Question

change each rational number to a decimal by performing long division. $$ \frac{11}{3} $$

Short answer

The decimal representation of \( \frac{11}{3} \) is \( 3.6\overline{6} \).

Step-by-step solution

Set Up the Long Division Problem

Divide the numerator (11) by the denominator (3). Set it up with 11 under the division bracket and 3 outside.

Determine How Many Times 3 Fits into 11

3 fits into 11 three times since 3 * 3 = 9, which is the largest multiple of 3 less than or equal to 11.

Subtract and Bring Down the Next Digit

Subtract the result (9) from 11, giving a remainder of 2. Because there are no more digits to bring down, we'll introduce a decimal point and add a zero making the remainder 20.

Divide 20 by 3

3 fits into 20 six times since 3 * 6 = 18. Write 6 after the decimal point and subtract 18 from 20, getting a remainder of 2 again.

Identify Repeating Sequence

As you repeat the process of bringing down zeros and dividing, you will continue to get a remainder of 2, indicating that the decimal repeats. Thus, 0.6 is the repeating sequence.

Key concepts

Rational Numbers

Rational numbers are numbers that can be written as fractions. Any number that can be expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \), is considered rational.
For example, the fraction \( \frac{11}{3} \) is a rational number because both 11 and 3 are integers, and 3 is not zero. Rational numbers can be found in all sorts of mathematical operations and are fundamental to understanding many mathematical concepts, including fractions, proportions, and parts of a whole.
One key property of rational numbers is that they can often be converted into different forms, such as decimals, using a method called long division. This conversion can sometimes result in decimals that have specific characteristics, such as repeating patterns.

Decimal Conversion

Decimal conversion is the process of transforming a fraction into its decimal equivalent. This is often done using long division, a methodical process that helps us break down the division of the numerator by the denominator.
In the example of \( \frac{11}{3} \), we use long division to determine how 11 divides by 3. Initially, 3 fits into 11 three times, leaving us with a remainder of 2. By placing a decimal point and adding zeros to the remainder, we can continue the division to find more decimal places.
During this process, each step follows the same pattern of division:

• Divide the current number
• Multiply and subtract to find the next remainder
• Add a zero if necessary and repeat

This is a powerful tool for converting fractions into decimals, and it's the basis for understanding many calculations in both mathematics and day-to-day applications.

Repeating Decimals

Repeating decimals are decimals that have digits that repeat infinitely after a certain point. They occur when the long division of a fraction results in a cycle of repeated remainders and quotients.
In the case of \( \frac{11}{3} \), the decimal expansion begins with 3.666... with the digit "6" repeating indefinitely. We represent repeating decimals using a bar notation over the digits that repeat, such as \( 3.\overline{6} \).
Understanding repeating decimals helps when working with non-terminating decimals. It's essential to recognize where the repeat begins and how to convey it efficiently, particularly while planning further calculations or presenting results.
Key points to recognize repeating decimals include:

• Look for the same remainder appearing repeatedly
• Count the digits in a cycle to identify the repeat length
• Apply bar notation to signify the repeating sequence briefly