Integers

In this article we explain the integers through its concept, definition, examples, usefulness, representation, operations in this field and its properties and characteristics.

What are integers?

Integers are the elements of the set that includes natural numbers (1, 2, 3, ...), the negatives of natural numbers (-1, -2, -3, ...), and zero (0).

The negatives of natural numbers are called negative integers and are identified with the minus (-) symbol in front of them, which should be read; for example, -10 is read as "minus ten." Natural numbers are called positive integers and can be identified with the plus sign (+) in front of them; however, if nothing is written in front, it is assumed that the number is positive. Zero is neither positive nor negative but neutral.

Some examples of integers are: -40, 2, 0, 27, -34, 18, -245, 5, -9, 14. Fractional numbers like 1/2 or real numbers like π are not integers.

What are they used for?

The concept of integers arose from the need to represent quantities that cannot be expressed with natural numbers, such as the absence of objects or quantities below zero. This way, the utility of numbers for counting things is extended.

Some examples of their use in everyday life are:

• Temperatures: Integers are commonly used to represent temperatures. For example, 5°C represents a temperature above zero, positive, while -5°C represents a colder temperature that is below 0°C, i.e., it is negative.
• Bank balance: A positive balance indicates that there is money in the account, for example, $100. However, a negative balance indicates a debt, for example, -$50 means that $50 is owed to the bank.
• Profits and losses: If a company has a profit of $1000, this is represented with the positive integer +1000 or simply 1000; however, if it has a loss of $500, it is expressed with the negative integer -500.
• Altitudes: Integers are used to represent altitudes above sea level. For example, Mount Aconcagua is the highest mountain in South America, with an altitude of approximately 6960 meters above sea level. The Dead Sea, located between Israel and Jordan, is 430 meters below sea level, which can be represented with the negative integer -430 meters.

From the mathematical perspective, integers allow for subtraction where the minuend is less than the subtrahend, i.e., the operation *a-b* where *a* is less than *b,* for example: *5-10=-5.* This was something that could not be done with natural numbers.

Set of integers

Every natural number is also an integer, symbolized as *\mathbb{N}⊆\mathbb{Z}.* Every integer is rational and real, but not all rationals and reals are integers. For example, the numbers *2/3,* *-7/2,* *\sqrt{2}*, and *\pi* are not integers.

There are two common ways to make a classification of integers:

• Positive, negative, and zero: positive integers are those greater than zero, for example: 1, 7, 24, 5, 30, etc. Negative integers are those less than zero, for example: -1, -30, -17, -12, -75, etc. Zero is the integer that is neither positive nor negative.
• Even and odd: if an integer can be divided by 2 exactly, i.e., leaving zero remainder, it is said to be an even integer, for example: -4, -2, 0, 2, 4, 6, etc. If the division is not exact, the number is said to be an odd integer, for example: -3, -1, 1, 3, 5, etc.

The set of positive integers is represented by the letter Z with a superscript "+". This set is identical to the set of natural numbers, *\mathbb{N}.*

*\mathbb{Z}^+=\{1,2,3,4,5,6,...\}*

The set of negative integers is represented using the letter Z with a superscript "-":

*\mathbb{Z}^-=\{-1,-2,-3,-4,-5,-6,...\}*

In either of these two subsets, zero can be included by using the subscript "0", resulting in *\mathbb{Z}^+_0~* and *~\mathbb{Z}^-_0.*

Characteristics and properties

The set of integers has the following properties:

• It is ordered and infinite, which means that new integers can always be found, and two different integers can be compared to determine which one is smaller or larger.
• It has no first or last element, meaning there is no integer smaller than all others or one that is the largest. This sets it apart from the set of natural numbers, which does have a first element, the number 1.
• Every integer has a successor and a predecessor. An integer and its successor are called consecutive. The successor of an integer can be obtained by adding 1, and its predecessor by subtracting 1. For example, the integer -5 has -4 as its successor and -6 as its predecessor.
• There is always a finite number of integers between two integers. That's why it's said to be a discrete set, or not dense. There are no natural numbers between an integer and its successor.

Graphical representation

We can represent integers on a number line. For this, zero is placed at a point, positive integers are located to the right of it with equal distances between them, and negative integers are located to the left of zero.

Integers become larger as we move to the right, and smaller as we move to the left. Positive integers are larger the farther they are from zero, while negative integers are larger the closer they are to zero.

An integer is greater than another if it is to the right of it on the number line. For example, *3* is greater than *-4,* symbolically *3>-4,* because *3* is to the right of *-4.*

The absolute value of an integer is defined as its distance from zero on the number line and is symbolized by two vertical bars around the number. The absolute value can be calculated as follows:

• If the integer is positive or zero, then its absolute value is the same number.
• If the integer is negative, then its absolute value is its opposite number.

For example: *|0|=0,* *~~|5|=5,* *~~|-7|=7.*

Operations

The basic operations with integers are addition, subtraction, multiplication, and division. We will see how to perform them and their properties below.

Addition

To perform the addition of two integers, we must pay attention to their signs:

Same sign: if both integers have the same sign (positive or negative), the result is another integer with the same sign whose absolute value is equal to the sum of the absolute values of the original numbers. For example:

*2+3=5*

*-4+(-2)=-6*

Different sign: if the integers have different signs, the result is obtained by subtracting the absolute value of the smaller number from the absolute value of the larger number. The sign of the result is the same as the sign of the larger number.

*8+(-9)=-1*

*-6+9=3*

Subtraction

If *a* and *b* are two integers, the subtraction of *a* minus *b* is represented as *a-b;* *a* is the minuend and *b* is the subtrahend. Subtraction can be seen as the addition of the minuend plus the negation of the subtrahend. Thus, the subtraction *a-b* is solved by calculating *a+(-b)* and using the rules of addition we just saw.

*4-3=4+(-3)=1*

*-5-(-3)=-5+3=-2*

*10-(-5)=10+5=15*

Multiplication

Multiplication of integers follows the standard rules of multiplication. The product of two integers will have a positive sign if both numbers have the same sign, and it will have a negative sign if the numbers have different signs. This is known as the rule of signs.

*7\cdot 2=14*

*(-7)\cdot 2=-14*

*3\cdot (-2)=-6*

*(-2)\cdot (-4)=8*

Division

Division is a mathematical operation that tells us how many times one number (called the divisor) is contained in another number (called the dividend). The result of division is called the quotient.

When working with integers, it may happen that the result of a division is not an integer. To ensure that the result is an integer, it must happen that the dividend is a multiple of the divisor. If this happens, the division will be exact.

The same rule of signs as in multiplication is followed: if the dividend and divisor have the same sign, the result is positive; if they have different signs, the result is negative. For example:

*10÷2=5*

*6÷(-3)=-2*

*-8÷2=-4*

*-12÷(-6)=2*

Limitations of integers

Integers, although they solve the problem of subtraction that arose with natural numbers, are still not sufficient for certain situations related to division:

• Exact division of two integers cannot be performed if the dividend is not a multiple of the divisor, as the result is not an integer. For example, divisions like *4÷3* and *7÷5* have no solution in integers.
• Integers cannot be used to represent quantities such as half a chocolate, a third of a pizza, or half a meter.

From another point of view, equations like *2x=3* cannot be solved, as there is no integer *x* that satisfies that equality. To overcome this, other types of numbers are introduced: rational numbers and later real numbers.

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