Types of Numbers
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, at other times a decimal is preferable. Usually from 2 in the afternoon to 8 in the evening is optimal for using decimals. (All right, we're pulling your leg. It isn't actually dependent on the time of day. Although decimals are quite nice at sunset.)
The first sort of decimals we are going to look at are just abbreviations for a fraction whose denominator is a power of 10, such as 10, 100 (10 x 10), 1,000 (10 x 10 x 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 around the number 10. For example, when we write the number 12, we really mean 10 + 2. When we write 5,673, 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), MCX is 1,000 (M) plus 100 (C) plus 10 (X). The point of the decimal notation is just to continue this in the opposite direction - to indicate the amount of a number that is less than 1. We want to record how many , , etc. our numbers have. So that if we ever lose them, we'll be able to identify and claim them.
0.3 is an abbreviation for , pronounced "three tenths." , "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 you determine how much of a tip you're going to leave.
If we start with a fraction whose denominator is a power of 10, for example , here's what we do to get the decimal abbreviation:
Count the number of zeros in the denominator: in this case, that would be 3
Write down the numerator, followed by a dot: 691.
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 is some batting average there, sport.
But what happens if the numerator has fewer digits than the denominator has zeros?
There are 3 zeros in the denominator. We write down the numerator, followed by a dot: 7.
It appears as if 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. Be gentle though - no need to ram the zero in there. This number has already been through a lot in the last few seconds.
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 will 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, 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:
The number of decimal places is determined by counting the number of digits that can be found 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 31. 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.