Study Notes on Whole Numbers and Decimals

INTRODUCTION TO WHOLE NUMBERS

WEEK 1 – PART ONE

• Overall focus on understanding whole numbers, including their representation, ordering, writing in various forms, rounding, and basic arithmetic operations (addition, subtraction, multiplication, and division).

REFERENCES

• Textbook Reference: Aufmann, R. N. & Lockwood, J. S. (2014). Basic College Mathematics: An Applied Approach, Brooks/Cole, Cengage Learning, Belmont, CA.

• Graphics/Charts used from the above references unless otherwise indicated.

• Other images used in accordance with Fair Use policies per US Copyright law, Section 107.

• Copyright owners with concerns should contact Pierce Mortuary Colleges, Inc. for adjustments.

TO IDENTIFY THE ORDER RELATION BETWEEN TWO NUMBERS

• Graphical Representation on a Number Line:

  • A whole number is represented by placing a heavy dot directly above the number on the number line. This visual aid helps in understanding the relative size of numbers.

  • The order of numbers can be visually observed:

    • A number on the left is less than the number to its right (indicated by <). For instance, 3 is to the left of 5, so 3 < 5.

    • A number on the right is greater than the number to its left (indicated by >). For instance, 7 is to the right of 2, so 7 > 2.

EXAMPLES OF ORDER RELATION

• Explicit Comparisons:

  • Four is less than seven: 4 < 7

  • Twelve is greater than seven: 12 > 7

EXAMPLE 1

• Place the correct symbol (< or >) between two numbers:

  • a. 39 ___ 24

  • Solution: 39 > 24 (Because 39 is to the right of 24 on a number line)

  • b. 0 ___ 51

  • Solution: 0 < 51 (Because 0 is to the left of 51 on a number line)

TO WRITE WHOLE NUMBERS IN WORDS AND IN STANDARD FORM

• Standard Form:

  • A whole number is expressed using digits 0-9. This is the common numeral system we use daily.

  • Each digit’s position determines its place value, contributing to the overall value of the number.

PLACE-VALUE CHART

• The chart displays the first 12 place values, organized into periods (ones, thousands, millions, billions, etc.), with an example:

  • For the number 37,462:

    • The digit 3 is in the ten-thousands place, representing $3 \times 10,000 = 30,000$.

    • The digit 7 is in the thousands place, representing $7 \times 1,000 = 7,000$.

    • The digit 4 is in the hundreds place, representing $4 \times 100 = 400$.

    • The digit 6 is in the tens place, representing $6 \times 10 = 60$.

    • The digit 2 is in the ones place, representing $2 \times 1 = 2$.

  • Each period (group of three digits separated by commas) corresponds to the name of the place value, making large numbers easier to read.

WRITING NUMBERS IN WORDS

• Example:

  • 3,786,451,294 is read as “three billion seven hundred eighty-six million four hundred fifty-one thousand two hundred ninety-four.”

• Writing a whole number in standard form involves replacing period names with commas and using zeros as placeholders when a place value is empty.

  • Four million sixty-two thousand five hundred eighty-four is written as 4,062,584.

  • The zero in this number serves as a placeholder for the hundred-thousands place, indicating that there are no hundred-thousands.

EXAMPLE 2

• Write 25,478,083 in words.

• Solution: “Twenty-five million four hundred seventy-eight thousand eighty-three.”

TO ROUND A WHOLE NUMBER TO A GIVEN PLACE VALUE

• Rounding Basics:

  • When approximating numbers for simplicity or estimation (e.g., stating the distance to the moon as approximately 240,000 miles instead of 238,900 miles), the concept of rounding helps represent numbers more simply.

  • A number is rounded to a specific place value by considering the digit immediately to the right of the target place value.

ROUNDING RULES

• To round a number to a specific place value, look at the digit to its immediate right:

  • If the digit to the right is 5 or greater (5, 6, 7, 8, 9), round up the digit in the target place value and change all digits to its right to zero.

  • If the digit to the right is less than 5 (0, 1, 2, 3, 4), keep the digit in the target place value the same and change all digits to its right to zero.

• Example:

  • 37 is closer to 40 than to 30. When rounding to the nearest ten, the digit in the ones place (7) is 5 or greater, thus yielding 40.

  • 673 rounds to:

    • Nearest ten: The digit in the ones place (3) is less than 5, so the tens digit (7) remains the same, and the ones digit becomes 0. Result: 670.

    • Nearest hundred: The digit in the tens place (7) is 5 or greater, so the hundreds digit (6) is rounded up to 7, and the tens and ones digits become 0. Result: 700.

HOW TO ROUND

• To round 13,834 to the nearest hundred:

  • Identify the hundreds place digit, which is 8.

  • Look at the digit immediately to its right (in the tens place), which is 3.

  • Since 3 is less than 5, the hundreds digit (8) remains the same. All digits to the right (3 and 4) change to zero.

  • Solution: 13,834 rounded to the nearest hundred is 13,800.

EXAMPLE 3

• Round 525,453 to the nearest ten-thousand.

• Solution:

  • Identify the ten-thousands place digit, which is 2.

  • Look at the digit immediately to its right (in the thousands place), which is 5.

  • Since 5 is 5 or greater, the ten-thousands digit (2) is rounded up to 3. All digits to the right (5, 4, 5, 3) change to zero.

  • 525,453 rounded to the nearest ten-thousand is 530,000.

TO ADD WHOLE NUMBERS

• Addition Defined:

  • The process of finding the total, or sum, of two or more numbers (called addends).

  • Example scenario: Maryka carried 4 urns to a service and later carried 3 more urns. To find the total number of urns she carried, we add them: $4 + 3 = 7$ urns.

BASIC ADDITION FACTS

• A table of basic addition facts (sums of single-digit numbers) should be memorized for efficient and accurate addition of larger numbers. Mastery of these facts is a fundamental prerequisite for more complex arithmetic.

PROPERTIES OF ADDITION

1. Commutative Property:

   • Addition can be performed in any order, and the sum will remain the same. The order of the addends does not affect the sum.

     • Example:

     • $4 + 8 = 12$ and $8 + 4 = 12$

     • $9 + 6 = 15$ and $6 + 9 = 15$

2. Associative Property:

   • The way numbers are grouped in addition (using parentheses) does not affect the sum. This property is useful when adding three or more numbers.

     • Example:

     • $(3 + 2) + 4 = 5 + 4 = 9$

     • $3 + (2 + 4) = 3 + 6 = 9$

     • Therefore, $(3 + 2) + 4 = 3 + (2 + 4)$.

3. Addition Property of Zero:

   • Adding zero to any number does not change the number; the number retains its identity. Zero is known as the identity element for addition.

     • Example:

     • $4 + 0 = 4$

     • $0 + 7 = 7$

ADDING LARGE NUMBERS

• To add large numbers, arrange them vertically, aligning similar place values (ones under ones, tens under tens, etc.) in columns. Begin adding from the rightmost column (ones place) and carry over any tens to the next column to the left.

EXAMPLE 4

• Find the total of 17, 103, and 8.

• Solution: ``` 17 103

  • 8
    -----
    ```

  1. Add the digits in the ones column: $7 + 3 + 8 = 18$. Write down 8 in the ones column and carry over 1 to the tens column.

  2. Add the digits in the tens column, including the carried-over 1: $1 (carried) + 1 + 0 = 2$. Write down 2 in the tens column.

  3. Add the digits in the hundreds column: $1$. Write down 1 in the hundreds column.
     The total is 128.

TO SOLVE APPLICATION PROBLEMS

• Reading the Problem: Always read the problem carefully to fully understand what is being asked. Identify the given quantities and keywords that indicate the necessary operations (e.g.,