Please complete your responses to the questions 1-6 at the end of the page on a separate piece of paper.  

In mathematics different types of numbers are grouped together and given names.  It is important to have a feeling for this organization.  Consider the following definitions:

The sets of numbers:

Names	Sets	Notes and examples
natural numbers	{1, 2, 3, . . .}	1. See note 1. on set notation below:
whole numbers	{0, 1, 2 , 3, . . .}
integers	{0, ±1, ±2, ±3,. . .}
rational	{ p/q | p and q are integers, q¹0}	2.
real	{x | x can be written as a decimal}	3.
irrational	{x | x is a nonrepeating and nonterminating decimal}	p » 3.14159. . , e » 2.71828, Ö2

                          
Notes:

1. The definitions above are written in set notation.  The {} symbols called braces indicate the closing and opening of a set or collection of numbers.  The three dots after the three indicate that the pattern continues. See page 47 in your workbook for more examples.

2.  Read as the fraction p over q, where p and q are both integers. 

Rational numbers can be written in decimal form, but they always end or repeat. 

3. Read as all numbers x, such that x can be written as a decimal. 

  Look at the list of numbers.  What do you notice?  Notice that as you go down the list a new set will contain the set of numbers above it.  For example, the whole numbers contain the natural numbers.  In fact, the whole numbers are the natural with one new number, zero.  As you go down the list the numbers get more "complicated."  The  progression of numbers is much the way we learn about numbers as we grow up.  We start as small children with the natural numbers when we count our fingers and toys.  Then we make an intellectual leap and learn about "all gone" and the concept of zero, which takes us to  the whole numbers.  Then at some time in our development we learn about debt and negative numbers, and we start to use integers.  The same sort of progression happens in math classes.  You start doing math with whole numbers, then you learn about operation with negative and positives, and then fractions.  Notice the integers are completely contained by the rational numbers.  Any integer can be written as a rational by putting a one under it.  The one exception to this progression is the irrational numbers.  They are one their own.  The rationals and the irrationals are disjoint sets, which means you are one, but not the others.  These two sets together make the real numbers, which we will use in this class.  Here is a diagram to help you visualize the sets of numbers. 

The real number system consists of two subsets of numbers, the rational numbers and the irrational numbers. Contained within the rational numbers are other useful subsets of numbers. The subsets of real numbers are nested as indicated by the picture below:

1. Determine if the following statements are true or false and give a short reason why:
    
   
1. Every integer is a rational number.
2. Every rational number is an irrational number.
3. Every natural number is an integer.
4. Every integer is a natural number.
    
   

2. Consider the following set of numbers:

    

    a)    natural numbers:

    b)    whole numbers:

    c)    integers:

    d)    rational numbers:

    e)    irrational numbers:

    f)    real numbers:

5.   The following data set represents the mass in grams of various U.S. coins.
 

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