overview

Considering a person receiving $3$ candies and given $3$ candies. These two are represented as $3$ in whole numbers. But, received $3$ and given $3$ are two different quantities.

It is noted that received $3$ and given $3$ together make $3-3=0$. So minus sign is adopted to provide the missing information in the whole numbers.

Taking "receiving" as the direction of reference, the integers are defined as

• $\text{received:}3=+3$

• $\text{given:}3=-3$

give 2 or take 2

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 $26}^{\circ$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 $2$.

receive and give

Considering whole numbers only. 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 the same?

Though the values of the numbers are same, the number 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.

*It is noted that the $2$ received and $2$ given are different quantities. $2$ 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 $-2$

directed numbers

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 $0}^{\circ$C or below $0}^{\circ$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*.

terminology

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

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

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.

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

example

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

The answer is '$-7$'.

summary

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

**Positive Integers** : Numbers that are aligned to the chosen direction are positive integers and are given with $+$ symbol or without that.

eg: $+2$ or $2$.

**Negative Integers** : Numbers that are opposed to the chosen direction are negative integers and are given with $-$ symbol.

eg: $-2$

Integer form $2$ is $\text{received:}2$ in directed whole number form.
Integer form $-2$ is $\text{given:}2$ in directed whole number form.

Outline

The outline of material to learn integers is as follows.

Note: * click here for detailed outline of Integers (directed numbers) *

→ __Introduction to Directed Numbers__

→ __Handling Direction__

→ __Ordinal Property__

→ __Sign and Absolute Value__

→ __Comparing Integers__

→ __Predecessor & Successor__

→ __Largest & Smallest__

→ __Ascending & Descending__

→ __Addition: First Principles__

→ __Addition: Simplified Procedure__

→ __Subtraction: First Principles__

→ __Subtraction: Simplified Procedure__

→ __Multiplication: First Principles__

→ __Multiplication: Simplified Procedure__

→ __Division: First Principles__

→ __Division: Simplfied Procedure__

→ __Numerical Expressions with Integers__

→ __PEMA / BOMA__