartering became

popular in the 1930s during the Great

Depression. It was used to obtain food and

various other services. It was done through

groups or between people who acted similar

to banks. If any items were sold, the owner

would receive credit (borrow) and the

buyer's account would be debited

(lending).

INFACT MONEY WASN’T ENOUGH

This system of credits and debits introduced the

concept of borrowing in trade.

If I have nothing and I am borrowing money from

someone, it means I have a negative balance.

And, if I have a lot and someone else is

borrowing from me, I have positive balance…

The positives and negatives in mathematics refer to positive numbers

and negative numbers.

These numbers are symbolized as

…-4,-3,-2,-1 etc.

The collection of natural numbers, zero and natural

numbers with –ve sign (called as negative numbers) is defined

as the set of integers…

This symbol

used for the

set of

integers…

With the flow of time, the

social relationships of

people improved. People

started sharing goods

with each other. This

sharing habit raise a

question in mathematics.

Once, a boy was eating an apple. He

gave a part of the apple to his sister.

How much apple has he eaten?

And how much part his sister

received?

Note that, part of the whole is

called as FRACTION. Thus, the

sister had a fraction of the Apple and the boy

also had another fraction of the same

Apple…

Sister’s

portion

Boy’s

portion

= ?

=?

=?

The number written at the

above part of the fraction is

called as NUMERATOR and

the number written below the

bar is called as

DENOMINATOR.

NUMERATOR

DENOMINATOR

… -327, -243, -137, -91, -86, -44, -30, -23, -5, -4, 0,

2,10, 34, 62, 97, 129, 294,892…

A handful of integers are written in the box

above.

Let us choose any two of them and write them

in the form of a fraction…

Eg:

__ __

, etc…

A fraction in which numerator

and denominator are chosen

from the set of integers,

denominator non zero, is called

a RATIONAL NUMBER…

And, the collection of all such

rational numbers is called as

the set of RATIONAL

NUMBERS.

We observe that any number can also be written in the form of

a decimal number. Further, A decimal number can be

categorized as follows…

DECIMAL

NUMBERS

TERMINATING NON-TERMINATING

RECURRING

DECIMALS

NON- RECURRING

DECIMALS

TERMINATING Eg: 4.5, 3.0, 22.75, 984.60, 2019.25 etc…

A number whose decimal representation

terminates.

NON-TERMINATINGA number whose decimal representation

does not terminates…

Eg: 7.333…, 6.494949…, 87.126126…,

3.142345667543277889665430045664…

78.909090…, 79.12356635679879… etc

3.142345667543277889665430045664…,

79.12356635679879…

7.333…, 6.494949…,

87.126126…,78.909090…

RECURRING

NON-RECURRING

DECIMAL

NUMBERS

TERMINATING NON-TERMINATING

RECURRING

DECIMALS

NON- RECURRING

DECIMALS

Terminating decimals and non-terminating recurring

decimals together called as RATIONAL NUMBERS.

And Non-Terminating Non-Repeating decimals are

called IRRATIONAL NUMBERS.

RATIONAL

NUMBERS (Q)

IRRATIONAL

NUMBERS(QC )

REAL NUMBERS

(R)

ALTOGETHER

PREPARED BY :RICHA BHARDWAJ