Converting Fractions to Recurring Decimals

To write the fraction \(\frac{1}{300}\) as a recurring decimal, we need to perform the division of 1 by 300. A recurring decimal is a decimal representation of a number whose digits are periodic (repeat at regular intervals).

Step-by-Step Division of 1 by 300

Let's perform long division:

We want to calculate \(1 \div 300\).

• Start with 1. Can 300 go into 1? No. Write down 0, add a decimal point, and add a zero to 1 to get 10.
• Can 300 go into 10? No. Write down 0 after the decimal point, and add another zero to 10 to get 100.
• Can 300 go into 100? No. Write down 0 after the previous 0, and add another zero to 100 to get 1000.
• Can 300 go into 1000? Yes. \(300 \times 3 = 900\). So, 300 goes into 1000 three times with a remainder. Write down 3.
• Calculate the remainder: \(1000 - 900 = 100\).
• Now we have a remainder of 100. Add another zero to the remainder to get 1000.
• Can 300 go into 1000? Yes, again three times (\(3 \times 300 = 900\)) with a remainder of 100. Write down 3.

We can see that the remainder is 100 again, and this process will repeat indefinitely. Each time we get 100, we add a zero to make it 1000, and 300 goes into 1000 three times with a remainder of 100. This means the digit '3' will repeat infinitely in the decimal representation.

Decimal Representation of \(\frac{1}{300}\)

Based on the long division, the decimal representation of \(\frac{1}{300}\) is 0.003333... .

Recurring Decimal Notation

To indicate that a digit or a sequence of digits repeats indefinitely, we use a bar over the repeating part. In this case, only the digit '3' repeats.

So, 0.003333... is written as \(0.00\bar{3}\).

Summary of the Conversion

The division of 1 by 300 yields a decimal where the digits '0', '0' appear after the decimal point, followed by an infinite sequence of the digit '3'. This is a recurring decimal.

The resulting recurring decimal is \(0.00\bar{3}\).

Therefore, \(\frac{1}{300}\) as a recurring decimal is \(0.00\bar{3}\).

Revision Table: Fractions and Decimals

Additional Information: Rational Numbers and Decimal Forms

Fractions like \(\frac{1}{300}\) are rational numbers. A rational number is any number that can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers, a numerator \(p\) and a non-zero denominator \(q\). An interesting property of rational numbers is that their decimal expansion either terminates or is recurring.

• If the prime factors of the denominator (in its simplest form) are only 2s and/or 5s, the decimal will terminate. For example, \(\frac{1}{4} = \frac{1}{2^2} = 0.25\) or \(\frac{3}{10} = \frac{3}{2 \times 5} = 0.3\).
• If the denominator has prime factors other than 2 or 5, the decimal will be recurring. For example, \(\frac{1}{3}\) (denominator 3) or \(\frac{1}{7}\) (denominator 7) or \(\frac{1}{6} = \frac{1}{2 \times 3}\) (denominator has factor 3).

In the case of \(\frac{1}{300}\), the denominator is 300. The prime factorization of 300 is \(300 = 3 \times 100 = 3 \times 10^2 = 3 \times (2 \times 5)^2 = 2^2 \times 3 \times 5^2\). Since the denominator contains the prime factor 3 (in addition to 2 and 5), the decimal representation must be recurring.

The division \(1 \div 300\) gives \(0.00\bar{3}\). The initial non-repeating zeros (0.00) occur because of the factors of 2 and 5 in the denominator, and the repeating digit (3) occurs because of the factor of 3.