# Decimals

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.

### Sample Problem

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?

### Sample Problem

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.