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Decimal Expansion of Rational Numbers

Decimal Expansion of Rational Numbers

 

A rational number is a number that can be represented in the form p/q where p and q are integers and q is not equal to zero. For example, rational numbers are ¼, -1/5, 0.4, -0.9, etc. Decimal representation can be used to denote rational numbers.

Some decimal values will be recurring, which means repeating and never-ending, and some will be non-recurring, which means non-repeating and terminating. For example, 0.5637… is a recurring decimal value, whereas 0.6 is a non-recurring decimal value. Rational numbers can also be denoted as fractions.

Decimal expansion of rational numbers

Decimal expansion of rational numbers means we can represent decimal numbers in the form of decimals. The fractional value of rational numbers can be converted into decimals. We can use the long-division method to represent rational numbers into decimals. We will get a quotient while performing long-division procedures, which will be the required decimal value.

A decimal value that is repeating and non-ending is an irrational decimal value. Whereas the decimal value, which is terminating and ending, is known as a rational decimal value. For example, 0.23084444…. is a recurring and never-ending value. Therefore, it is an irrational number. Whereas the number 0.456 is rational as it is a terminating number.

Decimal expansion of ½ and 1/3

Let us take two numbers, ½ and 1/3. Firstly, we will see ½.

By using the long-division method, we can find the decimal value of ½. We need to write 2 as the divisor and 1 as the dividend. We know 1 is not divisible by 2. Therefore, we need to put a decimal and write down 0 after 1. Once we have placed a decimal value, then we can use 0s like this anytime. Therefore, the dividend now becomes 10. 10 is divisible 2. 10 is 5 times 2. Therefore, the quotient becomes 0.5, leaving no remainder at the end of the division. Thus, 0.5 is non-repeating and ends at 5.

Consequently, it is a rational number. Now coming to 1/3. The number has been represented in the p/q form. The first confirmation is done that is a rational number. Taking 3 as the divisor and 1 as the dividend, we need to divide 1 by 3. Placing the decimal in the quotient, we can write down 0 after 1. Therefore, the number 1 now becomes 10. We know, 9 is divisible by 3, and this is the largest number before 10 available for division. Therefore, placing 3 after the decimal, we will get 1 as the remainder.

As told, we can put 0s any number of times. Therefore, 1 again now becomes 10. Placing another 3 after 3, we will get another 1 as the remainder. Thus, we need to repeat the steps a few more times. We see the number is non-ending and repeating. But, these numbers are fixed and have the same number in a repetitive order. Therefore, 1/3 is also a rational number.

Points to Ponder

1. If we can express a number in the form p / (2n x 5m) where p belongs to real numbers and m, n are whole numbers; then the given rational number will have a terminating decimal value.
2. Terminating decimals means the decimal values that end after a certain number of decimal values.
3. Every non-terminating but repeating number will be a rational decimal value.

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