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Subsets of Real Numbers

In this course, we will usually be dealing with the set of real numbers, which has many frequently used subsets.  When you were a young child, one of the first things you probably did with numbers was count.  This most natural set of numbers is called the set of counting or natural numbers:

  =  {1, 2, 3, 4, …}

If we include the number zero, we call the set the set of whole numbers:

W =  {0, 1, 2, 3, …}

Notice that neither the natural numbers nor the whole numbers included negatives or fractions. 

The set of Whole numbers together with their negatives is called the set of integers, often denoted by the capital letter, .

 = {…-3,-2, -1, 0, 1, 2, 3…}

The set of Rational numbers is the set of numbers of the form  where a and b are integers and  .  As a decimal, a rational number will terminate (end) or repeat.  This set is often denoted by the letter,  Q.

If a number is not rational, it is irrational.  As a decimal, an irrational number does NOT TERMINATE and DOES NOT REPEAT.  Some common irrational numbers are ,   ,   ,  , e.   We’ll use the letter I denote the set of irrational numbers.

All these subsets taken together compose the set of Real numbers, denoted by R.

We can represent the relationship among these subsets with the following diagram, called a Venn diagram and named after John Venn, an English mathematician.