(My math colleagues had to help me out on this one!) Mathematicians classify numbers into types or number systems. As you learn these various number systems, it's important to remember that numbers can be more than one number type. Or in math geek-speak, number systems can be subsets of other number systems. But before we get too complex (pun intended), let's start from the beginning.
How do you classify numbers, as in rational numbers, integers, whole numbers, natural numbers, and irrational numbers? I am mostly stuck on classifying fractions.
When you first learned to count, you started with 1, 2, 3 and kept going until you couldn't remember what came next or grew tired of counting. These positive counting numbers (1, 2, 3, 4, ...) are called natural numbers. The ... means the number list continues on infinitely.
If you add the number 0 to the natural numbers, you get the whole numbers (0, 1, 2, 3, ...). You also get an example of how a number can be classified as more than one type. For example, the number 2 is both a natural number and a whole number. In fact, all natural numbers are whole numbers, but not all whole numbers are natural numbers. Why? The number 0 is a whole number but not a natural number.
Integers include 0, the natural numbers, and the negatives of the natural numbers: (..., -3, -2, -1, 0, 1, 2, 3, ...). Again, the ... signifies the numbers go on to infinity — this time in both directions. All whole numbers (and therefore, all natural numbers) are integers, but not all integers are whole numbers. Starting to see the pattern here?
You asked about classifying fractions. Fractions are nothing more than ratios of integers. Numbers that can be written as fractions a/b, where a is an integer and b is a natural number, are called rational numbers. Remember that even an integer like 5 can be written as a fraction by dividing it by 1: 5/1. So you can see that all integers are rational numbers. Since decimals that end and repeat can be written in this form (0.66... = 2/3), they also are rational numbers.
If a decimal number doesn't repeat or end, it is not rational. It is classified as an irrational number. An irrational number can't be written as a fraction a/b, where a is an integer and b is a natural number. Pi (3.1415...) is a common example of a number that is irrational. Irrational numbers and rational numbers are two distinct classifications — a rational number (and integers, whole numbers, or natural numbers) can't be irrational.
Rational numbers and irrational numbers together make up the real numbers. Real numbers and imaginary numbers like i (the square root of –1) together comprise the complex numbers. But that, I suppose, is a lesson for another day.