Pre-Algebra Review Topic Rational and Irrational Numbers Topic Index | Algebra Index | Regents Exam Prep Center

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 :

 6   or can also be written as 6.0 -2  or can also be written as -2.0 can also be written as 0.5 can also be written as -1.25 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 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.

 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:

 Rational number in decimal form Rational number in fractional form 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 and compare their decimal values.

Examples:

 Which of the given numbers is greater? Using full calculator display to compare the numbers. 1. .6666666667  > .25 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:

 = 3.141592654….. = 1.414213562….. and 0.12122122212…

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.

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