A rational number is a number that can be expressed as a fraction or ratio
(rational). The
numerator and the denominator of the fraction are both integers.
When the fraction is divided out, it
becomes a terminating or repeating decimal.
(The repeating decimal portion may be one number
or a billion numbers.)
Rational numbers can be ordered on a number line.
Examples of
rational numbers
are :
6 or  | can also be written as | 6.0 |
| -2 or  | can also be written as | -2.0 |
 | can also be written as | .5 |
 | can also be written as | -1.25 |
 | can also be written as |  |
 | can also be written as | 0.666666666...  |
 | can also be written as | 0.38181818...  |
 | can also be written as | 0.62855421687... the decimals will repeat after 41 digits |
Be careful when using your calculator to determine if a decimal number is irrational. The calculator may not be displaying enough digits to show you the repeating decimals, as was seen in the last example above. | ||
Hint:
When asked to write a rational number (given in decimal
form) as a fraction, "say" the decimal using place value to help form the
fraction.
| 2 | . | 3 | 4 | 5 | 6 |
| o | a | t | h | t | ten- |
| n | n | e | u | h | t |
| e | d | n | n | o | h |
| s | t | d | u | o | |
| h | r | s | u | ||
| s | e | a | s | ||
| d | n | a | |||
| t | d | n | |||
| h | t | d | |||
| s | h | t | |||
| s | h | ||||
| s |
Examples: Write each rational
number as a fraction:
| 1. 0.3 |  |
| 2. 0.007 |  |
| 3. -5.9 |  |
Hint: When checking to see which
fraction is larger, change the fractions to decimals by dividing.
Examples: Which of the given
numbers is greater?
| 1. |  | .6666666667 > .25 | Using full |
| 2. |  | -2.333333333 > -3.666666667 | |
An irrational number cannot be expressed as a fraction.
Irrational numbers cannot be represented as terminating or repeating
decimals.
Irrational numbers are non-terminating, non-repeating
decimals.
Examples of irrational numbers are:
| ||||||
Note: Many students think that
is
the terminating decimal, 3.14, but it is not. Yes, certain math
problems ask you to use
as 3.14, but that problem is rounding the value of
to make your calculations easier.
is actually a non-ending decimal and is an irrational number.
Rational and irrational numbers are real
numbers.
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