This page introduces integers as directed numbers. The positive numbers are aligned to the chosen direction and the negative numbers are opposed to the chosen direction.

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The number of flowers in the picture is 5.

This is an example of using a whole number to count or measure quantities.

Some early humans naturally used hands to count similar to the numbers `1, 2, 3, cdots`. Natural numbers or Counting numbers are `1, 2, 3, cdots`

Whole numbers are `0` along with natural numbers.

Let us recall what is a whole number.

A whole number is a count or measurement of a quantity.

• Number of pencils is `20`.

• Length of a pencil is `14`cm

• Mass of a iron ball is `2`kg

• Volume of a bottle of water is `1` liter

• Time taken to finish a work is `30`mins

• Temperature of water is `26^@`C

• Price of a pen is `15` coins

Consider a girl and her brother sharing candies. She receives `2` candies from her brother. The number of candies received is `2`.

On another day, The girl gives `2` candies to her brother. The number of candies given is `2`.

A girl received `2` candies from her brother. This is represented by the number `2`.

On another day, she gives `2` candies to her brother. This is represented by the number `2`.

Note that both the numbers are represented as `2`. Are these two quantities same?

No, 2 received is different from 2 given.

Consider the two situations.

A girl receives `2` candies from her father. On another day, she receives another `2` candies from her father. Effectively, she has received, `2+2=4` candies from her father.

The girl receives `2` candies from her brother. On another day, she gives `2` candies to her brother. Overall, she has received `2` candies and returned `2` candies. This makes total of, `2-2=0` candies received from her brother.

From the example of a girl receiving `2` candies and giving `2` candies, it is noted that the `2` received and `2` given are different.

`2` received is aligned to the direction of receiving. And `2` given is opposed to the direction of receiving. They together make `0`.

A girl receives `10` candies from her brother. Later, she gives `2` candies to her brother.

She has `10-2=8` candies.

It is noted that the given is subtracted.

The two inherent properties of the numbers

• The quantities received and given represent two different values.

• When combined, the quantities received and given are subtracted.

In accordance with these, *negative numbers* are introduced. Minus sign is adopted to provide the missing information in the numbers.

For Example:

`2` received is `+2` or `2`

`2` given is `-2`

The count of candies is understood in two different forms

• `5` candies received.

• `5` candies given.

Similarly other quantities have two different forms.

• Height above and below ground level

• Distance to reach a city and distance covered after crossing the city

• Length or Mass or Volume : amount available to use and amount used in excess

• Temperature above `0^@`C or below `0^@`C

• Temperature to heat or cool down

• Time remaining or time passed since start

• Money credited or debited

All these applications are modeled with numbers given in the form

`3` and `-3`

which are equivalently given as `text(received:)3 = 3` and `text(given:)3 = -3` to explain some fundamentals in arithmetics.

Such numbers with positive and negative direction are called *Directed Numbers*.*Integers are directed whole numbers*.

The word "negative" means: absence of something; not having or does not exist.

*familiarize with the terminology *

negative

The word "positive" means: presence of something; having or existing.

*familiarize with the terminology *

positive

The word "directed" means: aimed in a particular direction.

The positive numbers represent numbers aligned to a direction and negative numbers represent numbers opposed to the direction.

*familiarize with the terminology *

directed

**Integers** : Directed whole numbers are called integers.

*Solved Exercise Problem: *

If `2` degree above `0` is `+2`, what is `7` degree below `0`?

- `+7`
- `-7`
- `-7`

The answer is '`-7`'.

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