The Real Numbers
All of the numbers that you are
currently familiar with are part of
the set of real numbers.
Natural or Counting Numbers
Man first used numbers to keep track of sheep,
goats and other countable possessions.
The Natural or Counting Numbers are the ones
you use to count.
{ 1, 2, 3, …}
Natural or Counting Numbers
There are no fractions or decimals in the counting
numbers.
You wouldn’t say you had three and a half sheep.
You’d say you had 3 sheep and dinner!
Whole Numbers
One day a very philosophical man contemplated the
question:
What number would I use if I had no sheep?
…and that was the birth of the number zero!
Whole Numbers
The set of whole numbers are the counting numbers plus
their new friend, zero
It is easy to remember which are the whole numbers
because zero is the only number that looks like a hole!
When you see the word WHOLE,
{ 0, 1, 2, …}
think “hole”
The Integers
Next, someone invented checking accounts and within the hour
someone had to invent the negative numbers!
The integers are the negative counting numbers and the whole
numbers (still no fractions or decimals).
{…-2, -1, 0, 1, 2, …}
The Rational Numbers
The word ratio means fraction.
Therefore rational numbers are any numbers which can be
written as fractions.
2
3
3
4
5
1
1
5
Integers are Rational Numbers
2
3
3
4
5
1
1
5
Like the 5 in our example, any integer can be made into a
fraction by putting it over 1. Since it can be a fraction, it
is a rational number.
Changing fractions to decimals
It’s easy to change a fraction to a decimal, so rational
numbers can also be written as decimals.
Rational numbers convert to two different types of
decimals:
Terminating decimals – which end
Repeating decimals – which repeat
Terminating decimals
To convert a fraction to a decimal, divide the top by the bottom.
To convert ½ to a decimal you would do:
.5
2 1.0
There is no remainder. The answer just ends – or terminates.
Repeating decimals
To convert a fraction to a decimal, divide the top by the bottom.
To convert 1/3 to a decimal you would do:
.333
3 1.000
=

.3
There is a remainder. The answer just keeps repeating.
Repeating decimals

.3
.09
The bar tells us that it is a repeating decimal.
The bar extends over the entire pattern that repeats.
Rational numbers as decimals
Rational numbers can be converted from fractions to either
• Terminating decimals or
• Repeating decimals
Rational numbers
The subsets of real numbers that we’ve discussed are
“nested” like Russian dolls.
Venn Diagrams
Rational numbers
Venn diagrams
illustrate how
sets relate to
each other.
Natural numbers
Whole numbers
Integers
Subsets are
drawn inside the
larger set.
Irrational Numbers
In English, the word “irrational” means not rational illogical, crazy, wacky.
In math, irrational numbers are not rational.
They usually look wacky!
5
3
17

…and their decimals never end or repeat!
Irrational Numbers
There is one trick you need to watch out for!
Numbers like 25 and
81
They look wacky but because the number in the house is a
perfect square, they are really the integers 5 and 9 in
disguise!
Sort of like the wolf at Grandma’s house!
Rounding or truncating
Some decimals are much longer than we need. There are
two ways we can make them shorter.
Truncating – just lop the extra digits off.
Rounding – use the digit to the right of the one we want to
end with to determine whether to round up or not. If that
digit is 5 or higher, round up.
Truncating
  3.1415926...
Truncating – just lop the extra digits off.
If we want to use  with just 4 decimal places.
We’d just chop off the rest!
3.1415/926…
3.1415
Truncate ~ tree trunk ~ chop!
Rounding
  3.1415926...
If we want to round  to 4 decimal places.
We’d look at the digit in the 5th place
9 is “5 or bigger” so the digit in the 4th spot goes up
3.14159
3.1416
Real numbers
The set of real numbers consists of two infinite,
non-overlapping sets.
Rational
numbers
Irrational
numbers
Every real number is either rational or irrational,
but it can’t be both!
Vocabulary
Natural numbers
Whole numbers
Integers
Rational numbers
Irrational numbers
Real numbers
Truncating
Rounding