Problem 11 - Algebra 2 and Trigonometry

Question

In \(8-12,\) write each rational number as a repeating decimal. $$ \frac{2}{15} $$

Step-by-Step Solution

Answer

\(\frac{2}{15} = 0.1\overline{3}\)

Step 1: Set Up Long Division

To convert \(\frac{2}{15}\) into a decimal, we need to perform long division. Divide 2 by 15. Since 15 is larger than 2, start by considering 20 instead, by adding a decimal point and a zero, turning 2 into 2.0.

Step 2: Perform the Division

Now divide 20 by 15. 15 goes into 20 once, which gives us a quotient of 1. Write 1 as the first digit after the decimal point. Subtract 15 from 20 to get a remainder of 5.

Step 3: Bring Down a Zero

To continue the division, bring down another 0 from 2.0 to make the remainder 50.

Step 4: Continue Dividing

15 goes into 50 three times (15 * 3 = 45). Write 3 next to the 1 in the quotient. Subtract 45 from 50 to get a new remainder of 5.

Step 5: Identify the Repeat

Bring down another 0, making it 50 again. Notice that we have repeated the same division as before: 15 goes into 50 three times. This pattern will continue, producing the repeating decimal.

Key Concepts

Long Division

Long division is a method used to divide numbers and find quotients in a step-by-step manner. It's especially helpful for dividing when the divisor is larger than the dividend. In the case of dividing 2 by 15, as shown in the exercise, you may initially notice that 15 cannot directly divide into 2. Instead, you convert 2 into a number like 20 by appending a decimal point and adding a zero. This makes it easier to deal with larger numbers and helps continue the division process smoothly.

During long division, you focus on portions of the dividend, dealing with them sequentially:

• Set up the problem by placing 15 outside of the division bracket and 2 (or 2.0) inside.
• Proceed with division by removing multiples of 15 from the adjusted number until a remainder is achieved that's smaller than 15.
• If there is a remainder, you bring down zeros and keep dividing until you achieve a precise result or identify a repeating pattern.

Long division is like breaking a big problem into smaller parts, which makes it less overwhelming and allows for accuracy, especially when it comes to converting fractions into decimals.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, where the numerator and denominator are whole numbers, and the denominator is not zero. In the context of the exercise with \(\frac{2}{15}\), it's clear that both 2 and 15 are integers, confirming that this is indeed a rational number.

A unique characteristic of rational numbers is their ability to be represented as either terminating or repeating decimals when divided. Repeating decimals occur because of the finite number of possible remainders, eventually causing repetition during the division process. In the example of \(\frac{2}{15}\), performing long division reveals a repeating decimal pattern.

Understanding that rational numbers can be expressed in different forms further emphasizes their versatility:

• They can be whole numbers, fractions, or decimals.
• When converted to decimals, they may end at a certain point (terminate) or continue with a recurring cycle (repeat).

Hence, every rational number has a predictable decimal representation that falls into one of these categories.

Decimal Representation

Decimal representation refers to the expression of numbers in terms of digits and positions of the decimal point. This is crucial for understanding how fractions like \(\frac{2}{15}\) are represented as decimals.

Through long division, the fraction \(\frac{2}{15}\) can be converted into a decimal format. After establishing the division setup, the decimal representation unfolds as repeating cycles in our quotient, showcasing a repeating decimal pattern. In this context, "repeating decimals" are those that continue indefinitely with a sequence that repetitively arises during division.

Key aspects of decimal representation include:

• Identifying whether a fraction converts to a terminating or repeating decimal.
• Understanding that repeating decimals are denoted with a line or bar over the repeating digits. For example, dividing 2 by 15 results in a decimal representation where 0.1333... (with "3" repeating) simplifies to 0.1\overline{3}.
• Recognizing that the division process continues until a pattern is discovered, at which point the decimal representation is solved.

Grasping decimal representation provides clarity and precision in interpreting rational numbers in everyday calculations.