Rational Numbers & Decimal Representations

Bymypaperr

Lesson 2: Terminating and Recurring Decimals

1. Lesson Overview

When a rational number \(\frac{p}{q}\) is converted into a decimal by dividing \(p\) by \(q\), it always results in either a terminating decimal or a recurring decimal. In this lesson, you will learn how to identify both types and how to convert recurring decimals back into fractions.

2. Definitions and Rules

A. Terminating Decimals

Definition: A decimal number that ends (has a finite number of digits after the decimal point).

Rule for Identification: A fraction in its simplest form produces a terminating decimal if and only if the prime factors of its denominator are only 2s, only 5s, or a combination of 2s and 5s.

Examples:

\[\frac{1}{4} = 0.25 \quad (\text{Denominator } 4 = 2 \times 2)\] \[\frac{3}{10} = 0.3 \quad (\text{Denominator } 10 = 2 \times 5)\]

B. Recurring Decimals (Non-Terminating Repeating Decimals)

Definition: A decimal number in which a single digit or a group of digits repeats infinitely.

Notation: A dot is placed above the repeating digit. If a block of digits repeats, dots are placed above the first and last digits of the repeating block.

Examples:

\[\frac{1}{3} = 0.3333\dots = 0.\dot{3}\] \[\frac{5}{6} = 0.8333\dots = 0.8\dot{3}\] \[\frac{2}{7} = 0.285714285714\dots = 0.\dot{2}8571\dot{4}\]

3. Worked Examples (Step-by-Step)

Example 1

Without performing long division, determine whether each fraction gives a terminating decimal or a recurring decimal:

(a) \(\frac{7}{20}\)    (b) \(\frac{5}{12}\)

Solution:

(a) Given fraction: \(\frac{7}{20}\)

Step 1: Check if the fraction is in its simplest form.
\(\frac{7}{20}\) cannot be simplified further.
Step 2: Find the prime factors of the denominator (\(20\)).
\[20 = 2 \times 2 \times 5 = 2^2 \times 5\]
Step 3: State the conclusion.
Since the prime factors of the denominator contain only \(2\) and \(5\), \(\frac{7}{20}\) is a terminating decimal.

(b) Given fraction: \(\frac{5}{12}\)

Step 1: Check if the fraction is in its simplest form.
\(\frac{5}{12}\) cannot be simplified further.
Step 2: Find the prime factors of the denominator (\(12\)).
\[12 = 2 \times 2 \times 3 = 2^2 \times 3\]
Step 3: State the conclusion.
Since the denominator contains a prime factor other than \(2\) or \(5\) (which is \(3\)), \(\frac{5}{12}\) is a recurring decimal.

Example 2 (O/L Exam Standard)

Express \(0.\dot{7}\) as a fraction in its simplest form.

Solution:

Step 1: Let the recurring decimal equal \(x\).
\[x = 0.7777\dots \quad \text{--- (Equation 1)}\]
Step 2: Since only one digit repeats, multiply both sides by \(10\).
\[10x = 7.7777\dots \quad \text{--- (Equation 2)}\]
Step 3: Subtract Equation 1 from Equation 2.
\[10x - x = 7.7777\dots - 0.7777\dots\] \[9x = 7\]
Step 4: Divide both sides by \(9\).
\[x = \frac{7}{9}\]

Example 3 (O/L Exam Standard)

Express \(0.\dot{3}\dot{6}\) as a fraction in its simplest form.

Solution:

Step 1: Let the recurring decimal equal \(x\).
\[x = 0.363636\dots \quad \text{--- (Equation 1)}\]
Step 2: Since two digits repeat, multiply both sides by \(100\).
\[100x = 36.363636\dots \quad \text{--- (Equation 2)}\]
Step 3: Subtract Equation 1 from Equation 2.
\[100x - x = 36.363636\dots - 0.363636\dots\] \[99x = 36\]
Step 4: Solve for \(x\) and simplify the fraction by dividing numerator and denominator by \(9\).
\[x = \frac{36}{99}\] \[x = \frac{4}{11}\]

Example 4

Express \(1.2\dot{4}\) as a mixed number in its simplest form.

Solution:

Step 1: Separate the whole number and the decimal part.
\[1.2\dot{4} = 1 + 0.24444\dots\]
Step 2: Let \(x = 0.24444\dots\)
\[x = 0.24444\dots\]
Step 3: Multiply by \(10\) so the repeating digit starts immediately after the decimal point.
\[10x = 2.4444\dots \quad \text{--- (Equation 1)}\]
Step 4: Multiply Equation 1 by \(10\) (so \(100x\) in total).
\[100x = 24.4444\dots \quad \text{--- (Equation 2)}\]
Step 5: Subtract Equation 1 from Equation 2.
\[100x - 10x = 24.4444\dots - 2.4444\dots\] \[90x = 22\]
Step 6: Simplify \(x\).
\[x = \frac{22}{90} = \frac{11}{45}\]
Step 7: Add back the whole number \(1\).
\[1.2\dot{4} = 1\frac{11}{45}\]

4. Practice Exercises for Students

  1. Express the following fractions as decimals using dot notation where necessary:
    1. \(\frac{5}{8}\)
    2. \(\frac{4}{9}\)
    3. \(\frac{7}{11}\)
  2. Convert the following recurring decimals into fractions in their simplest form:
    1. \(0.\dot{5}\)
    2. \(0.\dot{1}\dot{8}\)
    3. \(2.\dot{3}\)
Show Answers

1. (a) \(0.625\)    (b) \(0.\dot{4}\)    (c) \(0.\dot{6}\dot{3}\)

2. (a) \(\frac{5}{9}\)    (b) \(\frac{2}{11}\)    (c) \(2\frac{1}{3}\) (or \(\frac{7}{3}\))

Similar Posts

Leave a Reply

Your email address will not be published. Required fields are marked *