Properties of Addition and Subtraction of Integers

Introduction

Natural numbers, their additive inverses, and zero are all collectively known as a set of Integers. We get a whole number when we subtract a small number from a larger number. However, there are no whole numbers that can represent the difference between a large number and a smaller number, such as 12 – 37. We created integers to describe such differences. Integer Addition follows all the rules of algebra, whereas Integer Subtraction doesn’t.

Addition and Subtraction of Integers

In addition, and subtraction of integers, we will learn how to add and subtract integers with the same and different signs. Certain rules must be followed when performing operations on integers.

When you add two positive integers, you get a positive integer, but when you add two negative integers, you get a negative integer sum. Adding two different signed integers, on the other hand, results in subtraction only, with the result having the same sign as the larger number. For example,

Addition with the same sign

3 + 5 = 8 or (- 7) + (- 5) =- 12

Addition with a different sign

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

As for subtraction, when you subtract the integers, you add the additive inverse of the second integer, i.e., you can simply just change the sign of the second integer and add the two numbers using the rules of addition. For example,

7 – 5 = 7 + (- 5) = 2

9 – (- 2) = 9 + 2 = 11

(- 3) – (- 5) = (- 3) + 5 = 2

(- 6) – 7 = (- 6) + (- 7) =- 13

Properties of Integer Addition and Subtraction

Integers have a few properties that govern how they operate. These principles or properties can be used to solve a wide range of equations. Integers are any positive or negative numbers, including zero, to refresh your memory. These integers’ properties will aid in quickly simplifying and answering a series of integer operations.

All addition, subtraction, multiplication, and division properties and identities apply to all integers. The set of positive, zero, and negative numbers represented by the letter Z is known as the set of integers. Integers have the following five operational properties:

• Closure Property
• Commutative Property
• Associative Property
• Distributive Property
• Identity Property

Closure Property

According to the closure property, the set of all integers is closed under addition and multiplication, i.e., the addition or multiplication of any two integers will always result in an integer. Subtraction of integers also follows the closure property; division however does not follow the same rule for integers. For example, 2 divided by 5 is not an integer.

If a and b are two integers, then if,

c = a + b and d = a × b

Then both c and d are also integers

Changing b with – b,

e = a – b is also an integer.

Commutative Property

According to the commutative property of addition and multiplication, the order of terms does not affect the result. Let a and b be two integers, then by the commutative law:

a + b = b + a

Also,

a × b = b × a

But subtraction and division do not follow the same rules.

Associative Property

According to the associative property of addition and multiplication, it doesn’t matter how numbers are grouped; the result is the same. Regardless of the order of the terms, parenthesis can be used.

a + (b + c) = (a + b) + c

Also,

a × (b × c) = (a × b) × c

However, again, subtraction and division are not associative for Integers.

Distributive Property

The distributive property explains how one mathematical operation can distribute over another within a bracket. To make the calculations easier, the distributive property of addition or the distributive property of subtraction could be used. In this case, integers are multiplied or divided by each number in the bracket before being added or subtracted again.

a × (b ± c) = (a × b) ± (a × c)

Identity Property

When zero is added to any integer no matter the order, the result is the same number, according to the additive identity property of integers. Zero is known as additive identity.

Let a be an integer

Then, since 0 is known as the additive identity

a + 0 = a = 0 + a

Similar to this, we have a multiplicative identity. When a number is multiplied by 1 in any order, the product is the integer itself, according to the multiplicative identity property for integers.

Again, let a be an integer,

Then, since 1 is known as the multiplicative identity

a × 1 = a = 1 × a

Again, like most other properties, subtraction and division do not follow the identity property.

Solved Examples

Example 1: Simplify the following expressions

a. (- 2) + (- 4) – 5 + 13

b. 4 × 27 + 4 × 31 – 4 × 53

Solution:

a. (- 2) + (- 4) – 5 + 13

First, we will use parentheses to separate different operations

⇒ ((- 2) + (- 4) – 5) + 13

Here we have separated the only positive term, but we still have 3 terms in the parentheses,

⇒ (((- 2) + (- 4)) – 5) +13

Now solving the operations in the parentheses one by one,

⇒ ((- 6) – 5) + 13

⇒ (- 11) + 13

⇒ 2

b. 4 × 27 + 4 × 31 – 4 × 53

First, we can see that 4 is a common factor in all the products. So we will factor it out,

⇒4 × (27 + 31 – 53)

Now we will separate the terms inside the parentheses using centric parentheses

⇒ 4 × ((27 + 31) – 53)

No solving the parentheses one after the other

⇒ 4 × (58 – 53)

⇒ 4 × 5

⇒ 20

Summary

This article gives an insight into the properties of addition and subtraction of integers. According to the addition closure property, the sum of any two integers will always be an integer. According to the subtraction closure property, the sum of, or the difference between, any two integers will always be an integer. The commutative property of integer addition states that the order of addition of integers does not matter, the result remains the same regardless. Subtraction, on the other hand, is not commutative for integers. According to the associative property of addition, the order in which numbers are grouped does not affect the outcome. The nature of integer subtraction is not associative.

Frequently Asked Questions (FAQs)

1.What are Integers? How are they different from other types of Numbers, such as Natural Numbers and whole numbers?

Ans. Integers are numbers that have no decimal value, whole values represent them. The integers contain both positive and negative numbers, along with the number 0. The positive integers are known as natural numbers, whereas the positive integers with 0, aka the non-negative integers, are known as whole numbers.

2.What are the rules for Adding Integers?

Ans. The addition of integers follows two simple rules,

1. Adding two integers of the same sign results in the addition of the value of those numbers, and the result has the same sign as the additives.
2. Adding two integers of different signs results in a difference in the value of integers, and the result has the same sign as the number with the higher value.

3.Which Properties of Integer Addition can be applied to Integer Subtraction?

Ans. The following properties of integer addition can be applied to integer subtraction,

• Closure: The closure property states that the result of the operations is in the same set as the input, i.e., subtracting two integers will always result in an integer.
• Distributive: Distributive property states that the operation of multiplication can distribute over operations such as addition or subtraction, i.e., the product of a number and a difference of two numbers is the same as the difference of the product of the two numbers with the first.

4.What is an Inverse?

Ans. The inverse of a number is another number, which, when operated under a certain operation with the original number, results in the identity of that operation. That is, for addition, the inverse is defined as the number which when added together with the original number results in 0, and for multiplication, multiplies to give 1.