Chapter 19 - Speed And Ease In Decimals

Fractions are one way of expressing quantities of less than 1, or more than 1 but not reaching an exact digit. Decimals are another way of doing the same thing.

Of the two, decimals are usually by far the easier and more convenient way to express fractional quantities. If our measuring systems were based on our ten-base counting system (as is the Continental system of meters, grams, litres, and so on) we would perhaps face fractions only a very few times in our lives. But since we have inherited a jumbled group of weights and measures broken down variously into twelfths (feet), sixteenths (pounds), sixtieths (hours), fourths (gallons) and even 5,280th's (miles), we face fractions all the time.

Only in our U. S. money system are we blessed with a commonsense decimal progression. In all our other measurements, we cling to outrageous counting bases.

Even for these, however, decimal fractions are usually accurate enough. They are not as perfect an expression of many quantities as are fractions, which can express any conceivable quantity with exact preciseness, but the difference is so slight as to be meaningless in most cases. In fact, many of the quantities we consider "hard" or "exact" are only approximations to begin with. 5 apples is precisely 5 apples, but 5 inches or 5 pounds is only as (approximately) close to 5 inches or 5 pounds as our measuring equipment can determine at the time.

Actually, these are two completely different types of numbers. One is a precise quantity; the other is a declaration of comparison to an artificial standard such as acres or gallons. Think for a bit about the essential differentness of the two approaches to numbers, for the sake of your number sense.

As far as preciseness of decimals goes, ⅓ is a prime example. There is no decimal equivalent, nor can there ever be. The fraction ⅓ expresses a certain quantity with complete accuracy. The decimal O.3333333333333333333 approaches ⅓, but it is not ⅓. No matter how many 3's you add, you never quite reach ⅓. .33 is accurate to 1 part in 100, however, while .333 is accurate to 1 part in 1,000. For most practical needs, this is more than enough accuracy.
A decimal is a shorthand way of expressing a fraction that has a bottom of 10, 100, 1,000, or some other multiple of 10. We use the decimal point, the little period, to indicate that the digits following it do not express a whole-number quantity, but a fraction whose bottom is a multiple of 10. The number .3 is the same as 3/10- 1.3 is the same as l3/10-The point tells us when to stop figuring in whole numbers and begin noting the fraction.

The first lesson usually taught in reference to decimals is how to read them properly. Do you read 0.33 as 33 tenths, hundredths, or thousandths? There is a beautifully simple and reliable trick that removes any possible confusion. Merely pretend that the decimal point itself is a 1, followed by as many 0's as there are digits after the point. This imaginary number is the bottom of your fraction.

Thus 0.3 must be 3/10, since the "1" (point) is followed by one digit, and 10 is ten. 0.33 must be 33/100, because there are two digits after the point and two 0's in 100.

See if you can read 0.4567 with this method. The top of the fraction is 4567, of course. The bottom is a 1 followed by four 0's. So the bottom must be 10,000.

If there are any zeros immediately following the point, count them as digits too in figuring the bottom. 0.03 is 3/100, not 3/10, because there are two digits after the point.

A surprising number of people have trouble determining the proper "bottom" to decimal-form fractions, because they never learned this simple trick. For practice in using it, read the following decimals as fractions by saying aloud both the top and bottom of each fraction, just as we might say .3 as 3/10:

Mixed Numbers

Expressing mixed numbers (a whole number plus a fraction) is much easier in decimals than it is in fractional form. Whatever part of the number is in front of (to the left of) the point is a whole number. The part to the right of the point expresses the fraction.
So 22.4 is read as "22 and 4/10ths."

It is often read, too, as "22 point 4." There is nothing wrong with this. It is a short hand way of reading and saying the number, but it does not drive home the actual quantity involved as firmly as does reading the decimal as a full fraction.

1.43 is read as "1 and 43/100ths." Read aloud the number 45.67.

If you read it properly as "45 and 63/100 (hundredths)," go ahead to read the following numbers. Say the full number followed by the fractional part in terms of tenths, hundredths, thousandths, or whatever:
 
16.4     892.674           3.21     0.45

32.0     100.001           6.08     21.30

If you hesitated over any of these, particularly the one-thousandth or eight-hundredths, it would be a good idea to review the last few pages before going on.

Adding Decimals

There is no trick at all to adding numbers with decimals in them if you keep the basic rule in mind: line up the points. If you were adding 10,342 to 61, you would line up the right-hand ends of these numbers. The point in a number with a decimal fraction is just as clearly and firmly the end of the whole number as is the end when the digits come to a stop there.

Using this rule, set up the following numbers for addition:

65.3 2.13 100.2 .935

Cover the arrangement below with your pad until you have done your part.

Using the points as the ends of the whole numbers, you line up the above numbers for addition like this:

65.3

2.13 100.2 .935

That is really all there is to it. Elementary, but very important. Once you have lined up the numbers properly, you simply go ahead and add. Ignore the points, except to put a point in your answer in line with the column. Tens carried back across the point as you add behave just as if there were no point there, which is one of the great advantages of using decimals. They enable you to handle the fractional parts right along with your whole numbers, instead of creating them separately.

Having dismissed addition this easily, we can say the same thing about subtraction: keep your points in line, and "borrow" (or slash) across the point as if it were not even there. One example will make this clear:

In both addition and subtraction, you can pretend the points are invisible—as long as you line them up, and make sure to put one in your answer in line with the others.

Multiplying Decimals

When you come to multiplying decimals, you do not bother to line up the points because you have another way of placing the point properly in your answer. Refer back to the chapter on no-carry multiplication, if need be, to refresh your understanding of the following rule:

Add the digits in the two numbers multiplied. Starting with the very left top digit (including the 0 if it is a 0), count this many digits for the answer.

In multiplying decimals, add only the following special qualifier to the general rule: —to the left of the point.

That "to the left of the point" applies both to the numbers multiplied, and to the answer as well. It works like this:

How did we place that point in the answer? Each of the two numbers multiplied has two digits to the left of the point. So our answer should have four digits before the point. We start at the very left of the top line with the 0 that does not show up in the final answer but that does have to be counted.

In every other aspect of the problem, we simply ignore the points altogether. You can prove it out by nines-remainders or elevens-remainders, ignoring the points for this purpose too except that you start at the point in figuring odd and even digits for an eleven-remainder. If you use continuous subtraction, just keep right on subtracting as you go past the point.

In the example above, you could also have used the classic "point off as many places from the right as there are places to the right of the point in the two numbers multiplied." To rely on this method, however, would rob you of the rapid-estimating nature of no-carry multiplying. Work from left to right instead of right to left, and you can do just as much of any problem as you need to in order to get the accuracy required in that particular situation.

Dividing Decimals

For dividing decimals, we cannot improve on the usual rule: move the point in the divider (if any) all the way to the right. Put a point in your answer as many places to the right of the point in the number divided (if any) as you moved the point in the divider.

If this means adding O's to the number divided in order to move your point far enough, go ahead and add them.

Try it yourself. Where will the point in the answer appear for each of the following problems?

Here are two examples:

Each or these is a little different, but all operate on exactly the same system.

1. The point in the answer will be between the 7 and 8 of the number divided, because we move the point one place to the right.

2. The point in the answer will be directly above the point in the number divided, since there is no point in the divider.

3. The point in the answer will be after the final 6 in the number divided. Moved two places.

4. The point in the answer will be between the 3 and 9 in the number divided. Moved one place.

Other than placing your decimal point properly in the answer, there is no more to dividing with decimal numbers than there is to any division. Once you have determined the right place for the point, simply ignore all the points in the original problem. Your answer will be correct.

Only one other aspect needs special mention. We demonstrated it before, but it should be spelled out too. If you have to move the point in your answer way beyond the end of the number divided, simply do it. Fill in with O's as needed. For instance:

Decimal Remainders

Depending on the particular problem and the particular field in which your answer will be used, you may work out a division problem that has a remainder in either fractional or decimal form.

The making of a decimal remainder is very simple. It makes no difference how many O's you add after the last digit to the right of the point, any more than it makes any difference how many O's you add to the left of a whole number.

00045.2 is the same as 45.2000
 
There is one special meaning to 0's following the last digit to the right of a decimal point, however, and you should be aware of it. By common agreement, the 0 you place to the right means that the number is accurate to this point.

The number 4.6 might be a rounded-off number anywhere from 4.56 to 4.64. But the number 4.60 means that any rounding off was done beyond the 0. The convention in mathematics goes further, incidentally, and often places a plus or minus sign at the end of a number that has been rounded off, to indicate that it is not a precise quantity.

To make a decimal remainder, then, you simply keep mentally bringing down 0's as long as you have to in order to get an exact answer or the accuracy you need. With your mastery of shorthand division, you do not even have to note the 0's in the number divided; just bring down imaginary ones:

If the final division here had not come out even, you would keep bringing down imaginary 0's until you had no remainder, or had as complete an answer as you needed. If you divide 3 into 10, you will never get a complete answer. But at some point you will have as complete an answer as you need.

Converting from Fractions

A fraction, as we have said, is only a special way of writing a division problem. It expresses a specific quantity, but one that (except by decimals) we have no other way of showing with the numbers available than as a division of two known numbers. ⅜ is the same as 3 ÷8 or 8 /3. The fraction has a different purpose from the division, however; it says, in effect, "this is a quantity," rather than "here is a problem," because for many purposes 3/8 is more convenient than other expressions of that quantity.

Often, however, you want to convert a fraction to a decimal form. The method is simplicity itself. Simply carry out the implied division, and use a decimal remainder.

To convert ⅜ to a decimal, for instance, you do this:

The decimal equivalent of ⅜ is 0.375. In this case, it is an exact equivalent, and it should sound familiar: 375 is one of the basic aliquots.

Now you convert 6/7 to a decimal. Get out your pad and cover the answer. Express 6/7 as a decimal accurate to the nearest 10,000th.

Here is how the conversion looks in shorthand division:

The nearest 10,000th means four places after the point. We worked it out to five places so we could round off, and the last 4 indicates that the rounded-off form is .8571.

Sometimes you find it necessary to convert decimals back to fractions for particular purposes. In some problems, fractions are easier to handle. This, in fact, is part of the basis of the aliquot short cut.

For decimals other than aliquots, the process for converting to a fraction is to write it in fractional form and then see if it can be reduced. The decimal .1 can be written as 1/10 and .45 can be written as 45/100.

You reduce this resulting fraction exactly as you reduce any other fraction: divide both top and bottom by any number that will divide both exactly, if there is any. Try reducing the example above, .45.

45 is exactly divisible by 5 or by 9. 100, however, is divisible by 5 but not by 9. Dividing both top and bottom by 5, we reduce 45/100 to 9/20. No further reduction is possible.

Convert the following decimals to fractions:

.25       .8125   .625     .96875

The last one, admittedly, is a dilly. But it can be reduced quite substantially. Cover the reductions with your pad until you are satisfied.

Your answers should read ¼, 13/16, 5/8, and 31/32-The next chapter will take up decimals in another and quite special form, percentage. Before going on to that chapter, reflect for a moment or two on the entire decimal method of expressing fractions—and its firm foundation on the point made several times before in this book that each digit decreases in importance by a factor of 10 as it moves each place to the right. This is true right across the decimal point—which is the end of the whole number.

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