Decimal numbers for modern times
Chapter 9: Decimals: They Have Their Place Pennies are the root of all money
3. Place the decimal point
Because the multiplicand has two decimal places and the multi- plier has four decimal places, set the decimal point six places from the right. The answer is 1.147185
But wait, there’s more! Decimal multiplication gives you a little multiply-by- ten bonus. The rule is simple: To multiply a number by ten, just shift the decimal point one place to the right. To multiply the answer from the exercise (1.147185) by 10, shift the decimal:
1.147185 × 10 = 11.47185
To multiply by 100, shift the decimal two places to the right, and so on.
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Division is an important decision
Normal division of decimals isn’t normal. It’s the problem child of the Four Ops because the results have variations, but you’ll come to love it.
Here are a few things to remember about dividing decimals:
✓ As with other operations, your calculator or smartphone may not have enough room to hold the answer. If you’re not worried about the digits in the more miniscule decimal places, this limitation shouldn’t cause a problem.
✓ With a spreadsheet, format the cells to allow for a large number of decimal places — about eight places is a safe number.
✓ With manual calculating, be prepared to shift the decimal point in the divisor (number you’re dividing by) and the dividend (number you’re dividing into), so the math looks more “normal.” For example, you can express 3.00 ÷ .35 as 300 ÷ 35 by shifting the decimal point two places to the right in both the divisor and the dividend. The answer is 8.5714.
When you are dividing a larger decimal into a smaller one, you can do the same thing. For example, you can express 1.04 ÷ .3.25 as 104 ÷ 325 by shifting the decimal point two places to the right in both the divisor and the dividend. The answer is 0.32.
You can expect three kinds of results when you divide decimals:
✓ When you divide a number by a bigger number, expect a smaller decimal.
For example:
1.75 ÷ 3 = .583333333
✓ When you divide a number by a smaller number, expect a bigger decimal.
For example:
6.8 ÷ 0.3 = 22.666666
✓ When you divide a number by certain other numbers, you may get an infinite series of repeating decimals. For example:
1 ÷ 7 = 0.142857142857142857142857142857142857 . . .
In this situation, be prepared to do some rounding (which we cover later in this chapter).
Like decimal multiplication, decimal division cuts you a break when you’re dividing by ten. To divide a number by ten, just shift the decimal point one place to the left. Look at this example:
0.58 ÷ 10 = 0.058
To divide by 100, shift the decimal two places to the left, and so on.
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Chapter 9: Decimals: They Have Their Place
Decimal Conversion
Common fractions are useful, but you often need to convert them to decimal numbers; machines with digital input, especially computers, are often frac- tion hostile.
Machinists and cabinetmakers using computer-aided design aren’t the only folks affected by fraction intolerance. Graphic design work uses computers, and that means entering decimals. For example, if you’re designing a cover for a book with a spine thickness of
inches, you need to make allowance for that dimension in your design.
Convert
inches to 0.9375 inches for your art program to understand what you want.
Decimal numbers are very useful, to be sure, but they have their limits.
Sometimes you need to convert them to fractions. For example, if you want to mail an item that weighs 0.5625 pounds, that’s great, but the post office doesn’t do decimal pounds. It does ounces. You need to convert 0.5625 pound to
pound, which is 9 ounces. The following sections clue you in on both conver- sion processes.
Converting fractions to decimals
Turning a fraction into a decimal number is easy. In fact, you probably know some conversions by heart. For example, the easiest conversions are
The rule for converting fractions to decimals is to simply divide the denomi- nator into the numerator to get the answer. In the first example, divide 1 by 4 to get 0.25. Use a calculator or smartphone, a spreadsheet program, or pencil and paper.
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For example, to convert
to a decimal number, just divide 15 by 16.
15 ÷ 16 = 0.9375 The answer is 0.9375.
Your job or school probably celebrates Act Like A Sumerian Day. No? Too bad!
Well, if you want observe it anyway, you can do your conversions on a clay tablet, just like the Sumerians did in the good old (really old) days. In the 4th century BC, Sumerian scribes wrote characters on wet clay tablets with a reed stylus. No trees were destroyed, and they could recycle the tablets by soaking them in water.
Early in their careers, machinists learn basic decimal conversions for selecting drill bits. Many tables show drill bit diameters as fractions with their decimal equivalents, ranging from
inches to
inches. However, the world is changing. Fractional bit sizes are still common in America, but most of the rest of the world now uses metric sizes.
Converting decimals to fractions
Converting a decimal number to a fraction is just as easy as going from a fraction to a decimal (see the preceding section). Maybe easier. Try it for yourself and see if you can’t do it in 0.5 of the time. If that sounds stupid, that’s because it should be “half the time.” That number 0.5 needs a decimal- to-fraction conversion.
The conversion rule is simple: Set the decimal up as a fraction and reduce it to its simplest terms. For example, to convert 0.75 to a fraction, set up the fraction as follows:
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Chapter 9: Decimals: They Have Their Place
The basic answer is
To reduce the fraction, divide both the top and bottom by a common factor.
Decimal fractions always have 10, 100, 1,000, 10,000, and so forth as a denom- inator, so try dividing by factors of 10, such as 5 or 2. You may be able to do more division to get to the simplest terms. The most reduced answer is:
You can find greatest common factor calculators on the Internet with a quick search.
How do you know how big to make the denominator? The number of decimal places in the decimal number you have to convert tells you how many zeroes to use.
1 decimal place = (one zero)
2 decimal places = (two zeroes)
3 decimal places = (three zeroes)
4 decimal places = (four zeroes) 5 decimal places = (five zeroes)
Round, Round, Get Around, I Get Around
Sometimes, the answer to a decimal calculation isn’t useful because it has too many decimal places. In this situation, you need to do some rounding, replacing the answer with another value that’s very close to the original. You round up and round down, depending on the original answer and the number of decimal places you want to round to.
Rounding is a common practice. You do it with money all the time and don’t even think about it. For example, if you pay $5.98 for an item and someone asks you what it cost, you probably say, “Oh, about six dollars.”
Here are the rules for rounding: