IRRATIONAL NUMBER

                                   TERMINATING AND REPEATING DECIMALS:

TERMINATING DECIMALS: To express a vulgar fraction p/q in the form of a decimal, we divide p by q. If after a finite number of steps, no remainder is left, then the decimal so obtained is called a termination decimal. For example,

(i)                 1/4 = 0.25,

(ii)               5/8 = 0.625 and

(iii)              2.3/5 = 2.6 are all termination decimals.

AN IMPORTANT OBSERVATION: A fraction p/q is a termination decimal only when prime factors of q are not any number other than 2 and 5.

For example, 1/2, 3/7, 7/20 and 13/25 are terminating decimals because their denominators have no prime factors other than 2 and 5.

REPEATING OR RECURRING DECIMALS: A decimals in which a digit or a set of digits repeats periodically is called a repeating or a recurring or a periodic or a circulating decimal.

(i)                 In repeating decimals if only one digit is repeated, then a (͘ ) or (ˉ) is placed above it.

For example: 2/3 = 0.666…… =0.͘6͘ or 0. ˉ6

(ii)               In a repeating decimal, if two or more digit are repeated then a dot is put on each of the first and the last repeating digits (or a bar vinculum) is drawn on the entire block of repeating digits.

SPECIAL CHARACTERISTICS OF RATION NUMBERS:

(i)                 Every rational number is expressible either as a terminating or as repeating decimal.

(ii)               Every terminating decimal is a rational number.

(iii)              Every repeating decimal is a rational number.

IRRATIONAL NUMBERS:

IRRATIONAL NUMBERS: A number which can neither be expressed as terminating decimal nor as a repeating decimal, is called an irrational number.

Thus, nonterminating, nonrepeating decimals are irrational number. Clearly, 001001000100001…is a nonterminating, nonrepeating decimal, and therefore, it is irrational.

The square root of each nonperfect square natural number is nonterminationg, nonrepeating decimal, and therefore, each one of them is irrational number. In other words, if Öm is a positive integer which is not a perfect square then Öm is an irrational number. Thus, Ö2, Ö3, Ö5….etc.,are all irrational numbers.

The cube root of each nonperfect cube natural number is an irrational number.

π is a number whose exact value is not equal to 22/7. In fact, π has a value which is a nonterminating, nonrepeating decimal. So, π is an irrational number, while, 22/7 is a rational number.