1.12: Whole Numbers (Summary)
- Page ID
- 21665
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Key Terms
| coordinate | A number paired with a point on a number line. |
| counting numbers | The counting numbers are the numbers 1, 2, 3, …. |
| difference | The result of subtracting two or more numbers. |
| dividend | The number being divided when dividing two numbers. |
| divisor | The number dividing the dividend when dividing two numbers. |
| number line | A number line is used to visualize numbers. The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left. |
| origin | The point labeled 0 on a number line. |
| place value system | Our number system is called a place value system because the value of a digit depends on its position, or place, in a number. |
| product | The result of multiplying two or more numbers. |
| quotient | The result of dividing two numbers. |
| rounding | The process of approximating a number. |
| sum | The result of adding two or more numbers. |
| whole numbers | The whole numbers are the numbers 0, 1, 2, 3, …. |
Key Concepts
1.1 - Introduction to Whole Numbers
- Name a whole number in words.
- Starting at the digit on the left, name the number in each period, followed by the period name. Do not include the period name for the ones.
- Use commas in the number to separate the periods.
- Use place value to write a whole number.
- Identify the words that indicate periods. (Remember the ones period is never named.)
- Draw three blanks to indicate the number of places needed in each period.
- Name the number in each period and place the digits in the correct place value position.
- Round a whole number to a specific place value.
- Locate the given place value. All digits to the left of that place value do not change.
- Underline the digit to the right of the given place value.
- Determine if this digit is greater than or equal to 5. If yes—add 1 to the digit in the given place value. If no—do not change the digit in the given place value.
- Replace all digits to the right of the given place value with zeros.
1.2 - Add Whole Numbers
• Addition Notation: To describe addition, we can use symbols and words.
| Operation | Notation | Expression | Read as | Result |
|---|---|---|---|---|
| Addition | + | 3 + 4 | three plus four | the sum of 3 and 4 |
- Identity Property of Addition: The sum of any number a and 0 is the number. $$\begin{split} a + 0 &= a \\ 0 + a &= a \end{split}$$
- Commutative Property of Addition: Changing the order of the addends a and b does not change their sum. $$a + b = b + a \ldotp$$
- Add whole numbers.
- Write the numbers so each place value lines up vertically.
- Add the digits in each place value. Work from right to left starting with the ones place. If a sum in a place value is more than 9, carry to the next place value.
- Continue adding each place value from right to left, adding each place value and carrying if needed.
1.3 - Subtract Whole Numbers
| Operation | Notation | Expression | Read as | Result |
|---|---|---|---|---|
| Subtraction | - | 7 - 3 | seven minus three | the difference of 7 and 3 |
- Subtract whole numbers.
- Write the numbers so each place value lines up vertically.
- Subtract the digits in each place value. Work from right to left starting with the ones place. If the digit on top is less than the digit below, borrow as needed.
- Continue subtracting each place value from right to left, borrowing if needed.
- Check by adding.
1.4 - Multiply Whole Numbers
| Operation | Notation | Expression | Read as | Result |
|---|---|---|---|---|
| Multiplication | × | 3 × 8 | three times eight | the product of 3 and 8 |
| • | 3 • 8 | |||
| () | 3(8) |
- Multiplication Property of Zero: The product of any number and 0 is 0. $$\begin{split}a \cdot 0 &= 0 \\ 0 \cdot a &= 0 \end{split}$$
- Identity Property of Multiplication: The product of any number and 1 is the number. $$\begin{split} 1 \cdot a &= a \\ a \cdot 1 &= a \end{split}$$
- Commutative Property of Multiplication: Changing the order of the factors does not change their product. $$a \cdot b = b \cdot a \ldotp$$
- Multiply two whole numbers to find the product.
- Write the numbers so each place value lines up vertically.
- Multiply the digits in each place value.
- Work from right to left, starting with the ones place in the bottom number.
- Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.
- If a product in a place value is more than 9, carry to the next place value.
- Write the partial products, lining up the digits in the place values with the numbers above. Repeat for the tens place in the bottom number, the hundreds place, and so on.
- Insert a zero as a placeholder with each additional partial product.
- Add the partial products.
1.5 - Divide Whole Numbers
| Operation | Notation | Expression | Read as | Result |
|---|---|---|---|---|
| Division | ÷ | 12 ÷ 4 | Twelve divided by four | the quotient of 12 and 4 |
| \(\dfrac{a}{b}\) | \(\dfrac{12}{4}\) | |||
| \(b \overline{) a}\) | \(4 \overline{\smash{)}12}\) | |||
| a / b | 12 / 4 |
- Division Properties of One:
- Any number (except 0) divided by itself is one. $$a \div a = 1$$
- Any number divided by one is the same number. $$a \div 1 = a$$
- Division Properties of Zero:
- Zero divided by any number is 0. $$0 \div a = 0 $$
- Dividing a number by zero is undefined. $$a \div 0 = undefined$$
- Divide whole numbers.
- Divide the first digit of the dividend by the divisor. If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on.
- Write the quotient above the dividend.
- Multiply the quotient by the divisor and write the product under the dividend.
- Subtract that product from the dividend.
- Bring down the next digit of the dividend.
- Repeat from Step 1 until there are no more digits in the dividend to bring down.
- Check by multiplying the quotient times the divisor.