Integer

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Mathematics Integers Integers

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The integers are a very fundamental mathematical set on which arithmetic is based. They are the familiar "whole numbers"; that is, they do not include fractional quantities. They include zero and negative whole numbers--the natural numbers are the set that includes only positive whole numbers. That is, the integers are the set  ⋯ − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … {\displaystyle \dots -3,-2,-1,0,1,2,3,\dots } .

Mathematicians denote the set of integers with an ornate capital letter:  I {\displaystyle \mathbb {I} } . They are the 2nd item in this hierarchy of types of numbers:

* The "natural numbers"-- N {\displaystyle \mathbb {N} } , 0, 1, 2, 3, ...
* The "integers"-- I {\displaystyle \mathbb {I} } , positive, negative, and zero
* The "rational numbers"-- Q {\displaystyle \mathbb {Q} } , or fractions, like 355/113
* The "real numbers"-- R {\displaystyle \mathbb {R} } , including irrational numbers
* The "complex numbers"-- C {\displaystyle \mathbb {C} } , which give solutions to polynomial equations

## Definition of the integers

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Integer represents a set of signed numbers namely Negative numbers , Zero and Positive numbers . Integer is denoted as I . So Negative number is -I and Positive number is +I

I = − I , 0 , + I {\displaystyle I={-I,0,+I}}

Where

I {\displaystyle I} . Integer
− I {\displaystyle -I} . Negative integer
+ I {\displaystyle +I} . Positive integer

Negative number is defined as a number has value less than zero

− I < 0 {\displaystyle -I<0}

Positive number is defined as a number has value greater zero

+ I > 0 {\displaystyle +I>0}

Example Integers of octal base

I = − 9 , − 8 , − 7 , − 6 , − 5 , − 4 , − 3 , − 2 , − 1 , 0 , + 1 , + 2 , + 3 , + 4 , + 5 , + 6 , + 7 , + 8 , + 9 {\displaystyle I={-9,-8,-7,-6,-5,-4,-3,-2,-1,0,+1,+2,+3,+4,+5,+6,+7,+8,+9}}

## Mathematical operations

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### Differfent integers

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( − I ) + 0 = − I {\displaystyle (-I)+0=-I}
( + I ) + 0 = + I {\displaystyle (+I)+0=+I}
( + I ) + ( − I ) = 0 {\displaystyle (+I)+(-I)=0}
( − I ) + ( + I ) = 0 {\displaystyle (-I)+(+I)=0}

### Like positive integer

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( + I ) + ( + I ) = + 2 I {\displaystyle (+I)+(+I)=+2I}
( + I ) − ( + I ) = 0 {\displaystyle (+I)-(+I)=0}
( + I ) × ( + I ) = I 2 {\displaystyle (+I)\times (+I)=I^{2}}
( + I ) / ( + I ) = 1 {\displaystyle (+I)/(+I)=1}

### Like negative integer

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( − I ) + ( − I ) = − 2 I {\displaystyle (-I)+(-I)=-2I}
( − I ) − ( − I ) = 0 {\displaystyle (-I)-(-I)=0}
( − I ) × ( − I ) = I 2 {\displaystyle (-I)\times (-I)=I^{2}}
( − I ) / ( − I ) = 1 {\displaystyle (-I)/(-I)=1}

## Addition

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For all positive x and y, we extend addition to all integers as follows. The value of x+(-y) depends on which is larger, x or y.

If  x > y {\displaystyle x>y} ,  x + ( − y ) = x − y {\displaystyle x+(-y)=x-y}
If  x < y {\displaystyle x<y} ,  x + ( − y ) = − ( y − x ) {\displaystyle x+(-y)=-(y-x)}
Also,  ( − y ) + x {\displaystyle (-y)+x} =  x + ( − y ) {\displaystyle x+(-y)} , and  ( − x ) + ( − y ) = − ( x + y ) {\displaystyle (-x)+(-y)=-(x+y)}
Finally,  x + 0 = 0 + x = x {\displaystyle x+0=0+x=x} for all x, positive, negative, or zero.

## Negation

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This operator, written "-x", has already been defined for positive numbers. For negative numbers, we define

− ( − x ) = x {\displaystyle -(-x)=x} . Also,  − 0 = 0 {\displaystyle -0=0} .

## Subtraction

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We define subtraction for all numbers, positive, negative, or zero, in terms of negation and addition

x − y = x + ( − y ) {\displaystyle x-y=x+(-y)}

## Multiplication

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For negative numbers, we define

x × ( − y ) = − ( x × y ) {\displaystyle x\times (-y)=-(x\times y)}
( − x ) × y = − ( x × y ) {\displaystyle (-x)\times y=-(x\times y)}
0 × x = x × 0 = 0 {\displaystyle 0\times x=x\times 0=0}

* Ordering: If x and y are both positive numbers:

− x < − y {\displaystyle -x<-y} if and only if  x > y {\displaystyle x>y}
− x < 0 {\displaystyle -x<0} ,  − x < y {\displaystyle -x<y}
0 > − y {\displaystyle 0>-y} ,  x > − y {\displaystyle x>-y}
In all cases,

x < y {\displaystyle x<y} means x≠y and y > x

From these definitions, we can then prove all the various commutativity, associativity, and distributivity properties, and the trichotomy law. This is left as an exercise.

## Group and ring properties

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The integers, with their property of addition, constitute a group. Groups are extremely important objects in mathematics. A group requires a set (the integers), an operation (addition), an "inverse" operation (negation--the natural numbers don't have this, so they are not a group), and an "identity" operation (zero). To be a group, the operation must be associative, a property which the integers satisfy. Also, since addition

Furthermore, the integers have a multiplication operator, satisfying the distributive law, and having a multiplicative identity (1). This makes the integers a ring.

## References

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## See Also

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"Integers".

For a discussion of the philosophical and historical aspects of the various types of numbers, see Our_Playground:_The_Real_Numbers_and_Their_Development.