A stopwatch at the track meet, the price at a gas pump, the weight of a science sample: all of them use decimals. In this chapter you will see how place value works on both sides of the decimal point, how powers of 10 make huge and tiny numbers easy to handle, and how to read, write, compare and round decimals like a pro.
1. Place value patterns
The value of a digit depends on its place in the number. Moving one place to the left makes a digit worth 10 times as much. Moving one place to the right makes it worth \(\dfrac{1}{10}\) as much.
In \(352.614\) the digit \(5\) is in the tens place, so it is worth \(50\). The digit \(6\) is in the tenths place, so it is worth \(0.6\). The decimal point separates the whole-number places from the places smaller than one.
In \(6.666\) every digit is a 6, but not every 6 is worth the same.
The 6 in the ones place is worth \(6\). The 6 in the tenths place is worth \(0.6\), and \(6 = 10 \times 0.6\). The 6 in the hundredths place is worth \(0.06\), and \(0.06 = \dfrac{1}{10} \times 0.6\).
The place names on the right end in -ths (tenths, hundredths, thousandths). There is no “oneths” place: the center of the pattern is the ones place.
2. Powers of 10 and exponents
Multiplying by 10 again and again is such a common job that mathematicians invented a shortcut.
An exponent tells how many times a number is used as a factor. For example, \(10^3 = 10 \times 10 \times 10 = 1{,}000\). The number \(10\) is the base and \(3\) is the exponent.
| Power | \(10^1\) | \(10^2\) | \(10^3\) | \(10^4\) | \(10^5\) | \(10^6\) |
|---|---|---|---|---|---|---|
| Standard form | 10 | 100 | 1,000 | 10,000 | 100,000 | 1,000,000 |
| Zeros after the 1 | 1 | 2 | 3 | 4 | 5 | 6 |
Look at the pattern: the exponent is exactly the number of zeros after the 1. Each time the exponent goes up by one, the number becomes 10 times larger, just like moving one place to the left on the place value chart.
\(10^3\) does not mean \(10 \times 3\). It means three tens multiplied together: \(10 \times 10 \times 10 = 1{,}000\), not \(30\).
(a) Write \(100{,}000\) as a power of 10. There are 5 zeros, so \(100{,}000 = 10^5\).
(b) Write \(7 \times 10^4\) in standard form. \(10^4 = 10{,}000\), so \(7 \times 10^4 = 70{,}000\).
3. Reading and writing decimals to thousandths
Decimals are another way to write fractions whose denominators are 10, 100 or 1,000.
| Standard form | Fraction | Words |
|---|---|---|
| \(0.7\) | \(\dfrac{7}{10}\) | seven tenths |
| \(0.07\) | \(\dfrac{7}{100}\) | seven hundredths |
| \(0.007\) | \(\dfrac{7}{1000}\) | seven thousandths |
- Read the whole-number part.
- Say “and” for the decimal point.
- Read the digits after the point as one whole number.
- Finish with the name of the last place.
\(4.076\) is read “four and seventy-six thousandths”, because the last digit sits in the thousandths place. In the same way, \(0.305\) is “three hundred five thousandths”.
Writing goes the other way: “twelve and eight tenths” is \(12.8\), and “five hundredths” is \(0.05\) (the 0 holds the tenths place).
Adding zeros at the end of a decimal does not change its value: \(2.5 = 2.50 = 2.500\). They are called equivalent decimals.
4. Expanded form of decimals
Writing a number as the sum of the values of its digits is called expanded form. You can use multiplication to show each digit and its place.
Write \(36.082\) in expanded form.
\(36.082 = 3 \times 10 + 6 \times 1 + 0 \times 0.1 + 8 \times 0.01 + 2 \times 0.001\)
Without the zero term: \(36.082 = 30 + 6 + 0.08 + 0.002\). With fractions: \(8 \times \dfrac{1}{100} + 2 \times \dfrac{1}{1000}\) for the decimal part.
To go back, add the pieces: \(400 + 7 + 0.2 + 0.005 = 407.205\). Zeros in the expanded form tell you which place must hold a placeholder 0.
5. Comparing decimals
- Line up the decimal points (add zeros at the end if it helps).
- Start on the left and compare digits place by place.
- The first place where the digits differ decides which number is greater.
Compare \(0.47\) and \(0.456\). The ones and tenths match (0 and 4). In the hundredths place, \(7 > 5\). So \(0.47 > 0.456\), exactly as the number line shows: \(0.47\) is farther to the right.
Compare \(3.09\) and \(3.1\). Write \(3.1\) as \(3.10\). The tenths place gives \(0 < 1\), so \(3.09 < 3.1\).
A decimal with more digits is not automatically greater. \(0.456\) has three digits after the point, but it is smaller than \(0.47\).
6. Rounding decimals
- Underline the digit in the place you are rounding to.
- Look at the digit just to its right.
- If it is 5 or more, round up (add 1 to the underlined digit). If it is 4 or less, keep the digit.
- Drop the digits to the right.
Round \(3.462\). To the nearest tenth: the hundredths digit is 6, so round up: \(3.5\). The number line agrees, since \(3.462\) lies past the halfway mark \(3.45\).
To the nearest hundredth: the thousandths digit is 2, so keep: \(3.46\). To the nearest whole number: the tenths digit is 4, so \(3\).
Carrying happens too: \(9.96\) to the nearest tenth is \(10.0\), because \(9 + 1\) carries into the ones place.
7. Multiplying and dividing by powers of 10
Because each place is 10 times the one to its right, multiplying by 10 slides every digit one place to the left. Dividing by 10 slides every digit one place to the right.
To multiply by \(10^n\), move the decimal point \(n\) places to the right. To divide by \(10^n\), move it \(n\) places to the left. Use zeros as placeholders when you run out of digits.
| Start | \(\times 10\) | \(\times 10^2\) | \(\times 10^3\) | \(\div 10\) | \(\div 10^2\) |
|---|---|---|---|---|---|
| \(4.35\) | \(43.5\) | \(435\) | \(4{,}350\) | \(0.435\) | \(0.0435\) |
\(0.0638 \times 10^3\): move the point 3 places right, so \(0.0638 \times 1{,}000 = 63.8\).
\(5.9 \div 10^2\): move the point 2 places left and add a zero placeholder, so \(5.9 \div 100 = 0.059\).
The decimal point never really moves: the digits slide through the places. Picture the place value chart and push each digit one box left or right for every power of 10.
This is how metric conversions work. Since \(1\) meter \(= 100\) centimeters, \(3.5\) m \(= 3.5 \times 100 = 350\) cm. And since \(1\) liter \(= 1{,}000\) milliliters, \(2{,}450\) mL \(= 2{,}450 \div 1{,}000 = 2.45\) L.
Key takeaways
- One place to the left is worth 10 times as much; one place to the right is worth \(\dfrac{1}{10}\) as much.
- \(10^n\) is a 1 followed by \(n\) zeros; the exponent counts the factors of 10.
- Read a decimal by saying the whole part, “and”, then the digits and the name of the last place.
- Expanded form adds the value of each digit: \(36.082 = 30 + 6 + 0.08 + 0.002\).
- To compare, line up the points and find the first place where the digits differ.
- To round, look at the digit to the right: 5 or more rounds up, 4 or less stays.
- Multiplying by \(10^n\) moves the point \(n\) places right; dividing moves it \(n\) places left.
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