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What is an Integer in Math?

Learn the basics of integers and their properties in this comprehensive guide. Discover what integers are, how they are used in mathematics, and the different types of integers with a Learner tutor.
Mathematics
8 minutes
What is an Integer in Math?
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What is an Integer?

An integer is one of the basic building blocks of math, and it is crucial to understand what an integer is before performing any math function. An integer is a whole number. Integers can be positive, negative, or zero.

Once you understand what an integer is and how to use it, you can perform many different mathematical functions, such as addition, subtraction, multiplication, division, and more! An integer is used in most math problems and is a foundational tool in math.

Whether you are a young student or an adult hoping to improve your math skills, understanding an integer's role in math is extremely important and valuable!

Origin of the Integer

The word "integer" comes from the Latin words "in" and "tangere," which mean "untouched" or "complete." In mathematics, a number that is not a fraction or a whole number  is referred to as an "integer” in mathematics.

Numerous mathematicians have researched and developed the idea of integers throughout history. Ancient cultures like the Babylonians, Mayans, and Indians all came up with the idea of zero as a placeholder in math, which led to the creation of integers. Robert Kaplan's book "The History of Zero" says that in Mesopotamia around 500 BCE, zero was used as a placeholder.

In the 1800s, the German mathematician Richard Dedekind made a new sign for integers based on the German word for "numbers," "Zahlen." This symbol is still in use today to represent any number. The book "The Nature of Numbers" by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery says that Dedekind made this sign in 1888.

In addition, negative numbers were added to the conventional number system by mathematicians such as Auguste-Louis Cauchy, John Herschel, and John Wallis in the 19th century. "Negative Numbers and the Number Line" by John A. Paulos says that in 1821, Cauchy was the first person to think of negative numbers as real numbers. Negative numbers are integers and are useful in several complicated and simple equations. Negative numbers are the inverse of their positive equivalents, therefore negative 5 is the inverse of positive 5.

Types of Integers

There are three types of integers: positive, negative, and zero. Positive and negative integers include many different values, while zero is a single value. All three types of integers are essential to understand in order to solve complex math problems later on!

Positive

Positive integers are positive whole numbers. These are real numbers greater than zero and are used in many math problems.

Some examples of positive integers are 1, 29, 389, and even 1,000,000,000. Positive integers are any whole number greater than 0 that goes up to infinity. Integers must be absolute numbers, so a fraction like 1/2 is not an integer, even though it is positive.

Adding two positive integers together will always give another positive integer. Dividing two positive integers can give another positive integer, although it can also lead to a fraction that is not an integer.

Negative

Negative integers are negative whole numbers. A negative integer is one that has a value less than zero. This is the opposite of a positive integer, which has a value greater than zero.

Negative fractions are not integers, although they are negative. Any whole negative number is a negative integer. Some examples of negative integers are -1, -29, -389, and -1,000,000,000.

Adding together two negative integers will give you another negative integer as the solution. Subtracting negative integers can give you either a negative or positive integer, depending on the values of the integers you are working with.

Zero

Zero is a special integer, as it is neither positive nor negative and is the only integer of its kind. Zero is neutral and sits in between positive and negative integers. However, it is considered a whole number, meaning it is an integer.

Zero is special because it represents nothing. If you add zero to an integer, the solution is the integer. If you subtract zero from an integer, the solution is the integer. Multiplying with zero yields zero.

Many mathematical functions utilize zero as an important tool. 

Rules for Integers

When adding and multiplying integers, there are a few rules you need to know. Once you do, you can solve a variety of math problems quickly.

1. Adding together two positive integers will give you an integer as the solution.

For example, 5 + 6 = 11. 5 is a positive integer, 6 is a positive integer, and 11 is a positive integer.

2. Adding together two negative integers will give you an integer as the solution.

For example, -3 + -2 = -5. 

-3 is a negative integer, -2 is a positive integer, and -5 is a negative integer.

3. When you multiply one positive integer with another positive integer, your solution will also be an integer.

For instance, 3 x 2 = 6. 

3 is a positive integer, 2 is a positive integer, and 6 is a positive integer.

4. Multiplying a negative integer with another negative integer will give you an integer as the solution.

For example: -5 x -6 = 30. 

-5 is a negative integer, -6 is a negative integer, and 30 is a positive integer.

5. Adding an integer and its inverse will give you zero as the solution

For example: -4 + 4 = 0. 

-4 is the inverse of 4, and adding them together gives you the integer zero as the solution.

6. Multiplying an integer with its reciprocal will give you 1

For example, 5 x 1/5 = 1. 

1/5 is the reciprocal of 5, and when you multiply the two, you get one, which is a positive integer.

Basic Math Operations Using Integers

Integers are used in addition, subtraction, multiplication, and division.

Adding Integers

If you add two integers that have the same sign, you must first add the absolute values of the integers and then add the sign that accompanied the numbers to the final sum.

Example: (+6) + (+7) = +13

Example: (-4) + (-3) = -7.

If the two integers that you want to add have different signs, you should subtract the absolute values and then write down the difference using the sign of whichever number has a larger absolute value.

Example: (-5) + (+9) = +4

Example: (+1) + (-7) = -6

Subtracting Integers

To subtract two integers, the second number (the one being subtracted) should be changed, and then you can perform addition with these integers.

Example: (+5) - (+3) = (+5) + (-3) = +2

Example: (-10) - (+4) = (-10) + (-4) = -14

Addition and subtraction are very similar, which is why you can use the properties of addition to subtract two integers. This is useful because once you understand addition, you will also understand subtraction without having to do any extra work!

Multiplying Integers

If both integers that you are multiplying have the same sign, whether it is negative or positive, the answer will be positive. If both integers have different signs, the answer will be negative.

Example: (+3) x (+4) = +12

Example: (-8) x (-7) = +56

Example: (+5) x (-6) = -30

Dividing Integers

Division is a lot like multiplication, so if you know how to do one, it will help you a lot with the other. These two mathematical functions are linked, and one can be used to understand the other better.

If both integers that you are dividing have the same sign, the answer will be positive. If their signs are different, the answer will be negative.

Example: (+15) ÷ (+3) = +5

Example: (+16) ÷ (-4) = -4

Integers in the real world

Integers are important in the real world, and they are used in many places in your daily life that you may not even be aware of.

Temperatures are a place where integers are commonly used. In the Celsius system, zero represents freezing. Any positive integer temperature is above zero and represents the temperature being above freezing. Any negative integer temperature is below zero and represents the temperature being below freezing.

In sports games, the score is reported using integers. Generally, whichever team has a higher score is the winner. This is true in sports such as soccer, football, baseball, and more! Understanding what makes one integer bigger than another can help you understand who is winning! Positive integers that are further from zero are larger, while those closer to zero are smaller.

The value in your bank account can often be represented in integers, and having a positive integer means that you have money in the account, while having a negative integer means that you owe money. Understanding what this means is incredibly important and has real-life implications.

Find a math tutor!

Now that you understand what an integer is, it is time to learn how to use them! Integers are the basis of many math problems, and the door is now open to completing many different types of problems! Looking at learner.com can show you strategies and connect you to a math tutor.

There are many benefits to hiring and using a math tutor, and it is really useful to understand math concepts as they get more complex. The further you get into your mathematical journey, the more difficult it will become, and getting help is the best way to ensure that you do not fall behind in your quest for knowledge.

Utilizing a math tutor can help you understand how to use integers in functions such as multiplication, division, addition, subtraction, and even complex math problems! Integers are used in many types of math problems, such as in statistics, trigonometry, and even calculus! Understanding integers is the first step to understanding the complex world of mathematics!

Integers are used in many of the properties of math, which can be explored here. These are important properties that are very useful, and they all involve understanding what an integer is. Now that you understand what an integer is and some basics for how to use it, you can apply the concept to more complicated mathematical functions and start to show off your math skills to yourself and the world!

About the author:

Mike developed his passion for education as a math instructor at Penn State University. He expanded his educational experience launching and running an Executive Education business - training over 100,000 students per year. As the CEO of Learner, Mike focuses on accelerating learning and unleashing the potential of students. 

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