Numbers and Their Properties

Natural Numbers:

These numbers are sometimes referred to as the counting numbers. They are the numbers that lie within the set
{1 , 2 , 3 , ...}.

Whole Numbers:

These are the numbers that lie within the set {0 , 1 , 2 , 3 ,... }. Note that the whole numbers include 0.

Integers:

These are the numbers that lie within the set {... ,-3 , -2 , -1 , 0 , 1 , 2 , 3 , ...}. So the integers contain the whole numbers, and their additive inverses. Note: Zero is neither negative or positive.

Rational Numbers:

A rational number is a number of the form where p and q are integers and q 0. For example, is a rational number.

Irrational Numbers:

An irrational number is a real number that cannot be expressed as a fraction for any integer p and any integer q. For example, is an irrational number.

Real Numbers:

The real numbers are the set of all irrational and rational numbers. So a real number is either rational or irrational.

Example:

Show that the number 1.42 is rational.

Solution: In order to prove that 1.42 is rational, we need to show that the number follows the definition of a rational number. Notice that 1.42 = 1 + . We know that 142 and 100 are both integers, and certainly 100 does not equal 0. It follows from the definition that 1.42 is a rational number. The method employed here to show that 1.42 is a rational number is called a direct proof.

Example:

State the number system(s) that the number 10 belongs to.

Solution: Clearly, 10 is both a natural and a whole number. The set of integers is defined as the set {... ,-3 , -2 , -1 , 0 , 1 , 2 , 3 , ...}. Since 10 lies within this set, 10 is also an integer. We can also represent this number as 10 divided by 1. Since both 10 and 1 are integers, and 1 is nonzero, it follows that 10 is a rational number. Since 10 can be written as the quotient of two integers, 10 cannot be irrational. Since the real numbers are defined as the set of rational and irrational numbers, 10 is a real number. So the number 10 is a natural, whole, integer, rational, and real number.

Properties

The sets of numbers defined above follow a set of properties. For example, whenever we add or multiply two numbers from a set, we get a number back in the set. This is called closure. The sets above are said to be closed under addition and closed under multiplication. Notice that some of the sets defined above are not closed under subtraction. For example, the natural numbers are not closed under subtraction (3 + (-5) = -2 which is not a natural number). Also, division in the natural numbers doesn't even make sense since fractions are not natural numbers. This is why we need the real numbers. Because all of the properties mentioned are properties of the real numbers. We list some properties of the real numbers below.

 Let a,b, and c be any real number.

Distribution actually involves both addition and multiplication properties. It allows you to essentially convert a multiplcation with the sum of two variables into a sum of products. Below are the left and right distribution properties.

      a(b + c) = ab + ac    and    (a + b)c = ac + bc

Homework

Decide whether the given number is natural, whole, an integer, rational, irrational, or real:

1.   2

2.   -2

3.   0

4.  

5.  

6.   2.7182818284...

7.   Show that 0.5 is a rational number.

State the property illustrated.

8.   2 = 1

9.   3(1) = 1

10.   5(1 + 3) = 5 1 + 5 3.

11.   5 - 13 = -8 over the real numbers

12.   6 3 = 18 over the real numbers

13.   (3 2)4 = 3(2 4)

14.   5 - 5 = 0

True or False.

15.   A rational number can be a whole number.

16.   A rational number can be an irrational number.

17.   An irrational number can be a rational number.

18.   A whole number is always a natural number.

19.   Irrational and rational numbers are both real numbers.

20.   Every whole number is an integer.

21.   Every natural number is a whole number.

22.   Every natural number is a whole number, a rational number, and a real number.