Question

change each rational number to a decimal by performing long division. $$ \frac{11}{13} $$

Short answer

The decimal form of \( \frac{11}{13} \) is 0.846153\bar{846153}.

Step-by-step solution

Set Up the Division Problem

To change the fraction \( \frac{11}{13} \) into a decimal, we'll perform the division \( 11 \div 13 \). Start by writing 11 as the dividend inside the division bracket and 13 as the divisor outside.

Understand the Division Setup

Since 11 is smaller than 13, it does not contain 13 even once, so our starting integer part of the quotient is 0, and we'll need to include decimals. Since the integer part doesn't work, add a decimal point to the quotient and a 0 next to 11, making it 110.

Divide 110 by 13

Determine how many times 13 fits into 110. Since 13 times 8 is 104 (the closest multiple of 13 without exceeding 110), place 8 in the quotient after the decimal. Subtract 104 from 110 to get a remainder of 6.

Bring Down Another 0

Take the remainder 6, bring down the next 0 to make it 60. Now divide 60 by 13.

Divide 60 by 13

Determine how many times 13 fits into 60. 13 fits 4 times into 60 because 13 times 4 is 52, which is less than 60. Add 4 to the quotient, making it 0.84. Subtract 52 from 60 to get a remainder of 8.

Bring Down Another 0 Again

Now, bring down another 0 to make it 80 and divide it by 13 again.

Divide 80 by 13

Determine how many times 13 fits into 80. 13 times 6 is 78, which is less than 80. Add 6 to the quotient, so it becomes 0.846. Subtract 78 from 80 to get a remainder of 2.

Continue and Observe Repeating Patterns

Continue the steps by bringing down a 0 (making it 20) and dividing by 13 again. You will notice a repeating pattern, 13 fits into 20 once, leaving a remainder that brings back previous numbers, indicating it repeats.

Write the Final Decimal

Since the numbers repeat after a certain point, we recognize it as repeating decimals. The fraction \( \frac{11}{13} \) as a decimal is approximately 0.846153 with a repeating pattern of 846153, represented as 0.846153\bar{846153}.

Key concepts

Rational Numbers

Rational numbers are a fundamental part of mathematics. They are numbers that can be expressed as the quotient of two integers, where the numerator and the denominator are integers and the denominator is not zero. For example, the number \( \frac{11}{13} \) is a rational number.
These numbers are significant because they encompass a wide range of numerical values, including both whole numbers and fractions. They allow us to perform precise calculations and express numbers that are not whole, such as those found in everyday measurements.
One key feature of rational numbers is that when expressed as decimals, they either terminate or repeat. This predictability makes them easier to manage compared to irrational numbers, whose decimals are infinite and non-repeating. Thus, knowing how to work with rational numbers and convert them into decimal forms is crucial for students.

Converting Fractions to Decimals

Converting fractions to decimals is a skill that can be quite handy. It involves dividing the numerator by the denominator to get a decimal.
For the fraction \( \frac{11}{13} \), this means performing the division \( 11 \div 13 \). Setting this up involves placing 11 inside the division bracket as the dividend, and 13 outside as the divisor.
The process of division will give you a decimal representation of the fraction. Sometimes this division results in a terminating decimal, which simply means it ends after a few digits. Other times, you get a repeating decimal, meaning a sequence of numbers continually repeats without end.
Understanding how to convert fractions to decimals is essential, especially in fields such as science and engineering, where precise numerical values are often necessary.

Repeating Decimals

Repeating decimals are fascinating mathematical phenomena! When you divide two numbers and the division doesn't end neatly, you may encounter repeating decimals.
For example, when you divide 11 by 13, you get 0.846153, and this cycle of numbers keeps repeating indefinitely. This particular decimal is referred to as a repeating decimal and is often expressed with a bar notation, like 0.846153\( \overline{846153} \), to indicate that the sequence repeats.
Repeating decimals occur when the remainder in your long division starts to become familiar, cycling in a predictable pattern. It's like a never-ending loop, showing the beautiful precision and structure in mathematics.
Understanding repeating decimals is important because they help you recognize and express patterns in numbers that aren't immediately obvious, enabling more efficient problem solving in advanced mathematical calculations.

Division Algorithm

The division algorithm is a method used to find the quotient and remainder when dividing two numbers. It's a systematic approach that breaks down complex division problems into manageable steps.
When converting the fraction \( \frac{11}{13} \) into a decimal, the division algorithm guides us through the long division process. It involves

• Determining how many times the divisor fits into the initial dividends
• Subtracting the result from the dividend
• Bringing down the next digit to form a new dividend

By repeatedly applying these steps, you can systematically solve the division and identify patterns such as repeating decimals.
This algorithm isn't just a technique for turning fractions into decimals; it's a foundational tool in mathematics that helps solve a variety of problems involving division, ensuring precision and clarity in calculations.