1.7.2: Key Concepts

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Key Concepts

1.1 Introduction to Whole Numbers

A chart titled 'Place Value' with fifteen columns and 4 rows, with the columns broken down into five groups of three. The header row shows Trillions, Billions, Millions, Thousands, and Ones. The next row has the values 'Hundred trillions', 'Ten trillions', 'trillions', 'hundred billions', 'ten billions', 'billions', 'hundred millions', 'ten millions', 'millions', 'hundred thousands', 'ten thousands', 'thousands', 'hundreds', 'tens', and 'ones'. The first 8 values in the next row are blank. Starting with the ninth column, the values are '5', '2', '7', '8', '1', '9', and '4'. — Figure 1.16

Name a whole number in words.
1. Step 1. 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.
2. Step 2. Use commas in the number to separate the periods.
Use place value to write a whole number.
1. Step 1. Identify the words that indicate periods. (Remember the ones period is never named.)
2. Step 2. Draw three blanks to indicate the number of places needed in each period.
3. Step 3. 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.
1. Step 1. Locate the given place value. All digits to the left of that place value do not change unless the digit immediately to the left is 9, in which case it may. (See Step 3.).
2. Step 2. Underline the digit to the right of the given place value.
3. Step 3. Determine if this digit is greater than or equal to 5. If yes—add 1 to the digit in the given place value. If that digit is 9, replace it with 0 and add 1 to the digit immediately to its left. If that digit is also a 9, repeat. If no—do not change the digit in the given place value.
4. Step 4. 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. $a + 0 = a$ $0 + a = a$
Commutative Property of Addition
- Changing the order of the addends $a$ and $b$ does not change their sum. $a + b = b + a$ .
Add whole numbers.
1. Step 1. Write the numbers so each place value lines up vertically.
2. Step 2. 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.
3. Step 3. 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.
1. Step 1. Write the numbers so each place value lines up vertically.
2. Step 2. 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.
3. Step 3. Continue subtracting each place value from right to left, borrowing if needed.
4. Step 4. Check by adding.

1.4 Multiply Whole Numbers

Operation	Notation	Expression	Read as	Result
$Multiplication$	$\times$ $\cdot$ $()$	$3 \times 8$ $3 \cdot 8$ $3 (8)$	$three times eight$	$the product of 3 and 8$

Multiplication Property of Zero
- The product of any number and 0 is 0.
  $a \cdot 0 = 0$
  $0 \cdot a = 0$
Identity Property of Multiplication
- The product of any number and 1 is the number.
  $1 \cdot a = a$
  $a \cdot 1 = a$
Commutative Property of Multiplication
- Changing the order of the factors does not change their product.
  $a \cdot b = b \cdot a$
Multiply two whole numbers to find the product.
1. Step 1. Write the numbers so each place value lines up vertically.
2. Step 2. Multiply the digits in each place value.
3. Step 3. Work from right to left, starting with the ones place in the bottom number.
4. Step 4. Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.
5. Step 5. If a product in a place value is more than 9, carry to the next place value.
6. Step 6. 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.
7. Step 7. Insert a zero as a placeholder with each additional partial product.
8. Step 8. Add the partial products.

1.5 Divide Whole Numbers

Operation	Notation	Expression	Read as	Result
$Division$	$\div$ $\frac{a}{b}$ $b a$ $a / b$	$12 \div 4$ $\frac{12}{4}$ $412$ $12 / 4$	$Twelve divided by four$	$the quotient of 12 and 4$

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.
1. Step 1. 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.
2. Step 2. Write the quotient above the dividend.
3. Step 3. Multiply the quotient by the divisor and write the product under the dividend.
4. Step 4. Subtract that product from the dividend.
5. Step 5. Bring down the next digit of the dividend.
6. Step 6. Repeat from Step 1 until there are no more digits in the dividend to bring down.
7. Step 7. Check by multiplying the quotient times the divisor.

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