In math
we use various types of numbers. Though you will study more as you get older,
the following groups (or systems) include most of the numbers we will work
with.
Natural numbers are also known as the "counting numbers". They are the
group of numbers that we count with.
The lowest of these numbers is 1. The group is as follows:
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . . }
There is no highest number in the natural numbers group (the group approaches
infinity).
Whole numbers are all nonnegative numbers that do no have a decimal or
fractional portion.
To be nonnegative simply means to not be negative. Therefore, whole
numbers are not negative.
The least of these numbers is 0. The group is as follows:
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . . }
There is no greatest number in the whole numbers group (the group approaches
infinity).
Notice that 0 is the only difference between the whole numbers and
natural numbers groups.
Integers are all numbers that do nt have a decimal or portion. Sometimes
they are described as both positive and negative whole numbers.
There is no least or greatest of the integers. The group is as follows:
{ . . . , 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, . . . }
If you take all the whole numbers and make them both positive and negative,
you end up with the integers group.
Rational numbers are all numbers that can be written as a fraction of
two integers.
The rational numbers group includes all integers as well as terminating
and repeating decimals.
This group includes most of the numbers that we are familiar with at
this point. Later, we will study others.
Terminating decimals are those that end (termintate). Repeating decimals
are those that repeat.
Decimals that do not repeat, but yet do not terminate (e.g. pi) are not
included in the rational numbers group.
Examples:
Irrational numbers are numbers that are not rational. In other words, they
are numbers that CANNOT be written as a fraction of two integers.
Another way to think about these numbers is to keep in mind what type of
decimals they are. Since rational numbers are all the terminating and
repeating decimals, irrational numbers are all the numbers that do not
terminate and do not repeat. That means, irrational numbers are decimals
that never end and never repeat.
At this point, there is only one number we know of thats like this  pi (Π).
Although when using 3.14 it seems that
pi terminates, it is important to remember that this is just an estimate for
pi (as is 22/7).
There are other numbers in this group as well, but it is too early at
this point to mention them. We will return to this group during our discussion
of
exponents and roots. The important thing to note is that this group does
not include any other number discussed so far, only pi.
A real number is any rational or irrational number. The real number group
is just a collection of all numbers that are rational and numbers that
are
irrational. In other words, every number that we have seen or worked
with is a real number. There are numbers that are not real numbers.
No, they are
not called 'fake' numbers, but if your were thinking 'imaginary' you
might be on the right track. :o)
Venn Diagram
A Venn diagram allows us to visual see the relationship
that one or more groups/entities has.
This Venn diagram shows the relationship between the 6 number groups we've
looked at.
From the diagram we see that the smallest group of numbers
is the Natural numbers. We also see that the Natural nubmers circle
is completely inside the Whole Numbers circle. It is drawn this way to
show that every Natural number is a Whole number.
Since there is space leftover inside the whole numbers circle, it means
there must be one or more whole numbers that are not natural.
Zero is the only number that would be considered a whole number, but not
a natural number.
Since the whole numbers circle is inside the Integers
circle, we have shown every whole number is an integer.
However, there is still space leftover inside the integers circle for
the numbers that are integers, but not whole.
This space is reserved for the negative integers (e.g. 2, 3, 4, etc.).
The ingegers circle is completely inside the rational
numbers rectangle. This means all integers are rational.
There is room leftover inside the rational numbers rectangle for all those
numbers that are rational, but not considered integers.
This space is reserved for all values with fractional portions (e.g. 1.25,
2.3, etc.).
The irrational numbers rectangle is not inside, nor does
it overlap the rational nubmers rectangle. This means they share
no common numbers. If a number is inside the rational numbers group,
then it cannot be a part of the irrational numbers group,
and vice versa. However both rectangles are inside the entire Real numbers
rectangle (made from both smaller rectangles).
This means all rational numbers and irrational numbers are real numbers.
However, you'll notice there is no space leftover.
This means that Real numbers are only rational or irrational. No other
type of number is considered Real.

