Rational Numbers MathBitsNotebook.com

Rational numbers are numbers that look like fractions (ratios), or can be written as a fraction, with the numerator and denominator being integers (denominator not equal to zero).

 In the diagram at the right, you can see that the Real Numbers have two large subsets called "Rational Numbers" and "Irrational Numbers". Let's take a closer look at the subset of Rational Numbers.
 A rational number is a number that can be expressed as a fraction (ratio) in the form where p and q are integers and q is not zero.

Examples:

A rational number can be expressed as a ratio (fraction)
with integers in both the top and the bottom of the fraction.

When a rational number fraction is divided to form a decimal value,
it becomes a terminating or repeating decimal.

 Some rational fractions may produce a large number of digits in their repeating patterns, which may exceed the size of the viewing screen on a calculator. The fraction 53/83 has a calculator display of 0.6385542169, which shows no repeating pattern, when in reality the pattern will repeat after 41 digits.

Signs and Rational Values (Fractions):
If p and q are integers (with q not zero), then .
Notice that the negative sign can be "out front", "top only", or "bottom only" and all three fractions represent the same value.
Putting a negative sign BOTH "top" and "bottom" will yield a different result.

Determining if a fraction is a terminating decimal:
• Fractions whose denominators are powers of 10 are have terminating decimal expansions (finite decimal expansions).
Consider:

• Fractions with
denominators that can be written as the product of 2's and/or 5's are equivalent to denominators that are powers of 10, thus making them terminating decimal expansions. Be sure to simplify the fraction, if needed, before applying this process.
Consider:

• Fractions which do NOT possess these properties will NOT be terminating decimals. Be sure to simplify the fraction before examining.
Consider:
The denominator of this fraction cannot be expressed as the product of 2's and/or 5's, so this fraction is not a terminating decimal. The decimal expansion is infinite = 0.0357142857...

To convert a repeating decimal to a fraction:

To show that the rational numbers are "dense":
(The term "dense" means that between any two rational numbers there is another rational number.)

Properties of Rational Numbers:
Since rational numbers are a subset of the real numbers, they possess all of the properties assigned to the real number system.

Closure of the Rational Numbers:
Under Addition (Subtraction): By definition, a rational number can be expressed as a fraction with integer values in the numerator and denominator (denominator not zero). So, adding two rationals is the same as adding two such fractions, which will result in another fraction of this same form since integers are closed under addition and multiplication. Thus, adding two rational numbers produces another rational number. Rationals are closed under addition (subtraction).

Under Multiplication: Again, by definition, a rational number can be expressed as a fraction with integer values in the numerator and denominator (denominator not zero). So, multiplying two rationals is the same as multiplying two such fractions, which will result in another fraction of this same form since integers are closed under multiplication. Thus, multiplying two rational numbers produces another rational number. Rationals are closed under multiplication.

Under Division:
The rationals are not closed under division because of the possibility of division by zero.
Zero is a rational number and division by zero is undefined.
It is true that the rationals are closed under division as long as the division is not by 0.

Number Line:
A number line is a straight line diagram on which every point corresponds to a real number.
Since rational numbers are real numbers, they have a specific location on a number line.

 NOTE: In mathematics, the word fraction is also used to describe mathematical expressions that are not rational numbers (where the numerator and denominator are not integers). For example, there are expressions that contain radicals such as , and expressions such as that are referred to as fractions. There are also algebraic fractions such as where the values of a and b are not known (assuming b ≠0).

Irrational Numbers

 As you can see in the diagram at the right, not ALL numbers fall into the "Rational Numbers" set. Numbers that are NOT rational numbers, are called Irrational Numbers. Irrational means not rational.
One of the most famous irrational numbers is π
which cannot be expressed as a fraction with integers in the top and bottom.
Remember that is an approximation of π, and does NOT EQUAL π .

Certain radical values are also irrational numbers.
For example, cannot be written as a simple fraction
with integers in the numerator and the denominator.

As a decimal, = 1.4.4213562373095048801688624 ...
which is a non-ending, non-repeating decimal, making irrational.