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Adding Integers: Definition & Examples

add integers

An integer is a number written without a fractional component. For example, 2, 3, 72, 901,-65,-876- these are all integers. The easiest way to define an integer is – it consists of an absolute value and a positive or negative sign.

Zero (0) is neither a positive integer nor a negative integer. The positive natural numbers are called positive integers(1,2,3 and so on), and their additive inverses are called negative integers (-1,-2,-3, and so on). This is very basic to be kept in mind when we add integers.

Major Functions of Arithmetic:

Arithmetic is mathematics’ one of the most fundamental and foundation parts. The base knowledge further enables us to understand different applications and studies such as algebra, trigonometry, mensuration, geometry, and other relevant subjects. The basics of arithmetics act as the driving force behind all other aspects of math.

Some common basic problem statements regarding integers are – how to add and subtract integers and how do you add integers with the same sign. These, along with some other functions, make the foundation of math.

The basic operations of arithmetics are:

  • Addition (sum; ‘+’)
  • Subtraction (difference; ‘-‘)
  • Multiplication (product; ‘×’)
  • Division (÷)

How to add integers will be thoroughly discussed further in the article. Meanwhile, let’s find out briefly about the other three functions.

Subtraction

The arithmetic operation of subtraction shows the difference between two numbers. The symbol ‘-‘ represents it. Subtraction is mostly used to find out what is left when things are taken away or, in other words, taking one number away from another number.

E.g., I had ten apples. 2 apples are taken away from me. So I will now have (10-2) = eight apples.

Multiplication

Repeated addition is known as multiplication. The symbol’ × represents it’. Multiplication as an arithmetic operation helps us find out the total when a number repeats itself multiple times. For example, three times four is 12. Mathematically, we can write it as three × 4 = 12. Multiplicand and multiplier are the terms used in the multiplication process. The product is the term we use for the result of the multiplication of the multiplicand and the multiplier.

E.g. I had two apples every day for 3 consecutive days. So I had 2 x 3 = 6 apples.

Division

Dividing something into equal parts or groups is known as division. One of the four fundamental arithmetic operations produces an equitable result of equal sharing. The inverse of multiplication is division. The symbol’ ÷ denotes it’.

E.g., For me, Andrew & Michael had 12 mangoes distributed evenly between us. So we individually had 12 ÷ 3 = 4 mangoes.

How to add integers

The addition is one of the four basic operations of arithmetic, subtraction, multiplication & division. Generally, the addition of two positive integers will result in a positive integer. The addition of two negative integers will result in a negative integer. Adding a negative and a positive integer will result in an integer that holds a larger value between the two negative numbers. How to add integers will be thoroughly discussed here.

The addition represents the values added to the existing value. For example, if I have three pens and four more pens are added to that list, I will have a cumulative seven pens as a result of the addition. Additions are subjected to both irrational and rational numbers. Thus addition applies to both real and complex numbers.

The three main types of additions are: 

  • The addition between two positive integers
  • The addition between two negative integers
  • The addition between one negative, one positive integer
Type of NumbersOperationResultExample
Positive + PositiveAdditionPositive (+)5 + 10 = 15
Negative + NegativeAdditionNegative (-)(-5) + (-10) = -15
Positive + Negative*SubtractPositive (+)(-5) + 10 = -5

When a positive and a negative number are added, the operation and sign of the result are determined by the greater number. In the preceding example, 5 + (-10) = -5 and 10 + (- 5) = 5; because ten is greater than five without a sign, the numbers will be subtracted, and the answer will indicate the sign of the greater number. This is an example of how to add and subtract integers for the same operation. 

Since we know that multiplying a negative sign by a positive sign results in a negative sign, if we write 10 + (-5), the ‘+’ sign here is multiplied by ‘-‘ inside the bracket. As a result, the answer is 10 – 5 = 5.

To summarize, how do you add integers with the same sign and opposing sign.

  • The addition of 2 positive integers will always result in a positive integer.
  • The addition of 2 negative integers will always result in a negative integer.
  • In the Addition of One positive and another negative integer, the higher absolute value of the two integers will determine the sign.    

Basic Examples:

  • 14 +15 =29
  • 7 + 5 = 12
  • (-3)+3 = 0 [ addition of equally valued two inverse integers will always be zero ]
  • (-50)+4 = (-46)
  • 10 + (-1) = 9
  • (-20)+ (-10) = (-30)

The following tables enlist some basic properties of addition. Here a, b, and c are considered integers.

Properties of addition and multiplication on integers  

  • Closure: a + b is an integer       
  • Associativity: a + (b + c) = (a + b) + c    
  • Commutativity: a + b = b + a       
  • Existence of an identity element: a + 0 = a          
  • Existence of inverse elements: a + (−a) = 0         
  • Distributive Property: a × (b + c) = (a × b) + (a × c)

Closure properties of an integer

The sum of 2 integers will always be an integer. If a, and b are two integers, their summation will also be an integer.

Examples: 5 + 6 = 11 ; -5 + 8 = 3.

Commutative property

The commutative property of addition states that the order of terms does not affect the result. Changing the terms, in addition, does not affect the sum or product.

Examples: 2 + 4 = 4 + 2 ; -6 + 8 = 8 + (-6)

Associative Property

The associative property of addition states that it makes no difference how numbers are grouped; the result is the same. The answer is the same regardless.

Examples: 2 + (3 + (-4)) = 1 = (2 + (−4)) + 3 

Distributive Property

The distributive property explains how one mathematical operation can distribute over another within a bracket. It can be either the distributive property of addition or the distributive property of subtraction. In this case, integers are added or subtracted first, then multiplied or divided by each number within the bracket, and then added or subtracted. 

Examples: −5 (3 + 2) = −25 = (−5 × 2) + (−5 × 3)

How to add Integers on a Number Line

The following principles govern the addition of integers on a number line:

• To add a positive number, move your cursor to the right (or positive) side of the number line.

• Adding a negative integer is accomplished by moving to the number line’s left side (or negative side). • Any of the given integers is the starting point for moving on the number line.

The example and stepwise guide are provided here:

  • The first step is to select a number line scale. For example, whether we want to plot numbers in multiples of 1, 5, 10, or 50 is determined by the integers provided. For example, if we need to add 10 and 20, we can use a number line with a scale of 10 to help us. 
  • If we have to add -4 and 9, we can take a scale of counting numbers starting from 1.
  • The next step is to find any integer on the number line. Preferably one with a larger absolute value. For example, if we need to add 3 and 22, it is preferable to find 22 on the line first and then take two jumps to the right rather than find three first and then take two jumps to the right.
  • The final step is to add the second integer to the previous step’s number by making jumps to the left or right. This depends on whether the number is positive or negative.

Practical implementation of how to add and subtract integers

  • I had ten cookies. I borrowed five from my dad. How many cookies do I have now?

        A: The answer is 10 + 5 = 15 (Practical example of how do you add integers with the same sign)

  • The room temperature was 20 degrees c. Mom asked me to increase it by 3 degrees. What will be the final temperature of the room upon increasing?

         A: It would be 20 + 3 = 23

  • Mike scored -3 in a quiz competition before the final. He scored ten more points to qualify for the finals. What was his total score?

          A: Add integers. That would be -3 + 10 = 7 (Example of how to add and subtract integers)

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