Looking Back

(1) We have learnt that for addition and subtraction of fractions, the fractions should be fractions.

(2) We have also learnt how to multiply fractions: Product of numeratorsProduct of denominators

(3) "of" can be used to represent . For example, 13 of 6 = 13 × 6 =

(4) The product of two proper fractions is than each of the fractions that are multiplied. The product of a proper and improper fraction is than the improper fraction and than the proper fraction. The product of two improper fractions is greaterlower than each of the fractions.

(5) A of a fraction is obtained by inverting the numerator and denominator.

(6) We have seen how to divide two fractions:

(i) While dividing a whole number with a fraction, we multiply the whole number with the of that fraction.

(ii) While dividing a fraction by a whole number, we multiply the fraction with the of the whole number.

(iii) While dividing one fraction by another fraction, we multiply the first fraction with the reciprocal of the second. So 35 ÷ 47 = 35 × =

(7) We also learnt how to multiply two decimal numbers. While multiplying two decimal numbers, we first multiply them as whole numbers. We then count the total number of digits to the right of the decimal point in both the decimal numbers being multiplied. Lastly, we put the decimal point in the product by counting the digits from its rightmost place.

(8) To multiply a decimal number by 10, 100, 1000 ... etc., we move the decimal point in the number to the by as many places as there are zeros in the numbers 10, 100, 1000 ...

(9) We have learnt how to divide decimal numbers:

(i) To divide a decimal number by a whole number, we first divide them as whole numbers. We then place the decimal point in the quotient as is in the decimal number. Note that here we are considering only those divisions in which the remainder is .

(ii) To divide a decimal number by 10, 100, 1000 or any other multiple of 10, we shift the decimal point in the decimal number to the by as many places as there are in 10, 100, 1000 etc.

(iii) While dividing two decimal numbers, first shift the decimal point to the by equal number of places in both, to convert the divisor to a number.

(10) Rational numbers are a bigger collection of numbers, which includes all integers, all positive fractional numbers and all negative fractional numbers. In all these we have a ratio of two integers, thus pq represents a number.

In this: (i) p, q are and (ii) q ≠

The set of rational numbers is denoted by .