Rational and Irrational Numbers
Both rational and irrational numbers are real numbers.
This Venn Diagram shows the relationships between sets of numbers. Notice that rational and irrational numbers are contained in the large blue rectangle representing the set of Real Numbers.
- A rational number is a number that can be expressed as a fraction or ratio.
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:
Hint: When given a rational number in decimal form and asked to write it as a fraction, it is often helpful to “say” the decimal out loud using the place values to help form the fraction.
Examples: Write each rational number as a fraction:
Hint: When checking to see which fraction is larger, change the fractions to decimals by dividing and compare their decimal values.
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 π is the terminating decimal, 3.14, but it is not. Yes, certain math problems ask you to use π as 3.14, but that problem is rounding the value of to make your calculations easier. π is actually a non-ending decimal and is an irrational number.