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Properties of Integers: Definition, Examples and Solutions

What is an integer?

An integer is a number with no decimal or fractional part from the set of negative and positive numbers, including zero. Integers include the numbers -4,3,70,99 and 3,043.

Z denotes a set of integers that includes the following:

  • Positive Integers: If an integer is greater than zero, it is considered positive. For example: 10, 20, 30, 40, 50.
  • Negative Integers: If an integer is less than zero, it is considered negative. For example: -10, -20, -30.
  • Zero is defined as neither a negative nor a positive integer. It’s a complete number -70, -60, -50, -40, -30, -20, -10, 0, 10, 20, 30

In this article, we will briefly look into the properties of integers with examples to understand the concept of Integers.

Integers on a Number Line

A number line is a graph that depicts the numbers in a straight line. This line compares numbers placed at equal intervals on an infinite line that extends horizontally on both sides.

The set of integers, like other numbers, can be represented on a number line.

Properties of Integers

The significant Properties of Integers are:

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

Property 1: Associative Property

Changing the grouping of two integers does not change the result of the operation, according to the associative property. When two integers are added or multiplied, the associative property applies.

x and y are any two integers:

x + (y + z) = (x + y) + z
x × (y × z) = (x × y) × z

Example: 1 + (2 + (-4)) = 0 = (1 + 2) + (−4);
1 × (2 × (−4)) =−6 = (1 × 2) × (−4)

Subtraction of integers is not associative, i.e. x − (y − z) ≠ (x − y) − z.

Property 2: Distributive Property

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

x × (y + z) = x × y + x × z

Example: −5 (3 + 1) = −20 = (−5 × 3) + (−5 × 1)

Property 3: Additive Inverse Property

The additive inverse property states that adding an integer to its negative value returns zero.

For any integer, x:
-x + (x) = 0

Property 4: Commutative Property

The commutative property states that swapping the positions of operands in operation does not affect the outcome. The commutative property governs the addition and multiplication of integers.

For any two integers, x and y:
x + y= y+ x
x × y= y× x]

Example : 3 + (−6) = −3 = (−6) + 3;
10 × (−1) = −10 = (−1) × 10

But, subtraction (x − y ≠ y − x) and division (x ÷ y ≠ y ÷ x) are not commutative for integers and whole numbers.

Example : 3 − (−6) = 9 ; (−6) – 3 = -9
⇒ 3 − (−6) ≠ (−6) – 3
Ex: 12 ÷ 2 = 6 ; 2 ÷ 12 = 1/6
⇒ 12 ÷ 2 ≠ 2 ÷ 12

Property 5: Identity Property

The identity property states that the result is the same when a zero is added to any number., and additive identity refers to a zero value.

For any integer a,
a + 0 = a

According to the integer’s multiplicative identity property, the integer itself is when a number is multiplied by 1. As a result, it is called a number’s multiplicative identity.

For any integer a,
a × 1 = a= 1 × a

If an integer is multiplied by 0, the result will be zero:
Y× 0 = 0 =0 × y

If an integer is multiplied by -1, the result will be the opposite of the number:
Y × (−1) = −y = (−1) × y

Property 6: Closure Property

For any mathematical operation, the set is closed, according to the closure property. Addition, subtraction, multiplication, and division of integers close to Z. x and y are any two integers:

X + y ∈ Z
X – y ∈ Z
X × y ∈ Z
x/y ∈ Z

Example 1: 1 – 4 = 1 + (−4) = −3;
(–5) + 5 = 0,

The results are integers.

When it comes to multiplying, the closure property asserts that the product of any two numbers will be an integer, i.e. if x and y are both integers, XY will be an integer as well.

Example 2: 6 × 5 = 30 ; (–5) × (4) = −20, which are integers.

The quotient of any two numbers x and y may not always be an integer, as a division of integers does not obey the closure principle.
Example 3: (−1) ÷ (−2) = ½ is not an integer.

Property 7: Multiplicative Inverse Property

The multiplicative inverse property states that the result is one when an integer is multiplied by its reciprocal.

Finding the multiplicative inverse of natural numbers is simple, but complex and real numbers are more complicated.
For any integer, x: x * 1/x = 1

Solved Example

Example 1: Show that -32 and 22 follow commutative property under addition.
Solution :
Let a = -32 , b = 22

Commutative property states that
a + b = b + a

L.H.S = a + b
= -32 + 22
= -10

R.H.S = b + a
= 22 + (-32)
= 22 – 32
= -10

So, L.H.S = R.H.S, i.e a + b = b + a

This means the two integers hold the actual commutative property under addition.
Example 2: Show that (-6), (-4) and (8) are associative under addition.
Solution : Let a = – 6; b = – 4 and c = 8

Associative property for addition states that

a + ( b + c) = ( a+ b ) + c
L.H.S = -6 + ( -4+ 8)
= – 6 + 4
= -2

R.H.S = (- 6 + (-4)) + 8
= (- 6 – 4) + 8
= -10 + 8
= – 2

So, L.H.S = R.H.S, i.e a + (b + c) = (a + b) + c

This proves that all three integers follow associative property under addition.

Conclusion

Understanding the different properties of Integers helps in solving problems related to integers easily. By following the examples and solutions in this article, meticulously practice to grasp the concept thoroughly.