Question

Use a calculator to find the decimal form of the rational number. If the number is a nonterminating decimal, write the repeating pattern.$$ \frac{2}{3} $$

Short answer

The decimal equivalent of \(\frac{2}{3}\) is 0.6, where 6 is the repeating digit.

Step-by-step solution

Assigning the Numbers

The given number is \(\frac{2}{3}\). Let's use a calculator to divide 2 by 3. This will give us the decimal equivalent of the fraction.

Division and Decimal Conversion

When we carry out the division, we get 0.66666667. Notice that the digit 6 repeats indefinitely after the decimal point making it a non-terminating decimal.

Identifying the Repeating Pattern

The repeating pattern here can be identified by looking at the digits after the decimal point. In this case, the number 6 is repeating. Therefore, we can write \(\frac{2}{3}\) as 0.6 with a bar over the 6 to signify that it repeats indefinitely.

Key concepts

Decimal Conversion

When you convert a fraction into a decimal, you're essentially trying to express it in another form. For instance, converting \(\frac{2}{3}\) into a decimal. Since fractions by nature are divisions waiting to happen, you perform the division: 2 divided by 3. The result of this division gives you the decimal equivalent of the fraction.

The method of Decimal Conversion primarily uses division. With a calculator, when you divide 2 by 3, you get approximately 0.66666667. The decimal we get here is an approximation. It helps provide a precise representation. Decimal Conversion is crucial for comparing fractions, doing calculations, or simply interpreting numbers in their alternate form.

Repeating Decimals

Repeating decimals occur when a particular digit or sequence of digits repeats infinitely. In the decimal representation of a fraction, if you notice that certain numbers continue to appear after every set of divisions, you've found a repeating sequence.

In the case of \(\frac{2}{3}\), the division yielded 0.66666667. The number "6" continues forever. To denote this repetition, mathematicians use a bar, referred to as a "vinculum", over the repeating digit. So, \(0.\overline{6}\) represents the repeating decimal of \(\frac{2}{3}\). This technique helps make long sequences of numbers manageable when writing or solving problems.

Nonterminating Decimals

Nonterminating decimals, as the name suggests, do not end. They differ from finite or terminating decimals that come to an exact stop after a few digits. When you look at fractions like \(\frac{2}{3}\), the division continues indefinitely without reaching a conclusion.

The result, 0.66666667, continues perpetually because it's a Nonterminating Decimal. These decimals can either be repeating (containing a specific sequence that continues) or non-repeating (with no discernible pattern). For rational numbers like \(\frac{2}{3}\), you're working with a repeating nonterminating decimal. Being aware of whether decimals terminate or not is important in mathematics to understand number properties and ensure precise arithmetic operations.