Integer
A Integer is an element of the numerical set that contains natural numbers; which are N={0,1,2,3,4, !{displaystyle mathbb {N} ={0,1,2,3,4,cdots }or N↓ ↓ ={1,2,3,4, !{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
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:
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In addition, to better differentiate them, a plus sign («+») is added in front of the natural numbers and they are called positive numbers.
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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”.
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The Number Line
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:
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Examples. |+5| = 5 |−2| = 2 |0| = 0.
The order of integers can be summarized as:
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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 the sum of two integers, the sign and absolute value of the result are determined separately.
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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:
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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:
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Subtraction:
Subtraction of integers is very easy, since it is now a particular case of addition.
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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.
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To remember the sign of the result, we also use the rule of signs:
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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:
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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:
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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
- 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.