Terminating and Repeating Decimals

Revisiting Decimal Expansions

Let's begin by recalling the different types of decimal expansions. As we've discussed, any real number can be written as a decimal. These decimals fall into distinct categories that correspond to the type of number they represent.

As you can see, rational numbers are associated with decimal expansions that are predictable: they either end (terminate) or they repeat a pattern forever. Irrational numbers are the ones whose decimal expansions go on forever without any repeating pattern.

The most direct way to see if a fraction gives a terminating or repeating decimal is to simply perform the long division.

In the case of 20/8, the process ends when we reach a remainder of 0, giving a terminating decimal (2.5). For 10/3, the remainder is never 0; we keep getting a remainder of 1, leading to a non-terminating, repeating decimal (3.333...).

While long division always works, it's not very efficient. For a fraction like 13/28, you might have to divide for a long time to determine the pattern. As an engineer, you're familiar with the value of finding an analytical shortcut to avoid brute-force computation. Let's find one.

The Key Insight: Working Backwards from Terminating Decimals

To find the rule, let's start with the answer and work backward. Consider a few simple terminating decimals and convert them into fractions:

• $0.7=\frac{7}{10}$
• $0.45=\frac{45}{100}$
• $0.625=\frac{625}{1000}$

Notice the pattern in the denominators: they are all powers of 10 (10¹, 10², 10³). This makes sense; our decimal system is base-10.

Now, let's apply a tool from our earlier lessons: prime factorization. What are the prime factors of 10?

• $10=2\times 5$
• $100=10^2=(2\times 5{)}^2=2^2\times 5^2$
• $1000=10^3=(2\times 5{)}^3=2^3\times 5^3$

This reveals the fundamental secret: any terminating decimal, when written as a fraction in its simplest form, will have a denominator whose prime factors are only 2s and/or 5s.

Why? Because to turn a fraction into a decimal, we are essentially trying to make its denominator a power of 10. If the denominator only contains factors of 2 and 5, we can always multiply the numerator and denominator by the missing factors of 2 or 5 to achieve this.

For example, consider $\frac{3}{8}$:

1. The denominator is $8=2^3$.
2. To make this a power of 10, we need three 5s to match the three 2s.
3. We multiply the top and bottom by $5^3$:
   $\frac{3}{8}=\frac{3}{2^3}=\frac{3\times 5^3}{2^3\times 5^3}=\frac{3\times 125}{(2\times 5{)}^3}=\frac{375}{10^3}=\frac{375}{1000}=0.375$
   The decimal terminates!

What if the denominator has a prime factor other than 2 or 5, like in $\frac{1}{3}$? We can never multiply 3 by any integer to turn it into a power of 10. This inability to clear out other prime factors is what forces the decimal to repeat.

The following video provides an excellent and thorough explanation of this core concept.

Formalizing the Rules

Our observations can be summarized by three key theorems. These formalize the shortcut we've discovered.

To summarize the theorems: For a rational number $x=p/q$ that is in its simplest form:

1. Terminating: If the prime factorization of the denominator $q$ is of the form $2^n\cdot 5^m$ (where $n,m$ are non-negative integers), then $x$ has a terminating decimal expansion. This means the only prime factors allowed in the denominator are 2 and 5.
2. Non-Terminating Repeating: If the prime factorization of $q$ contains any prime factor other than 2 or 5 (such as 3, 7, 11, etc.), then $x$ has a non-terminating, repeating decimal expansion.

The Practical Algorithm

Now we can define a clear, step-by-step process to solve this problem for any rational number.

Step 1: Simplify the Fraction
This is the most important first step and a common place for errors. Always reduce the fraction $p/q$ to its simplest form by canceling any common factors.
Example: For 6/15, you must first simplify it to 2/5.

Step 2: Prime Factorize the Denominator
Take the denominator of the simplified fraction and find its prime factors.

Step 3: Analyze the Prime Factors

• If the only prime factors you find are 2s, only 5s, or a combination of both, the decimal is terminating.
• If you find any other prime factor (3, 7, 11, 13, etc.), the decimal is non-terminating and repeating.

Let's watch a couple of videos that demonstrate this algorithm with clear examples.

Check Your Understanding

Let's apply the algorithm to a few examples.

1. Is the decimal expansion of $\frac{22}{7}$ terminating or repeating?

   • Step 1: Is 22/7 in simplest form? Yes.
   • Step 2: What is the denominator? 7. Its prime factor is just 7.
   • Step 3: The denominator has a prime factor (7) that is not 2 or 5.
   • Conclusion: Non-terminating and repeating.

2. Is the decimal expansion of $\frac{21}{150}$ terminating or repeating?

   • Step 1: Simplify the fraction. Both are divisible by 3. $\frac{21}{150}=\frac{7}{50}$.
   • Step 2: Prime factorize the new denominator: $50=2\times 25=2^1\times 5^2$.
   • Step 3: The prime factors are only 2 and 5.
   • Conclusion: Terminating. (This shows the importance of simplifying first!)

3. Is the decimal expansion of $\frac{13}{3125}$ terminating or repeating?

   • Step 1: Is it simplified? 13 is prime. Is 3125 divisible by 13? No. So, yes, it's simplified.
   • Step 2: Prime factorize the denominator: $3125=5\times 625=5\times 5^4=5^5$.
   • Step 3: The only prime factor is 5.
   • Conclusion: Terminating.

Conclusion

In this lesson, we've established a clear and efficient rule to determine the nature of a rational number's decimal expansion without performing long division. By connecting the type of decimal to the prime factors of the fraction's denominator, we've uncovered one of the elegant structures that underpins the number system.

Key Takeaways:

• A rational number $p/q$ has a terminating decimal expansion if and only if, in its simplest form, the denominator $q$ has only prime factors of 2 and/or 5.
• If the denominator $q$ of the simplified fraction has any prime factor other than 2 or 5, the decimal expansion will be non-terminating and repeating.
• The first and most critical step in this process is to simplify the fraction.

This concludes our foundational module on Real Numbers. We have journeyed from classifying numbers and finding HCF/LCM to proving irrationality and now, analyzing decimal expansions. In our next lesson, we will shift gears and begin our exploration of algebra with the module on Polynomials.