Whole+Numbers

= The whole numbers are the counting numbers and 0. The whole numbers are 0, 1, 2, 3, 4, 5, ... =

** The position, or __place__, of a digit in a number written in standard form determines the actual __value__ the digit represents. **

** This table shows the place value for various positions: **
 * __**Place (underlined)**__ || __**Name of Position**__ ||
 * **1 00__0__** || **Ones (units) position** ||
 * **1 0__0__0** || **Tens** ||
 * **1 __0__00** || **Hundreds** ||
 * **__1__ 000** || **Thousands** ||
 * **1 0__0__0 000** || **Ten thousands** ||
 * **1 __0__00 000** || **Hundred Thousands** ||
 * **__1__ 000 000** || **Millions** ||
 * **1 0__0__0 000 000** || **Ten Millions** ||
 * **1 __0__00 000 000** || **Hundred millions** ||
 * **__1__ 000 000 000** || **Billions** ||

Example: The number 721040 has a 7 in the hundred thousands place, a 2 in the ten thousands place, a one in the thousands place, a 4 in the tens place, and a 0 in both the hundreds and ones place.

The expanded form of a number is the sum of the values of each digit of that number. Example: 9836 = 9000 + 800 + 30 + 6. To compare two whole numbers, first put them in standard form. The one with more digits is greater than the other. If they have the same number of digits, compare the most significant digits (the leftmost digit of each number). The one having the larger significant digit is greater than the other. If the most significant digits are the same, compare the next pair of digits from the left. Repeat this until the pair of digits is different. The number with the larger digit is greater than the other.
 * Symbols** are used to show how the size of one number compares to another. These symbols are **<​** (less than), **>** (greater than), and **=** (equals.) For example, since 2 is smaller than 4 and 4 is larger than 2, we can write: **2 < 4**, which says the same as **4 > 2** and of course, **4 = 4**.

Example: 402 has more digits than 42, so 402 > 42.

Example: 402 and 412 have the same number of digits. We compare the leftmost digit of each number: 4 in each case. Moving to the right, we compare the next two numbers: 0 and 1. Since 0 < 1, 402 < 412.

Examples: Rounding 119 to the nearest ten gives 120. Rounding 155 to the nearest ten gives 160. Rounding 102 to the nearest ten gives 100. Similarly, to round a number to any place value, we find the number with zeros in all of the places to the right of the place value being rounded to that is closest in value to the original number. Examples: Rounding 180 to the nearest hundred gives 200. Rounding 150090 to the nearest hundred thousand gives 200000. Rounding 1234 to the nearest thousand gives 1000. Rounding is useful in making estimates of sums, differences, etc.
 * To round to the nearest ten means to find the closest number having all zeros to the right of the tens place. Note: when the digit 5, 6, 7, 8, or 9 appears in the ones place, round up; when the digit 0, 1, 2, 3, or 4 appears in the ones place, round down.**

Example: To estimate the sum 119360 + 500 to the nearest thousand, first round each number in the sum, resulting in a new sum of 119000 + 1000.. Then add to get the estimate of 120000.


 * Commutative Property of Addition and Multiplication **

Examples: 100 + 8 = 8 + 100 100 × 8 = 8 × 100
 * Addition and Multiplication are commutative: switching the order of two numbers being added or multiplied does not change the result. **


 * Associative Property **

**Addition and multiplication are associative: the order that numbers are grouped in addition and multiplication does not affect the result.**
Examples: (2 + 10) + 6 = 2 + (10 + 6) = 18 2 × (10 × 6) = (2 × 10) × 6 =120

**Distributive Property**

10 × (50 + 3) = (10 × 50) + (10 × 3) 3 × (12+99) = (3 × 12) + (3 × 99)
 * The distributive property of multiplication over addition: multiplication may be distributed over addition.** Examples:

Adding 0 to a number leaves it unchanged. We call 0 the additive identity.** Example: 88 + 0 = 88 Example: 88 × 0 = 0 0 × 1003 = 0
 * The Zero Property of Additon
 * Multiplying any number by 0 gives 0.**

**The Multiplicative Identity**

Example: 88 × 1 = 88
 * We call 1 the multiplicative identity. Multiplying any number by 1 leaves the number unchanged.**

1) Perform operations within parentheses. 2) Multiply and divide, whichever comes first, from left to right. 3) Add and subtract, whichever comes first, from left to right.
 * The Order of Operations **
 * The order of operations for complicated calculations is as follows:**

Example: 1 + 20 × (6 + 2) ÷ 2 = 1 + 20 × 8 ÷ 2 = 1 + 160 ÷ 2 = 1 + 80 = 81

Example 1: ** Evaluate each expression using the rules for order of operations. ** **Solution:** ||
 * **Order of Operations**
 * **Expression ** || **Evaluation ** || **Operation ** ||
 * 6 + 7 x 8 || = 6 + **7 x 8 ** || Multiplication ||
 * ^  || = 6 + 56 || Addition ||
 * ^  || = 62 ||   ||
 * 16 **÷ ** 8 - 2 || <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">= **<span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">16 ****<span style="color: black; font-family: 'Arial','sans-serif'; font-size: 9pt;">÷ ****<span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;"> 8 **<span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;"> - 2 || <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">Division ||
 * ^  || <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">= 2 - 2 || <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">Subtraction ||
 * ^  || <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">= 0 ||   ||
 * <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">(25 - 11) x 3 || <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">= **<span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">(25 - 11) **<span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;"> x 3 || <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">Parentheses ||
 * ^  || <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">= 14 x 3 || <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">Multiplication ||
 * ^  || <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 10.5pt;">= 42 ||   ||
 * In Example 1, each problem involved only 2 operations. Let's look at some examples that involve more than two operations.**
 * **Example 2:** || **Evaluate 3 + 6 x (5 + 4) ÷ 3 - 7 using the order of operations.** ||
 * [[image:images/tab.gif width="10" height="3" caption=" "]] ||
 * **Solution:** ||  || ** Step 1: ** || 3 + 6 x **(5 + 4)** **÷** 3 - 7 || = || 3 + 6 x 9 **÷** 3 - 7 || Parentheses ||
 * ** Step 2: ** || 3 + **6 x 9** **÷** 3 - 7 || = || 3 + 54 **÷** 3 - 7 || Multiplication ||
 * ** Step 3: ** || 3 + **54** ÷ **3** - 7 || = || 3 + 18 - 7 || Division ||
 * ** Step 4: ** || **3 + 18** - 7 || = || 21 - 7 || Addition ||
 * ** Step 5: ** || 21 - 7 || = || 14 || Subtraction ||  ||


 * ** Example 3: ** || **Evaluate 9 - 5 ÷ (8 - 3) x 2 + 6 using the order of operations.** ||
 * [[image:images/tab.gif width="10" height="3" caption=" "]] ||
 * **Solution:** ||  || ** Step 1: ** || 9 - 5 **÷** **(8 - 3)** x 2 + 6 || = || 9 - 5 **÷** 5 x 2 + 6 || Parentheses ||
 * ** Step 2: ** || 9 - **5** ÷ **5** x 2 + 6 || = || 9 - 1 x 2 + 6 || Division ||
 * ** Step 3: ** || 9 - **1 x 2** + 6 || = || 9 - 2 + 6 || Multiplication ||
 * **Step 4:** || **9 - 2** + 6 || = || 7 + 6 || Subtraction ||
 * ** Step 5: ** || 7 + 6 || = || 13 || Addition ||  ||
 * In Examples 2 and 3, you will notice that multiplication and division were evaluated from left to right according to Rule 2. Similarly, addition and subtraction were evaluated from left to right, according to Rule 3.**
 * When two or more operations occur inside a set of parentheses, these operations should be evaluated according to Rules 2 and 3. This is done in Example 4 below**


 * ** Example 4: ** || Evaluate 150 **÷** (6 + 3 x 8) - 5 using the order of operations. ||
 * ** Solution: ** ||  || ** Step 1: ** || 150 **÷** (6 + **3 x 8**) - 5 || = || 150 **÷** (6 + 24) - 5 || Multiplication inside Parentheses ||
 * ** Step 2: ** || 150 **÷** **(6 + 24)** - 5 || = || 150 **÷** 30 - 5 || Addition inside Parentheses ||
 * ** Step 3: ** || **150** ÷ **30** - 5 || = || 5 - 5 || Division ||
 * ** Step 4: ** || 5 - 5 || = || 0 || Subtraction ||  ||


 * ** Example 5: ** || **Evaluate the arithmetic expression below:



** || ** ||
 * ** Solution: ** || **This problem includes a** ****fraction bar**** **(also called a vinculum), which means we must divide the numerator by the denominator. However, we must first perform all calculations above and below the fraction bar BEFORE dividing.** ||
 * || **Evaluating this expression, we get:

77** ||
 * ** Example 6: ** || **Write an arithmetic expression for this problem. Then evaluate the expression using the order of operations.** ||
 * || **Mr. Smith charged Jill $32 for parts and $15 per hour for labor to repair her bicycle. If he spent 3 hours repairing her bike, how much does Jill owe him?** ||
 * **Solution:** || **32 + 3 x 15 = 32 + 3 x 15=32 + 45=
 * || **Jill owes Mr. Smith $77.** ||


 * ** Summary: ** || **When evaluating arithmetic expressions, the order of operations is:** * **Simplify all operations inside parentheses.**
 * **Perform all multiplications and divisions, working from left to right.**
 * **Perform all additions and subtractions, working from left to right.** ||
 * || **If a problem includes a fraction bar, perform all calculations above and below the fraction bar before dividing the numerator by the denominator.** ||