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Decimal Fractions

The decimal system can include fractions included identified by decimal point.

A decimal fraction comprises the decimal point followed by a collection of digits. Its value is less than 1.

Note that the zero to the left of the decimal point is optional.

Every decimal fraction can be represented by a fraction, whose numerator is given by the digits to the right of the decimal point and whose denominator is given by 1 followed by zeros equal in number to the digits to the right of the decimal point.

Decimals and Decimal Places

A decimal is the sum of a whole number and a decimal fraction.

The decimal point separates the whole number and the decimal fraction.

Note that a decimal fraction is special kind of decimal whose whole number is zero.

Also, a whole number is a special kind of decimal whose decimal fraction is zero.

In the decimal system, the position of the decimal point determines the place value of the digits. Thus, each digit in a decimal has a place value.

In the number 839.401, for example, here are the values of the different digits.

Equivalent Decimals

A fraction with a denominator that is a multiple of 10 is equivalent to a decimal.

To convert such a fraction to a decimal, a decimal point is inserted in the numerator such that the number of decimal places equals the number of zeros in the denominator.

While converting a fraction to a decimal, zeros need to be inserted before the numerator in case the numerator does not have enough digits.

Note A decimal does not change in value when zeros are inserted to its right.

Division of Decimals

divide the numbers in the same way as whole numbers (ignoring the decimal point); and

insert a decimal point in the result such that the quotient has the same number of decimal places as the dividend.

When a decimal is divided by 10, 100, 1000, ..., the decimal point moves to the left by 1, 2, 3, ... places.

Multiplication of Decimals

multiply the numbers in the same way as whole numbers (ignoring the decimal point); and

insert a decimal point in the result such that the decimal places in the product equal the sum total of the decimal places in all the numbers multiplied.

When a decimal is multiplied by 10, 100, 1000, ..., the decimal point moves to the right by 1, 2, 3, ... places.

Addition and Subtraction of Decimals

write the decimals one below the other such that the decimal points are vertically aligned;

add or subtract the numbers in the same way as whole numbers (ignoring the decimal point); and

insert a decimal point in the result directly under the decimal points of the numbers added or subtracted.