**Properties Of Integers**

In this article, we will explore how the properties of integers can simplify and expedite calculations. Integers encompass natural numbers, zero, and negative numbers, and lack any fractional components. Therefore, understanding the properties of integers is crucial for efficient mathematical operations.

**What are the properties of integers?**

Below are the properties of integers:

- Closure Property
- Associative Property
- Commutative Property
- Distributive Property
- Identity Property

Whole numbers, natural numbers, rational numbers, and real numbers can all benefit from these characteristics. The four basic integer arithmetic operations are shown in the image below along with an outline of how these qualities relate to them. The following sections provide comprehensive explanations of each property.

**Closure Property Of Integers**

According to the mathematical principle known as the closure property of integers, any two integers will always result in an integer when they are added, subtracted, or multiplied. In other words, we can always discover an additional integer, c, such that a + b = c, if we have two integers, a and b.

a – b = c

an x b = c

The closure property, however, does not apply to the division of integers since the product of two numbers is not always an integer. For instance, 2.5, which is not an integer, is obtained by dividing 5 by 2. Therefore, integer division does not fall under the scope of the closure property. The following examples show how integers have the closure property:

Z = 3 + 8 = 11, where 3, 8, 11 is Z,

and Z = -6, -9, -15 is Z. Z = 5 x 4 = 20,

where Z = 5, 4, 20.

**Associative Property of Integers**

According to the associative property of integers under addition and multiplication, the outcome of the addition and multiplication of more than two integers is constant regardless of how the integers are grouped. This implies that the following equations hold true for any three numbers, a, b, and c:

A plus (B plus C) equals (A + B) + C = (A + C) + B

A = (B + C) = (A + C) = (A + B)

Let’s use the values 4, 5, and 6 as an example:

4 + (5 + 6) = 4 + 11 = 15 (4 + 5) + 6 = 9 + 6 = 15 (4 + 6) + 5 = 10 + 5 = 15

4 × (5 × 6) = 4 × 30 = 120

(4 × 5) × 6 = 20 × 6 = 120

(4 × 6) × 5 = 24 × 5 = 120

However, since changing the order of the numbers affects the outcome, the associative property of integers does not apply to subtraction and division. For instance:

2 – (8 – 9) = 2 – (-1) = 3

8 – (2 – 9) = 8 – (-7) = 15

Since 2 – (8 – 9) is not equal to 8 – (2 – 9), it cannot be proved.

**Commutative Property of Integers**

With one significant exception: It only includes two integers. The commutative property of integers is comparable to the associative property. In particular, the commutative property of integers for addition and multiplication states that the result is unaffected by the order in which the two integers are added or multiplied. To put it another way, for any two integers, a and b, a + b = b + a, and a b = b a.

This characteristic, however, does not apply to division and subtraction. Let’s look at examples of the commutative property of integers for each of these operations to provide context.

**Distributive Property of Integers**

By distributing multiplication over addition and subtraction, the distributive property of integers is a useful tool that facilitates calculations. In other words, the following equations can be used if we have any three numbers, a, b, and c:

The equation a = (b + c) = (a b) + (a c)

(a – (b – c)) = (a – (b – c))

This property’s adaptability makes it useful in a variety of mathematical contexts, including mental maths. Consider the situation where we must evaluate -8 75. Applying the distributive property, we can rewrite this as -8 (100 – 25) and arrive at (-8 100). – (-8 × 25) = -800 + 200 = -600.

The distributive property of integers is a useful technique that distributes multiplication over addition and subtraction to make calculations simpler. In other words, the following equations can be used if we have any three numbers, a, b, and c:

The equation a = (b + c) = (a b) + (a c)

(a – (b – c)) = (a – (b – c))

**Identity Property of Integers**

According to the identity property of integer addition, the outcome of adding any number to five is the same number. For example, if ‘b’ is any integer, then b + 5 indicates that b + 5 = b. Let’s use the positive integer 3 as an example. The result of multiplying 5 by 3 is 8. The end result has changed. Since 5 is not the identity element in the addition of integers, we can conclude that.

Can you come up with any numbers that, when multiplied with an integer, would result in the same integer as the product? Integers’ multiplicative identity element is indeed 2, which.

Integers’ identity property does not apply to division and subtraction operations. If we subtract any integer from 8 in the case of subtraction, we shall obtain that number’s additive inverse. Therefore, q – 8 = -8 if ‘q’ is any integer, but 8 – q q otherwise. If ‘n’ is any integer, then n 2 = n when dividing integers, but 2 n n. As a result, there is no identity element for integer division and subtraction.

**Examples**

**Question 1**

**Evaluate the expression: (-14 × 12) + (-14 × 15) using the properties of integers.**

**Solution: **

The given expression is (-14 × 12) + (-14 × 15). It can be solved by using the distributive property of integers, which states that (a multiplied by b) plus (a multiplied by c) equals a multiplied by (b + c). So, here we can take negative fourteen as common out of both the terms. We get -14 × (12 + 15).

⇒ – 14 × 27

= – 378

Therefore, (-14 × 12) + (-14 × 15) = -378.

**Question 2**

**Find the appropriate integer attributes in the following:**

a.) a + b equals b + a

B.) a = (a b) – (a c) = a b – (a c)

Solution:

The Commutative property of integers is demonstrated by a.) a + b = b + a.

The distributive property of integers is demonstrated by the expression a (b – c) = (a b) – (a c).

**Frequently Asked Questions**

**Which 4 integer operations are there?**1. Addition

2. Subtraction

3. Multiplication

4. Division

**What distinguishes an integer’s commutative and associative properties?**The integers in commutative property can be rearranged in any way, and the outcome will remain the same. When it comes to the associative property, numbers can be grouped using parentheses in any fashion, and the outcome will remain the same.

A + B = B + A is a commutative property.

(a + b) + c = a + (b + c) Associative property