Count from one to 10 on your fingers: 1, 2, 3 . . . 10. Each of your fingers represents a number, and just as you can only have a whole finger, you can only represent a whole number on each finger. That's the meaning of of integers in math and algebra: Whole numbers. No fractions allowed! Integers are counting numbers, and they include 0.
Let's say you now want to count from -1 to -10, and to represent these numbers you put your fingers upside down. Count again: -1, -2, -3 . . . -10. The same rule applies. Each of your fingers represents a number, and just as you (hopefully) don't have a partial finger, you never have a partial number, or fraction. In other words, integers can be negative, but they can't be fractional. Any number with a fraction – and that includes decimal fractions – is not an integer.
The Arithmetic of Whole Numbers
Arithmetic is math at its most basic, and it involves four operations that most people use almost every day. They are addition, subtraction, multiplication and division. You can do arithmetic with both positive and negative integers, which are also known as signed numbers, or you can do it with absolute values, which means that you ignore the signs and assume the integers are all positive. Almost everyone learns the arithmetic rules of signed numbers in the first few years of elementary school:
Adding Integers – Add two positive or negative integers together to make a bigger number and keep the sign. When you have a positive and negative integer, you "add" them by subtracting the smaller one from the larger one and keeping the sign of the larger one.
Subtracting Integers – When you subtract two integers with the same sign, you end up with a smaller integer, and when you subtract two integers with opposite signs, you get a larger one. Subtracting a negative integer is the same as changing the sign of the integer to positive and adding it.
Multiplying and Dividing Integers – The rule for multiplication and division is easy to remember. When multiplying and dividing numbers with the same signs, the result is always positive. If the numbers have opposite signs, the result is negative.
Note that addition and subtraction are inverse operations, and so are multiplication and division. Adding an integer to 0 and then subtracting the same integer leaves you with 0. When you multiply any number except 0 by an integer, and then divide by the same integer, you're left with the original number.
Every Integer Can Be Factored Into Prime Numbers
Another way to consider integers is to recognize that each one is the product of prime numbers, which are integers that can't be factored any further. For example, 3 is a prime number, because you can't factor it, but 81 can be written as 3 • 3 • 3 • 3. In addition, there is only one way to factor a given number into its component prime numbers. This is known as the Fundamental Theorem of Arithmetic.
Integers and Whole Numbers in Algebra
In algebra, you use letters to represent numbers. The letters are called variables. When the variables represent integers, you apply the same rules that you apply in basic arithmetic. Remember, integers are whole numbers, so if you encounter a problem that specifies that the variables represent integers, they must be whole numbers. That means you can't input any fractions for them, but it doesn't mean that, after you perform the indicated operations, the results won't be fractional.
About the Author
Chris Deziel holds a Bachelor's degree in physics and a Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. He began writing online in 2010, offering information in scientific, cultural and practical topics. His writing covers science, math and home improvement and design, as well as religion and the oriental healing arts.
Hemera Technologies/AbleStock.com/Getty Images