Terminating Decimal

What Is a Terminating Decimal?

A terminating decimal is a decimal number that has a finite number of digits after the decimal point. In other words, it comes to a complete stop. The word "terminate" means "to end," and that's exactly what these decimals do. They don't go on forever.

You see terminating decimals all the time in real life, especially with money. For example, $\$1.50$ is a terminating decimal because it ends after the '0'. Here are a few more simple examples:

• $0.5$
• $0.25$
• $0.8$
• $3.125$

Each of these numbers has a final digit and doesn't continue. This is different from a non-terminating decimal, which goes on forever. A common example of a non-terminating decimal comes from the fraction $\frac{1}{3}$. If you type $1÷3$ into a calculator, you'll see $0.333333...$. The three dots, called an ellipsis, show that the 3s repeat infinitely. This is also called a repeating decimal. For this lesson, our focus is on the ones that neatly end.

How Do Fractions and Decimals Connect?

Every terminating decimal can be written as a fraction. This connection is the key to understanding them. Decimals are really just a special kind of fraction—one where the denominator is a power of ten, like $10$, $100$, $1000$, and so on.

Think about decimal place value:

• The first digit after the decimal point is the tenths place. So, $0.6$ is the same as $\frac{6}{10}$.
• The second digit is the hundredths place. So, $0.75$ is the same as $\frac{75}{100}$.
• The third digit is the thousandths place. So, $0.125$ is the same as $\frac{125}{1000}$.

This table shows the relationship clearly:

Since every terminating decimal can be written with a denominator that is a power of ten, this gives us a huge clue about which fractions will create terminating decimals when we convert them.

How Can You Spot a Terminating Decimal from a Fraction?

Is there a way to know if a fraction will become a terminating decimal without actually doing the division? Yes, there is! It's a simple, three-step rule that involves looking at the denominator of the fraction.

Let's break down how to use this rule:

1. Simplify the Fraction: First, reduce the fraction to its simplest form. This is a very important step that can't be skipped.
2. Find the Prime Factorization of the Denominator: Break down the new denominator into its prime factors. A prime factor is a prime number that divides the original number exactly. The first few prime numbers are $2,3,5,7,11,...$.
3. Check the Factors: Look at the list of prime factors. If the only numbers in your list are $2$s and $5$s (or just one of them), then the fraction will convert to a terminating decimal. If there are any other prime factors, like $3$, $7$, or $11$, it will be a non-terminating, repeating decimal.

Why Does the '2s and 5s' Rule Work?

The secret behind the rule lies in our number system, which is base-ten. As we saw earlier, terminating decimals are just fractions with denominators like $10$, $100$, $1000$, etc. Let's look at the prime factors of these special denominators:

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

Notice a pattern? Every power of ten is made up of an equal number of $2$s and $5$s in its prime factorization.

So, if a fraction's denominator (in simplest form) has only $2$s and $5$s as its prime factors, we can always multiply the numerator and denominator by the right combination of extra $2$s or $5$s to turn the denominator into a power of ten. And once the denominator is a power of ten, writing it as a decimal is easy!

Let's look at $\frac{3}{8}$. The denominator is $8$. The prime factorization of $8$ is $2\times 2\times 2$, or $2^3$. It only contains $2$s, so it will terminate. To turn the denominator into a power of ten, we need to have an equal number of $2$s and $5$s. We have three $2$s, so we need three $5$s. We can get them by multiplying the fraction by $\frac{5^3}{5^3}$ (which is just $1$).

$$\frac{3}{8}=\frac{3}{2^3}\times \frac{5^3}{5^3}=\frac{3\times 125}{2^3\times 5^3}=\frac{375}{(2\times 5{)}^3}=\frac{375}{{10}^3}=\frac{375}{1000}=0.375$$

This works because the original denominator was only made of $2$s. If the denominator had a prime factor of, say, $3$ (like in $\frac{1}{12}$), there's no whole number we could multiply by to turn that $3$ into a power of ten. That's why the '2s and 5s' rule is the ultimate test.

How Do You Convert Fractions to Terminating Decimals?

Once you know a fraction will terminate, there are two main methods to find its decimal form.

Method 1: Long Division

This method always works. You simply divide the numerator by the denominator. Since we know the decimal will terminate, the division process will eventually end with a remainder of $0$.

Method 2: Creating an Equivalent Fraction

This is the method we explored in the previous section. It's often faster if the denominator is easy to work with.

What Are Some Common Mistakes to Avoid?

Working with terminating decimals is straightforward once you know the rules, but a few common slip-ups can happen. Here’s what to watch out for:

• Forgetting to Simplify the Fraction First: This is the most common mistake when using the prime factor rule. For example, consider the fraction $\frac{6}{30}$. The denominator is $30$, and its prime factors are $2,3,$ and $5$. Seeing the $3$ might make you think it's a non-terminating decimal. However, you must simplify first! $\frac{6}{30}$ simplifies to $\frac{1}{5}$. The denominator is $5$, which has only $5$ as a prime factor, so it does terminate (it's $0.2$). Always simplify before you check the denominator.
• Confusing Factors with Prime Factors: When you break down the denominator, make sure you go all the way to its prime factors. For the denominator $20$, you might list the factors as $4$ and $5$. If you stop there, you might miss that $4$ is not prime. You must break $4$ down further into $2\times 2$. The prime factors are $2,2,$ and $5$.
• Decimal Point Placement Errors: When converting a fraction like $\frac{4}{1000}$ to a decimal, remember that the number of zeros in the denominator tells you the number of decimal places. $1000$ has three zeros, so you need three places after the decimal. The answer is $0.004$, not $0.4$ or $0.04$.
• Errors in Long Division: Long division requires careful work. It's easy to make a small subtraction mistake or forget to bring down a zero. Double-check your calculations as you go.

Quick Summary and Reference

Here are the most important points to remember about terminating decimals.

Key Concepts

• A terminating decimal is a decimal that ends. It has a finite number of digits.
• Every terminating decimal can be written as a fraction with a denominator that is a power of $10$ (like $10,100,1000$).

How to Tell if a Fraction Terminates

1. Simplify the fraction to its lowest terms.
2. Find the prime factors of the new denominator.
3. Check the factors: If they are only $2$s and/or $5$s, the decimal will terminate.

How to Convert a Fraction to a Decimal

• Method 1 (Long Division): Divide the numerator by the denominator until the remainder is $0$.
• Method 2 (Equivalent Fractions): Multiply the numerator and denominator by the necessary factors (of $2$s and $5$s) to make the denominator a power of $10$.