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To put it simple, integers are just extended natural numbers, the set of natural numbers, add 0 and negative natural numbers. The set of integers is denoted by:

but

which is exactly , while

and we see that it is and we say that the set of natural numbers consists of the set of positive integers (which is the set of natural integers), the set of negative integers, and the number zero.

Since integers are just extended natural numbers, they can be represented on number line. We extend it by mirroring all natural numbers over the starting point - making all integers to the left of the starting point negative.

Extend the line with natural numbers to the left:

A unit to the left, we get

One unit to the left of , we get

We repeat the process by applying the unit evenly to the left to obtain the remaining negative numbers.

The number is called the additive inverse of . Natural numbers are positive integers, and mirrored natural numbers are additive inverse.

Addition and multiplication of integers are defined similarly to addition and multiplication of natural numbers. In addition to these two basic arithmetic operations, we also define subtraction in the set of integers.

The subtraction is defined as the addition of the additive inverse: , where is called the minuend, is the subtrahend and the result of is the difference.

For integers, in addition to properties of arithmetic operations natural numbers, some rules also apply.

When adding to any integer, the result does not change.

Identity element:

for each of

The sum of any integer and its additive inverse equal .

The additive inverse of is .

The negation of a sum is equal to the sum of its additive inverses.

The number is the multiplication identity element.

Multiplying by gives the additive inverse of the number .

When multiplying any number by , the result is always equal to .

The product of two negative integers is positive.

The product of two positive integers is positive.

The product of a negative and a positive is negative.

When calculating, we follow the order of arithmetic operations (multiplication takes precedence over addition and subtraction), and in the case of brackets, we always eliminate these first. In doing so, we must pay attention to the signs:

If the brackets are preceded by a plus sign (+), the numbers in the brackets retain their sign:

and

If the brackets are preceded by a minus sign (-), the numbers in brackets change their sign:

and

In a set of integers, we can compare any two numbers with each other and determine whether they are the same (=) or which of them is greater (>) or less (<). The inequalities less than or equal to (<=) or greater than or equal to (> =) can also apply.

For any integers and , is greater than , which is written with symbols:

if and only if

or if lies to the right of on the number line.

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For any integers and , is less than , which is written with symbols:

if and only if

or if lies on the left side of on the number line.

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For any integers and , is equal to , which is written with symbols:

if and only if

or if coincides with on the number line.

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For any integers and , is less than or equal to , which is written with symbols:

if and only if

or if lies on the left side of or coincides with it on the number line.

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For any integers and , is greater than or equal to , which is written with symbols:

if and only if

or if lies on the right side of or coincides with it on the number line.

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If we add the same number on both sides of the inequality, the inequality is maintained:

if

then it is true for

"Less than" is transitive:

if and , then

When multiplying inequalities by a positive number, the inequality sign is retained:

if and then

When multiplying inequality by a negative number, the inequality sign is reversed:

if and then

"Less than or equal to" is reflexive:

if

"Less than or equal to" is antisymmetric:

if and , then

"Less than or equal to" is transitive:

if and , then

"Greater than or equal to" is reflexive:

if

"Greater than or equal to" is antisymmetric:

if and , then

"Greater than or equal to" is transitive:

if and , then

material editor: ISMAIL SHOBOLA