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Integers

In Mathematics, the term “integer” was derived from Latin. An integer denotes completeness. Integers are similar to whole numbers, but they can also include negative numbers.

What is an integer?

An integer is a number with no decimal or fractional element from the set of negative and positive numbers, including zero. -5, 0, 1, 5, 8, 97, and 3,043 are examples of integers. Z represents a group of integers that includes:

Positive Integers: If an integer is bigger than zero, it is considered positive. Example: 1, 2, 3…

Negative Integers: Integers that are less than zero are called negative integers. Example: -1, -2, -3 . . .

The integer zero is neither negative nor positive. It’s a whole number.

Z = {… -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, …}

Integers on a Number line:

A number line is a visual representation of numbers on a straight line. This line is used to compare numbers that are placed at equal intervals on an infinite line that extends on both sides, horizontally. Just like other numbers, the set of integers can also be represented on a number line.

Graphing Integers on a Number Line:

On a number line, positive and negative integers can be visually represented. A number line with integers helps with arithmetic procedures. When arranging integers on a number line, bear the following in mind:

  • The right horizontal side number is always greater than the left horizontal side number.
  • Because positive integers are greater than “0,” they are put on the right side of 0.
  • Negative numerals, smaller than “0,” are put to the left of “0.”
  • The center is kept at zero, which is neither positive nor negative.

Integer Operations;

Integers are related with four basic arithmetic operations:

  • Multiplication of integers
  • Addition of integers
  • Subtraction of integers

There are several guidelines to follow when performing these tasks.

Before we begin learning these integer operations methods, there are a few things to keep in mind.

The absence of a sign in front of a number indicates that it is positive. 5 means +5, for example.

An integer’s absolute value is a positive number, hence |2| = 2 and |2| = 2.

The process of obtaining the sum of two or more integers, where the value may increase or decrease depending on whether the number is positive or negative, is known as adding integers. When adding two integers, we get into the following situations:

  • Both integers have the same sign: Add the absolute values of the numbers and give the result the same sign as the given integers.
  • The first integer is positive, whereas the second is negative: Calculate the difference between the absolute values of the numbers, then add the original sign of the greater of the two.
  • Example: Adding two integers: Calculate the value of 2 + (-5).
  • Solution: 

Here, the absolute values of 2 and (-5) are 2 and 5 respectively.Their difference (larger number – smaller number) is 5 – 2 = 3. Now, among 2 and 5, 5 is the larger number and its original sign “-”. Hence, the result gets a negative sign, “-”. Therefore, 2 + (-5) = -3.

  • Example: Adding two integers: Calculate the value of -2 + 5.
  • Solution: 

Here, the enterlute values of (-2) and 5 are 2 and 5 respectively. Their difference (larger number – smaller number) is 5 – 2 = 3. Now, among 2 and 5, 5 is the larger number and its original sign “+”. Hence, the result will be a positive value. Therefore,(-2) + 5 = 3.

We can also use a number line to solve the problem above. The following are the guidelines for adding integers on the number line:

  • Always begin with “0.”
  • If the number is positive, move to the right.
  • If the number is negative, move to the left.
  • Let’s use a number line to find the value of 5 + (-10).
  • The initial number in the puzzle is 5, which is positive.
  • As a result, we begin at 0 and advance 5 units to the right.
  • The number we have moved to finally is -5.

Subtraction of Integers;

Subtracting integers is the process of finding the difference between two or more integers where the final value might increase or decrease depending on the integer being positive or negative. To carry out the subtraction of two integers:

  • Convert the operation into an addition problem by changing the sign of the subtrahend.
  • Apply the same rules of addition of integers and solve the problem thus obtained in the above step.

Example: Subtracting two integers: Calculate the value of 7 – 10.

Solution:

Converting the given expression into an addition problem, we get: 7 + (-10).

Now, the rules for this operation will be the same as for the addition of two integers.

Here, the absolute values of 7 and (-10) are 7 and 10 respectively.

Their difference (larger number – smaller number) is 10 – 7 = 3.

Now, among 7 and 10, 10 is the larger number and its original sign “-”.

Hence, the result gets a negative sign, “-”.

Therefore, 7 – 10 = -3.

Multiplication of integers:

Multiplication of integers is a similar process of repetitive addition where an integer is added a specific number of times. To carry out the multiplication of two integers:

  • Multiply their signs and get the resultant sign.
  • Multiply the numbers and add the resultant sign to the answer.

The different possible cases for the multiplication of two signs can be observed in the following table:

Product of SignsResultExample
+ × ++2 × 3 = 6
+ × –2 × (-3) = -6
– × +(-2) × 3 = -6
– × –+-2 × -3 = 6

Example: Multiplying integers on a number line: Calculate the value of -2 × 3 and -2 × -3 using a number line.

Solution:

We read 2 × -3 as “2 times -3”. We have to represent -3 on the number line 2 times. To do so, we will start from and move left by 3 units twice.Thus, 2 × -3 = -6.

Also, -2 × -3 is similar to -2 × 3, but 2 is replaced by -2. Hence, we follow the same number line process as above but in the opposite direction (i.e., to the right side).Therefore, -2 × -3 = 6.

Division of integers:

Division of integers means equal grouping or dividing an integer into a specific number of groups. To carry out the division operation between two integers:

  • Divide the signs of the two operands and get the resultant sign.
  • Divide the numbers and add the resultant sign to the quotient.

The different possible cases for the division of two signs can be observed in the following table:

Division of SignsResultExample
+ ÷ ++12 ÷ 3 = 4
+ ÷ –12 ÷ -3 = -4
– ÷ +-12 ÷ 3 = -4
– ÷ –+-12 ÷ -3 = 4

Rules of Integers:

Rules defined for integers are:

  • Sum of two positive integers is an integer.
  • Sum of two negative integers is an integer.
  • Product of two positive integers is an integer.
  • Product of two negative integers is an integer.
  • Addition operation between any integer and its negative value will give the result as zero
  • Multiplication operation between any integer and its reciprocal will give the result as one.

Properties of Integers

The major Properties of Integers are:

  • Closure Property
  • Associative Property
  • Commutative Property
  • Distributive Property
  • Additive Inverse Property
  • Multiplicative Inverse Property
  • Identity Property

Closure Property:

The closure property states that the set is closed for any particular mathematical operation. Z is closed under addition, subtraction, multiplication, and division of integers. For any two integers, a and b: 

  •           a + b ∈ Z
  • a – b ∈ Z
  • a × b ∈ Z
  • a/b ∈ Z

Associative Property:

According to the associative property, changing the grouping of two integers does not alter the result of the operation. The associative property applies to the addition and multiplication of two integers.

For any two integers, a and b:

  • a + (b + c) = (a + b) + c
  • a ×(b × c) = (a × b) × c

Commutative Property:

According to the commutative property, swapping the positions of operands in an operation does not affect the result. The addition and multiplication of integers follow the commutative property.

For any two integers, a and b:

  • a + b = b + a
  • a × b = b × a

Distributive Property:

Distributive property states that for any expression of the form a (b + c), which means a × (b + c), operand a can be distributed among operands b and c as (a × b + a × c) i.e.,

a × (b + c) = a × b + a × c

Additive Inverse Property:

The additive inverse property states that the addition operation between any integer and its negative value will give the result as zero.

For any integer, a:

a + (-a) = 0

Multiplicative Inverse Property:

The multiplicative inverse property states that the multiplication operation between any integer and its reciprocal will give the result as one.

For any integer, a: a × 1/a = 1

Identity Property:

Integers follow the Identity property for addition and multiplication operations.

Additive identity property states that: a × 0 = a

Similarly, multiplicative identity states that: a × 1/a = 1

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