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Study Guide

A **decimal** is another way of representing a real number. Many numbers that can be written as fractions can also be written as decimals, and vice versa. Sometimes it makes more sense to work with a fraction, but other times a decimal is the way to go. Usually from 2 in the afternoon to 8 in the evening is optimal for using decimals. All right, we're pulling your leg. It doesn't actually depend on the time of day. Although decimals *are* quite nice at sunset.

The first types of decimals we're going to look at are just abbreviations for a fraction whose denominator is a power of 10, such as 10, 100 (10 × 10), 1000 (10 × 10 × 10), etc. These are some pretty important fractions we're abbreviating, and they deserve our attention. There's a reason we say something's "a perfect ten," and not a perfect eleven or twelve. Unless we're talking about donuts.

Our system of numbers is based entirely on the number 10. For example, when we write the number 12, we really mean 10 + 2. When we write 5673, we mean that we have 5 thousands, 6 hundreds, 7 tens, and 3 ones.

The ancient Romans had this figured out: XII is literally 10 (X) plus 2 (II), and MCX is 1000 (M) plus 100 (C) plus 10 (X). The point of decimal notation is just to continue this in the opposite direction—to show the amount of a number that's *less than* 1. We want to record how many , , etc. our numbers have. That way, if we ever lose them, we'll be able to identify and claim them like lost baggage at the airport.

For example, 0.3 is an abbreviation for , pronounced "three tenths." And , or "fifty-seven hundredths," is abbreviated by 0.57. We can also think of this as .

The number directly to the right of the decimal tells you how many tenths you have. The number one more to the right says how many hundredths you have. Continuing still further down the line, we would abbreviate by 0.671, read as "six hundred seventy-one thousandths" (but this is also 6 tenths plus 7 hundredths plus 1 thousandth). If you keep going, the millionths and billionths place will also get some love, but most of the time you won't have to go to such extremes. Unless you're really, really exacting when figuring out how much of a tip to leave at a restaurant.

What is in decimal form?

If we start with a fraction whose denominator is a power of 10, like , here's what we do to get the decimal abbreviation:

1. Count the number of zeros in the denominator. In this case, we've got 3 zeros in 1000.

2. Write down the numerator, followed by a dot: 691.

3. Move the dot to the left as many places as you counted zeros in the denominator: 0.691.

We end up with 0.691, our decimal for "six hundred ninety-one thousandths." That's some batting average there, sport.

But what happens if the numerator has fewer digits than the denominator has zeros?

What is ^{7}/_{1000} in decimal form?

There are 3 zeros in the denominator. First we write down the numerator, followed by a dot: 7.

Hmm...it looks like we can only move the dot one place to the left. However, appearances can be deceiving. For each additional place we need the dot to move over, we can stick a zero in that place.

We end up with .007, meaning "seven thousandths." Or, "James Bond."

Getting the hang of it? Fortunately, you won't have to speak these out loud very often. Until they add an oral report section of the SAT, you should be in the clear on that one. You'll definitely need to know how to write them out, though, and to be able to move back and forth between decimals and fractions with relative ease.

Okay, time for a little low-stress vocabulary building. These are pretty easy to learn, and even easier to remember.

The dot within a decimal is called the **decimal point**. Or the "lost period" in some circles. But you only need to remember it as the decimal point.

When learning addition way back when, we learned that the places to the left of the decimal point are named by powers of ten:

The places to the *right* of the decimal point, called the **decimal places**, are named by fractions whose denominators are powers of ten:

We get the number of decimal places by counting the number of digits to the right of the decimal point. For example, the number of decimal places in 0.571 is three. The number of decimal places in 0.238756102356403297056123456032 is 30. Pray you never see that one again.

Before we move on to decimal arithmetic, a quick word on calculators. You may wonder why we should bother doing this stuff by hand when we could let our souped-up calculators do all the work. Here are three good reasons:

1. Some tests don't allow calculators, so you'd better learn how to do arithmetic without them.

2. Doesn't it make you feel smart when you can do the problem in your head, *and* get the right answer, before your friend is even done typing it into their calculator? It's always nice to be able to give a pal a friendly "in your face!" in situations like that.

3. What if you're ever stranded on a desert island with only pen and paper and have to perform a calculation in order to survive? You probably think this unlikely, but who knows? As it turns out, it may help to figure out exactly what fraction of your leg the shark swam off with.

### Converting Fractions into Decimals

To change a fraction into a decimal, divide the numerator by the denominator using long division. That's it? That's it.

### Converting Decimals into Fractions

Since a decimal is an abbreviation for a fraction whose denominator is a power of 10, we already know how to do this. We don't even have to dig deep to recall the process. We can dig pretty shallow.

Start with a decimal. For example: 0.456. The number to the right of the decimal point is the numerator of the fraction:

The denominator is the power of 10 with as many zeros as there were decimal places:

Finally, simplify the fraction to tidy things up:

Now, you have a choice when it comes to expressing yourself. You can either say you played video games for 0.456 of the day, or that you played video games for

^{57}/_{125}of the day. Either way...how about you go outside and toss a football around for a bit? It's such a nice day.### Comparing Decimals

When comparing decimal numbers to see which is bigger, first we look at the numbers before the decimal point. If one is greater than the other, we're done. Stick a fork in us. For example, 4.6248 is definitely bigger than 3.9998, because that 4 in front of the decimal point is bigger than 3.If the numbers out front are identical (like with 0.4 and 0.562), we compare the tenths place. Since 4 is smaller than 5, we'd say that 0.4 is smaller than 0.562, or 0.4 < 0.562.

If the whole number

*and*the tenths place digits are identical, just keep going till you can compare two digits in the same spot. For example, say we're comparing 0.2337 and 0.2318. Both decimals start with a 0, and both have a 2 in the tenths place and a 3 in the hundredths place. But when we get to the thousandths place, we have a 3 in 0.2337 and a 1 in 0.2318. Since 3 is bigger than 1, we can say 0.2337 > 0.2318.BTW, adding extra zeros to the end of a decimal doesn't change its value at all: 08 = 080 = 0800 = etc. It just makes it longer and more annoyed. Trust us, you do

*not*want to taunt or provoke a decimal.### Adding and Subtracting Decimals

To add decimals, remember that decimals are abbreviations for fractions.

For example, 0.3 + 0.04 + 0.001 translates to .

Since and , this means that .

Is this paragraph making you dizzy? Well, stop running in circles, sit down, and for goodness' sake, eat something!

The example in the previous paragraph represents the long way to do things. (Sigh of indescribable relief.) While it's mathematically correct to turn decimals into fractions, add the fractions, then turn them back into decimals, it's too much work. Solving decimal problems that way would be like deleting all your songs on iTunes and then uploading them again just so you can change the order of your playlist. By the way,

*David Hasselhoff Sings America*? Really?Instead, remember that adding zeros to the end of a decimal doesn't change its value, so 0.3 = 0.300. (This is the same as saying but without having to write the fractions.)

Also, we can rewrite the 0.04 part of the problem as 0.040. The last part, 0.001, is already written out to the thousandths place, so we can leave that one alone. Good thing, because all this decimal manipulation is wiping us out.

To add up the decimals, always add zeros to the ends of the decimals as needed so that all of them have the same number of decimal places, like we did above. Line up the numbers at the decimal point, and add them like whole numbers:

0.300

0.040

+ 0.001

= 0.341We're really adding up just like we did before, but without having to write the fractions. Good thing, because all this fraction writing is wiping us out. In hindsight, maybe we just didn't get enough sleep last night.

Subtraction is similar. To subtract one decimal number from another, first give the two decimals the same number of decimal places by adding zeros at the end as needed. Line up the numbers at the decimal point, and then rock some subtraction just like you would with whole numbers. If you have a different, "cool" way of subtracting whole numbers, then never mind.

### Multiplication and Division by Powers of 10

When you're multiplying or dividing a decimal by a power of 10, there are some sweet, sweet shortcuts.

First we'll show you how to solve one of these problems

*without*the shortcut. That should heighten your appreciation for it once we tell you about it.### Sample Problem

Transform both decimals into fractions, then multiply 'em together. We then write as a decimal to get 0.3

That's the long way. Too much work, right? We agree. Couple more examples and then you can start taking the shortcut. Just try not to get your legs cut up in the brambles.

### Sample Problem

Same deal here. We transform 0.4 into a fraction with 10 in the denominator, then multiply by 1000. That'll give us 4000 in the numerator and 10 in the denominator, which reduces to a nice, even 400.

### Sample Problem

Shortcut time. To multiply a decimal by a power of 10, all we do is move the decimal point to the right as many places as there are zeros in that power of 10. Multiplying by 100? Scooch it over two spots. Multiplying by 10,000? Move it over four. Multiplying by 52,398? Move it over...

*psych*. Not in base-10. No shortcut for you.When

*dividing*a decimal by a power of 10, we move the decimal point to the left instead. As a way to remember this, picture the volume control on your iPod. You rotate it to the right to increase (multiply) the volume, and to the left to decrease (divide) the volume. Or, if it's a decimal that you really, really like, you can always put it on repeat. Like so: 0.33333333...### Sample Problem

What's 0.6 ÷100?

Let's bust out that shortcut. Since we're dividing by 100, we just slide our decimal point over two spots to the left, since 100 has two zeros.

0.6 ÷100 = 0.006

Here's how it works without the shortcut, just to prove this thing works:

### Multiplying Decimals

When multiplying a decimal number by another decimal number, it always helps to be reminded what they'd look like in fraction form.

### Sample Problem

Just like with addition and subtraction, converting decimals to fractions and back again is pretty inefficient. Thankfully, we can get around that. Just plow right through those "detour" signs.

In the example 0.8 × 0.4, we multiplied two decimals with one decimal place each. When we wrote the numbers as fractions, we were multiplying two fractions that each had 10 in the denominator. The product of those fractions gave us a denominator of 100, so the corresponding decimal had two decimal places. Once again, we're just counting zeros. Better than counting crows.

Suppose

*a*,*b*, and*c*are three decimal numbers. How do you figure out the number of decimal places in the product of*a*×*b*×*c*? Just add the number of decimal places in*a*to the number of decimal places in*b*to the number of decimal places in*c*. Yeah, unfortunately you'll need to know how to multiply three numbers together. Curse those three-dimensional shapes.### Dividing Decimals

To divide one decimal by another decimal, we use long division. Don't you just love making use of existing skill sets?

First, a quick refresher on the names for the different parts of a division problem:

• The

**dividend**is the thing being divided up. The "dividee," if you will.• The

**divisor**is the thing that performs the dividing. Just remember that this word sounds like "operator" or "actor"; the one who operates or acts. And what a smooth operator he is, too.• The

**quotient**is the answer. Don't get excited—not to the meaning of life, just to a long division problem. Sheesh, way to jump the gun there.16.12 ÷ 4 = 4.03 ← quotient

↑ ↑

dividend divisorNow let's talk about what happens when we divide a decimal by a whole number. In this case, we put the decimal point for the quotient directly above the decimal point in the dividend...

...then perform long division......and that's it! Decimal point, set, match!

If we have a decimal divided by something that

*isn't*a whole number, we have to do slightly more work. Oh, relax. It builds character.### Sample Problem

4.08 ÷ 3.4 = ?

First, write this division problem as a fraction:

Now multiply by a cleverly disguised form of 1:

This means that 4.08 ÷ 3.4 and 40.8 ÷ 34 will give us exactly the same result. Since 40.8 ÷ 34 is a decimal divided by a whole number, we can now work this out with long division. Two heads may be better than one, but one decimal is definitely better than two.

40.8 ÷ 34 = 1.2

In general, if we have a division problem where the divisor isn't a whole number, we have to do three things:

1. Convert it to a new division problem where the divisor is a whole number.

2. Find the quotient for the new division problem.

3. Floss. This last one isn't directly related to division, but it's still very important.

To find the new division problem, we multiply by a cleverly disguised form of 1. This means we multiply both the dividend and the divisor by 10. We keep doing that until we have a division problem where the divisor is a whole number. Hopefully you won't have to do this so many times that you miss dinner. It's lasagna night.

Since multiplying a decimal number by 10 means moving the decimal point one place to the right, there's a nice way to summarize what we do to divide one decimal by another:

Count the number of decimal places in the divisor. Move the decimal point that many places to the right in the divisor, and that many places to the right in the dividend also. This produces a new division problem where the divisor is a whole number, and where the quotient is the same as the quotient in the original division problem. Everything stays the same, and everyone seems happy. Just like life in the suburbs.

### Infinite Decimals

So far, all the decimal arithmetic we've done has involved decimal numbers with a finite number of decimal places. However, sometimes decimal numbers are infinite. Make sure you don't confuse "infinite decimal" with "infinitesimal." Although a number can sometimes be both, they're not the same thing. Even though they sound

*exactly*the same when pronounced aloud. Thanks again, English.

**Infinite decimals**sometimes show up when we convert fractions into decimals.### Sample Problem

Convert the fraction

^{1}/_{3}into a decimal, using long division.We end up with a decimal that goes on forever. Literally. And we thought Mondays seemed long.

To show an infinite decimal, we write "..." at the end. This is also good for when you get bored writing all the digits of a lengthy finite decimal, or when your pen is running out of ink.

0.33333333...

Another way to write an infinite decimal with a repeating pattern is to draw a bar over the part that repeats.

0.333333333.... = 0.3There are also infinite decimals

*without*repeating patterns. These decimals represent the irrational numbers, and there's no way to know all the digits of any such number. And please don't take that as a personal challenge. We don't want to see you wasting the next thirty years of your life trying to memorize pi to an infinite number of digits before finally realizing it can't be done. We'd feel partly responsible.However, you

*can*see at least the first few digits of some famous infinite decimals:

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