Decimals, Percents, and Real Numbers
7.1 Terminating Decimals
Place Value and Decimal Expansion
Decimal notation extends base-10 place value to negative powers of 10. Recall that $10^{-1}$ means $\frac{1}{10}$, $10^{-2}$ means $\frac{1}{10^2}$, and so on. So we can represent something like $4\frac{7}{10}$ as $$4\frac{7}{10}=4+7\left(\frac{1}{10}\right) = 4 \left(10^0\right) + 7\left(10^{-1}\right)$$
That idea is a lot less mysterious once you realize the decimal point is just a marker that lets the same place-value pattern continue to the right. We are not inventing a new system; we are extending the one we already know.
Simon Stevin helped popularize decimal notation centuries ago, and that change made arithmetic with measurements and money much easier to write down. The digits still do the same job they always did.
Definition: Decimal Expansion by Powers of 10
A decimal like 1452.789 can be written as
$$1\left(10^3\right)+4\left(10^2\right)+5\left(10^1\right)+2\left(10^0\right)+7\left(10^{-1}\right)+8\left(10^{-2}\right)+9\left(10^{-3}\right).$$
Example: Expanded form
Write 123.456 as an expanded sum of powers of rational numbers.
Show Solution
Break by place value:
This is the same structure as whole numbers, just continued to the right of the decimal point.
The only difference is the scale attached to each digit. Tenths, hundredths, and thousandths are just negative powers of 10.
MyOpenMath: Write Expanded Form
Fraction to Decimal Conversion
The first type of decimal numbers we will look at are ones that terminate. That just means that they don't go on forever. In the terminating decimals, there are two cases.
- Denominator is already a power of 10.
- Denominator can be scaled to a power of 10 using equivalent fractions.
Case 1: Suppose we want to convert $\frac{133}{10}$ into its decimal form. Luckily for us, the denominator is already a power of $10$. We just need to split it up.
Example: Convert to decimal without calculator
Convert $\frac{794}{100}$ to a decimal.
Show Solution
MyOpenMath: Fraction to decimal conversion
Case 2: Suppose we want to convert $\frac{314}{25}$ into its decimal form. Unluckily for us, the denominator is no already a power of $10$. But, we can make it so using equivalent fractions.
We see the denominator has two $5$'s, we just need to give it two $2$'s so that they can become $10$.
and then the problem becomes exactly like case 1.
$$\frac{1256}{100}=12.56$$
Example: Convert to decimal without calculator
Convert $\frac{21805}{200}$ to a decimal.
Show Solution
That $200$ isn't a nice power of $10$. Factoring it we see $$200=2^3\cdot 5^2$$
We have three $2$'s but only two $5$'s. That number is going to need one more $5$ to turn everything into a nice round power of $10$.
MyOpenMath: Fraction to decimal conversion
When Decimals Terminate
Some decimals, as you may know do not terminate like the ones above. To begin, lets talk about another way we can find a decimal expansion. Consider the rational number $\frac{50}{4}$. If we try to compute this using long division
We end up with a remiander of 2. But, we can extend our division now by adding a decimal point and bringing down a $0$ to continue.
And the division finishes with $\frac{50}{4}=12.5$. On the other hand, if we try to do something like $\frac{1}{3}$:
As we can see, after a few repetitions it becomes clear that this process will never end. So $\frac{1}{3}=0.33333\ldots$ is non-terminating. There is no way to write $3$ as a power of $10$ by just multiplying it by a constant.
Theorem: Terminating Decimal Criterion
A reduced fraction $\frac{a}{b}$ has a terminating decimal expansion iff the prime factorization of $b$ contains no primes other than 2 or 5.
This criterion is a fast way to predict decimal behavior before doing any long division. It is one of the many cases in arithmetic where factoring tells you everything you need to know. We just have to be careful with that reduced part.
Example: Terminating or not?
Decide whether each terminates: $\frac78$, $\frac{11}{250}$, $\frac{21}{28}$, $\frac{37}{121}$.
Show Solution
Reduce and inspect denominator factors.
- $\frac78$: denominator $2^3$, only has 2's therefore terminates.
- $\frac{11}{250}$: denominator $2\left(5^3\right)$, had 2's and 5's, terminates.
- $\frac{21}{28}=\frac34$: at first glance that denominator has a 7, but the reduced denominator is $2^2$, terminates.
- $\frac{37}{30}$: denominator $2\cdot 3\cdot 5$, includes a factor other than a 2 or 5, so non-terminating repeating.
The last one looks almost harmless until the factor 3 shows up. Once that happens, termination is off the table.
MyOpenMath: Terminating test by factors
Ordering Terminating Decimals
To compare terminating decimals, align digits and scan left to right until the first mismatch. Because again, in our positional numbering system: the position is the most important part in dictating how large it is.
If one decimal has fewer digits, give it trailing zeros so the place values line up. That makes the comparison feel much more like comparing whole numbers.
Example: Order from least to greatest
Order: 0.25, 0.255, 0.254, 0.24, 0.025, 0.0025.
Show Solution
Pad with trailing zeros to compare cleanly:
- 0.0025
- 0.0250
- 0.2400
- 0.2500
- 0.2540
- 0.2550
So least-to-greatest is
$$0.0025<0.025<0.24<0.25<0.254<0.255.$$
The zeros do not change the values, but they help us compare the numbers in a clean, place-value way.