Rational Numbers

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We have defined rational numbers as those which can be expressed as fractions:

We now define decimal fractions.

The rational numbers can be expressed as terminating or recurring decimals.

Decimal fraction has denominator as 10 or a power of 10.

Examples of terminating decimal:

2.793 represents 2 units, seven-tenths, nine-hundredths and three-thousandths or using fraction notation:

(a)

(b)

Examples of recurring decimal:

(a)

(b)

Hence we state the classical definition of rational numbers:

Find a rational number between two rational numbers

To obtain the required number, add numerators and denominators as shown below:

Ans:

Find three rational numbers between

i.e.,

are three rational numbers between

We can obtain infinite number of rationals between two given rational numbers.

A recurring decimal can be expressed as a fraction

A recurring decimal is denoted by placing dots or a bar over recurring digits.

Express the following recurring decimals as equivalent fractions:

(a)

(b)

(c)

(a) Let x = 0.4444……

Multiply both sides by 10 ( one digit is recurring)

(b) Let x =

Multiply both sides by 100 ( two digits are recurring)

We can find equivalent fraction for recurring decimal

(c) Let x =