Integer

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A Integer is an element of the numerical set that contains natural numbers; which are ${displaystyle mathbb {N} ={0,1,2,3,4,cdots }}$ or ${displaystyle mathbb {N} ^{ast }={1,2,3,4,cdots }}$ depending on how they are defined, their opposites, and in the second definition, plus zero. Negative integers, such as −1 or −13 (read "less one", "less thirteen", etc.), are less than zero and are also less than all positive integers. To highlight the difference between positive and negative, you can write a "less" sign in front of negatives: -1, -5, etc. And if no sign is written to the number it is assumed to be positive.

The set of all integers is represented by the letter {Z}={..., -4, -3, -2, -1, 0, +1, +2, +3,,...} initial letter of the German word Zahlen («numbers», pronounced [ˈtsaːlən] ).

On a number line, negative numbers are to the left of zero and positive numbers are to its right.

Integers can be added, subtracted, multiplied and divided, following the model of natural numbers by adding rules for the use of signs.

Integers extend the usefulness of natural numbers for counting things. They can be used to account for losses: if 80 new first-year students enter a school in a certain year, but there are 100 final-year students who went on to secondary education, there will be a total of 100 − 80 = 20 fewer students; but it can also be said that said number has increased by 80 − 100 = −20 students.

Certain quantities, such as temperature or height, use values below zero. The height of Everest is 8848 meters above sea level, and on the contrary, the shore of the Dead Sea is 423 meters below sea level; that is, its height can be expressed as −423 m.

Introduction

Negative numbers are needed to perform operations like:

3 − 5 = ?

When the minuend is smaller than the subtrahend, subtraction cannot be done with natural numbers. However, there are situations in which the concept of negative numbers is useful, such as when talking about profits and losses:

Example: A man plays roulette two days in a row. If the first man earns 2,000 pesos and the next day loses 1,000, the man earned a total of 2,000 − 1,000 = $1,000. However, if the first day he earns 500 and the next day he loses 2,000, it is said that lost in total 2000 − 500 = $1500. The expression used changes in each case: won in total or lost in total, depending on whether the gains were greater than the losses or vice versa. These two possibilities can be expressed using the sign of the negative (or positive) numbers: in the first case, he won a total of 2,000 − 1,000 = + $1,000, and in the second, he won a total of 500 − 2,000 = − $1,500. understands that a loss is a negative gain.

Signed numbers

Main article: Sign (matures)

The natural numbers 0, 1, 2, 3,... are the ordinary numbers used to count. Adding a minus sign ("−") in front of them yields negative numbers:

A integer negative is a natural number like 1, 2, 3, etc. preceded by a sign less, « −». For example −1, −2, −3, etc. They read "less 1", "less 2", "less 3",...

In addition, to better differentiate them, a plus sign («+») is added in front of the natural numbers and they are called positive numbers.

A positive integer It's a natural number like 1, 2, 3, preceded by a sign more. “+”.

Zero is neither positive nor negative, and can be written with a sign plus or minus or without a sign interchangeably, since adding or subtracting zero is the same as doing nothing. All this collection of numbers are called “integers”.

Them integer numbers are the set of all integers with sign (positive and negative) along with 0. They are represented by the letter $Z$ also written in "blackboard" as $Z$ :
${displaystyle mathbb {Z} ={dots-2,-1,0,+1,+2,dots }}$

The Number Line

Main article: Numerical collection

Negative integers are less than all positive integers and less than zero. That is, every number that is located to the right is greater than the number that is located to the left. To understand how they are ordered, use the number line:

It is seen with this representation that negative numbers are smaller the further to the left, that is, the larger the number after the sign. This number is called the absolute value:

The absolute value of an integer is the distance from the origin (zero) to a given point. The absolute value of 0 is simply 0. It is represented by two vertical bars “UDOSITY”.

Examples. |+5| = 5 |−2| = 2 |0| = 0.

The order of integers can be summarized as:

The order of the integers is defined as:
Given two whole numbers of different signs, $+ a$ and $- b$ the negative is less than the positive: $- b . + a$ .
Given two integers with the same sign, the minor of the two numbers is:
The lowest absolute value, if the common sign is "+".
The highest absolute value, if the common sign is "−".
Zero, 0, is less than all positive and greater than all negatives.

Examples. +23 > −56 +31 < +47 −15 < −9 0 > −36

Operations with integers

Integers can be added, subtracted, multiplied, and divided, just as you can with whole numbers.

Add:

In this figure, the absolute value and the sign of a number are represented by the size of the circle and its color.

In the sum of two integers, the sign and absolute value of the result are determined separately.

Stop. sumar two integers, the sign and absolute value of the result is determined as follows:
If both sums have the same sign: that is also the sign of the result, and its absolute value is the sum of the absolute values of the sums.
If both of them have a different sign:
The sign of the result is the sign of adding with greater absolute value.
The absolute value of the result is the difference between the highest absolute value and the lowest absolute value, between the two sums.

Examples. (+21) + (−13) = +8 (+17) + (+26) = +43 (−41) + (+19) = −22 (− 33) + (−28) = −61

The addition of integers behaves in a similar way to the addition of natural numbers:

The sum of integers fulfills the following properties:
Associative property. Given three integers $a$ , $b$ and $c$ the sums $(a + b + c$ and $a + (b + c)$ They're the same.
Commutative property. Given two integers $a$ and $b$ the sums $a + b$ and $b + a$ They're the same.
Neutral element. All integers $a$ remain unchanged as they add 0: $a + 0 = a$ .

Example.

Associative property:
[ (−13) + (+25) ] + (+32) = (+12) + (+32) = (+44)
(−13) + [ (+25) + (+32) ] = (−13) + (+57) = (+44)
Commutative property:
(+9) + (−17) = −8
(−17) + (+9) = −8

In addition, the addition of integers has an additional property that the natural numbers do not have:

Opposite or symmetrical element: For every integer $a$ , there is another integer $- a$ which adds to the first results in zero: $a + (- a) = 0$ .

Subtraction:

Subtraction of integers is very easy, since it is now a particular case of addition.

La resta of two integers (minuendo less subtract) is done adding the minuendo plus the subtracting by changing sign.

Examples
(+10) − (−5) = (+10) + (+5) = +15
(−7) − (+6) = (−7) + (−6) = −13
(−4) − (−8) = (−4) + (+8) = + 4
(+2) − (+9) = (+2) + (−9) = −7

Multiplication and Division

Multiplying and dividing integers, like addition, requires separately determining the sign and absolute value of the result.

In the multiplication and in the division of two integers determine the absolute value and the result sign as follows:
The absolute value is the product (or quotient) of the absolute values of the factors (or the dividend and divider).
The sign is "+" if the signs of the factors (or the dividend and divider) are equal, and "−" if they are different.

To remember the sign of the result, we also use the rule of signs:

Rule of signs - Multiplication
(+) × (+)=(+)More like more.
(+) × (−)=(−)More for less than that.
(−) × (+)=(−)Less equally than less.
(−) × (−)=(+)Less equal to more.

Rule of signs - Division
(+): (+)=(+)More like more.
(+): (−)=(−)More the less equal to less.
(−): (+)=(−)Less like less.
(−): (−)=(+)Less at least equal to more.

Multiplication examples. (+5) × (+3) = +15 (+4) × (-6) = -24 (−7) × (+8) = −56 (−9) × (−2) = +18.

Division examples. (+15): (+3) = +5 (+12): (-6) = -2 (−16): (+4) = −4 (−18): (−2) = +9.

The multiplication of integers also has properties similar to that of natural numbers:

The multiplication of integers fulfills the following properties:
Associative property. Given three integers $a$ , $b$ and $c$ , products $(a \times b) \times c$ and $a \times b \times c)$ They're the same.
Commutative property. Given two integers $a$ and $b$ , products $a \times b$ and $b \times a$ They're the same.
Neutral element. All integers $a$ are unchanged by multiplying them by 1: $a \times 1 = a$ .

Example.

Associative property:

[ (−7) × (+4) ] × (+5) = (−28) × (+5) = −140
(−7) × [ (+4) × (+5) ] = (−7) × (+20) = −140
Commutative property:
(−6) × (+9) = −54
(+9) × (−6) = −54

The addition and multiplication of integers are related, just like the natural numbers, by the distributive property:

Distributive property. Given three integers $a$ , $b$ and $c$ , the product $a \times b + c)$ and the sum of products $(a \times b) + (a \times c)$ They're identical.

Example.

(−7) × [ (−2) + (+5) ] = (−7) × (+3) = −21
[ (−7) × (−2) ] + [ (−7) × (+5) ] = (+14) + (−35) = −21

The division of integers does not have the associative, commutative, or distributive properties.

Algebraic Properties

Main article: Properties of the integers

The set of integers, considered together with their operations of addition and multiplication, has a structure that in mathematics is called ring; and has a relation of order. Integers can also be built from natural numbers through equivalence classes.

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