CHAPTER 1 - WHOLE NUMBERS

OBJECTIVES

• Identify whole numbers.
• Read, write, and round whole numbers correctly.
• Solve problems involving arithmetic operations with whole numbers.
• Determine the least common multiple (LCM) and greatest common factor (GCF)
• Perform order of operations with whole numbers.

CHAPTER OUTLINE

1.1 Understanding Whole Numbers

1.2 Arithmetic Operations with Whole Numbers

1.3 Factors and Multiples

1.4 Order of Operations

1 Review Exercises

1 Self-Test Exercises

Introduction

Arithmetic is the elementary branch of mathematics that we use in everyday life, in such tasks as buying, selling, estimating expenses, and checking bank balances. When we count, we use arithmetic; when we perform the simple operations of addition, subtraction, multiplication, and division, we use principles of arithmetic. Arithmetic is woven into our general interaction with the real world, and as such, it forms the basis of all science, technology, engineering, and business.

Whole numbers are simply the numbers 0, 1, 2, 3, 4,... They include all counting numbers, also known as natural numbers or positive integers (1, 2, 3, 4,...), and zero (0).

All whole numbers are integers. However, whole numbers and integers are not the same because integers include counting numbers (positive integers) and their negatives (negative integers).

In this chapter, we will learn how to perform arithmetic operations with whole numbers, including powers and roots of perfect squares.

1.1 Understanding Whole Numbers

Place Value of Whole Numbers

All numbers can be made up using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Numbers may consist of one or more digits. When a number is written using the above digits, it is said to be in standard form.

For example, 7, 85, and 2,349 are examples of numbers in their standard form, where 7 is a single- (one) digit number, 85 is a two-digit number, and 2,349 is a four-digit number.

The position of each digit in a whole number determines the place value for the digit.

Exhibit 1.1-a illustrates the place value of each of the ten digits in the whole number: 3,867,254,129. In this whole number, 4 occupies the 'thousands' place value and represents 4 thousand (or 4,000), whereas 7 occupies the 'millions' place value and represents 7 million (or 7,000,000).

The place value of 'ones' is 100 ( = 1) and each position has a value of 10 times the place value to its right, as shown in Table 1.1.

Table 1.1: Place Value Chart of Whole Numbers

We read and write numbers from left to right. A number in standard form is separated into groups of three digits using commas. The vertical red lines in Table 1.1 denote the positions of the commas that separate the groups of three digits, starting from the place value for 'ones'.

For example, the ten-digit number in Exhibit 1.1-a is written as 3,867,254,129 in its standard form.

Numbers can also be written in expanded form, by writing the number as the sum of what each place value represents.

For example, the number 3,867,254,129 in standard form can be written in expanded form as follows:

3,000,000,000 + 800,000,000 + 60,000,000 + 7,000,000 + 200,000 + 50,000 + 4,000 + 100 + 20 + 9

Or,

3 billion + 800 million + 60 million + 7 million + 200 thousand + 50 thousand + 4 thousand + 1 hundred + 2 tens + 9 ones

Example 1.1-a: Identifying the Place Value of a Digit and the Amount it Represents

What is the place value of the digit 5 in each of the following numbers and what amount does it represent?

1. $2,543

2. $75,342

3. $6,521,890

4. $915,203,847

Example 1.1-b: Identifying the Digit of a Number Given its Place Value

In the number 5,320,948, identify the digit that occupies the following place values:

1. Hundred thousands

2. Ten thousands

3. Thousands

4. Tens

5. Hundreds

6. Millions

Example 1.1-c: Writing Numbers in Expanded Form

Write the following numbers in expanded form:

1. 698

2. 8,564

3. 49,005

4. 521,076

5. 9,865,323

6. 43,583,621

Reading and Writing Whole Numbers

To make it easier to read and write numbers, any number larger than three digits is separated into smaller groups of three digits, starting from the last digit of the number. Each of these groups of three digits has a name.

• The first group of three digits on the right is the “Units” group.
• The second group from the right is the “Thousands” group.
• The third group from the right is the “Millions” group.
• The fourth group from the right is the “Billions” group.
• The fifth group from the right is the “Trillions” group and so on, as shown in the following chart.

Follow these steps to write large numbers in word form:

1. Start from the group furthest to the left and write the number formed by the digits in that group, followed by the name of the group.
2. Moving to the next group (to the right), write the numbers formed by this next group, followed by its name. Continue to do this for each of the groups.
3. For the last group (i.e., the group furthest to the right), write the numbers formed by the group; however, for this group, do not write the name of it.

Note: When a group contains all zeros, that group is neither read nor written.

Also, commas and hyphens are used when expressing numbers in word form.

• Commas (,) are used between the groups to separate them.
• Hyphens (-) are used to express two-digit numbers in each group;
  i.e., 21 to 29, 31 to 39, 41 to 49,... 91 to 99.

For example, 2,835,197,000,642 expressed in word form using the above rules would be as follows:

When writing numbers in word form, the names of the groups remain in their singular forms, irrespective of the number preceeding; i.e., hundred, thousand, million, billion, trillion, etc.
For example:

Example 1.1-d: Writing Numbers in Word Form Given their Standard Form

Write the following numbers in word form:

1. 7 4 3

2. 5 , 0 0 6

3. 1 5 , 0 1 7

4. 8 0 0 , 6 2 9

5. 6 , 7 8 3 , 2 5 1

6. 5 2 , 6 3 0 , 0 4 2

Example 1.1-e: Writing Numbers In Standard Form Given their Word Form

Write the following numbers in standard form:

1. Two hundred five

2. Six thousand, four

3. Thirty-five thousand, eight hundred twenty-five

4. Eight hundred thousand, five

5. Two million, three hundred forty-two thousand, six hundred seventeen

6. Half of a million

7. One-quarter of a billion

Representing Whole Numbers on a Number Line

Whole numbers can be represented graphically as a point on a horizontal line, called the number line, as shown below

The arrowhead at the end shows that the line continues indefinitely in that direction.

The smallest whole number is zero (0). It is not possible to find the largest whole number because for any given number, there will always be another number greater than that number.

Writing numbers on a number line helps in comparing and identifying numbers that are smaller or larger than other numbers. Numbers that lie to the left of a number on the number line are less than (i.e., smaller than) that number, and numbers that lie to the right of a number on the number line are greater than (i.e., larger than) that number.

• 6 is greater than 2 (or 2 is less than 6).

• 5 is less than 7 (or 7 is greater than 5).

The signs used to show the relative position of two numbers (or quantities) are:

• ‘ > ’ read as “greater than”, meaning that the number on the left of the sign has a value greater than that on the right.
  For example, “6 is greater than 2” is written as 6 > 2.
  This is the same as “2 is less than 6” which is written as 2 < 6.
• ‘ < ’ read as “less than”, meaning that the number on the left of the sign has a value less than that on the right.
  For example, “5 is less than 7” is written as 5 < 7.
  This is the same as “7 is greater than 5” which is written as 7 > 5.

Example 1.1-f: Plotting Numbers on a Number Line and Using Signs to Show the Relative Positions of the Numbers

Plot the following numbers on a number line and place the correct sign of inequality, ‘>’ or ‘<’, in the space between the numbers.

1.   7 <> 11

2.   7 <>   5

3. 11 <>   5

4.   5 <> 12

5.   3 <>   5

6. 12 <> 11

Solution

1. 7 < 11

2. 7 > 5

3. 11 > 5

4. 5 < 12

5. 3 < 5

6. 12 > 11

Example 1.1-g: Writing a Statement to Represent ‘>’ or ‘<’

Write statements using the words “greater than” or “less than” for the following expressions:

1. 24 > 22

2. 36 < 39

3. 9 > 0

4. 0 < 5

Rounding Whole Numbers

Rounding numbers makes them easier to work with and easier to remember. Rounding changes some of the digits in a number but keeps its value close to the original. It is used in reporting large quantities or values that change often, such as population, income, expenses, etc.

For example, the population of Canada is approximately 37 million, or Henry’s car expense for this month is approximately $700.

Rounding numbers also makes arithmetic operations faster and easier, especially when determining the exact answer is not required.

For example, if you are required to estimate the area of a rectangular plot of land that measures 114 m by 97 m, you would have to multiply 114 × 97, which would result in 11,058 m². However, rounding the measurements to the nearest ten can provide a quick estimate.

Rounding Whole Numbers to the Nearest Ten, Hundred, Thousand, etc.

Rounding whole numbers refers to changing the value of the whole number to the nearest ten, hundred, thousand, etc. It is also referred to as rounding whole numbers to a multiple of 10, 100, 1,000, etc.

• Rounding a whole number to the nearest ten is the same as rounding it to a multiple of 10.
• Rounding a whole number to the nearest hundred is the same as rounding it to a multiple of 100.
• Rounding an amount to the nearest $1,000 is the same as rounding the amount to a multiple of $1,000.

For example, rounding the measurements of the above mentioned plot of land to the nearest ten (or multiple of 10):

• Rounding 114 to the nearest ten results in 110.

114 is closer to 110 than 120. Therefore, round down to 110.

• Rounding 97 to the nearest ten results in 100.

97 is closer to 100 than 90. Therefore, round up to 100.

Therefore, rounding the measurements to the nearest ten results in an estimated area of 110 m × 100 m = 11,000 m².

Example 1.1-h: Rounding Numbers Using a Number Line (Visual Method)

Round the following numbers to the indicated place value using a number line:

1. 624 to the nearest ten (multiple of 10).

2. 150 to the nearest hundred (multiple of 100).

3. 1,962 to the nearest hundred (multiple of 100).

Follow these steps to round whole numbers:

1. Identify the digit to be rounded (this is the place value for which the rounding is required).
2. If the digit to the immediate right of the required rounding digit is less than 5 (0, 1, 2, 3, 4), do not change the value of the rounding digit.
   If the digit to the immediate right of the required rounding digit is 5 or greater than 5 (5, 6, 7, 8, 9), increase the value of the rounding digit by one (i.e., round up by one number).
3. Change the value of all digits to the right of the rounding digit to 0.

Example 1.1-i: Rounding to Indicated Place Values

Round the following numbers to the indicated place value using a number line:

1. $568 to the nearest $10.

2. $795 to the nearest $10.

3. $5,643 to the nearest $100.

4. $19,958 to the nearest $100.

1.1 Exercises

For Problems 1 to 4, write (i) the place value of the underlined digit and (ii) the value it represents.

1. a. 4,792
   b. 5,352
   c. 45,721
2. a. 7,628
   b. 4,687
   c. 94,083
3. a. 319,526
   b. 7,825,500
   c. 16,702,555
4. a. 204,095
   b. 35,217,123
   c. 4,385,207

For Problems 5 to 10, write the numbers in their (i) expanded form and (ii) word form.

5. a. 407
   b. 2,056
6. a. 860
   b. 7,805
7. a. 29,186
   b. 464,448
8. a. 94,975
   b. 684,137
9. a. 2,604,325
   b. 15,300,604
10. a. 9,084,351
    b. 23,006,045

For Problems 11 to 16, write the numbers in their (i) standard form and (ii) word form.

11. a. 600 + 70 + 9
    b. 3,000 + 100 + 40 + 7
12. a. 400 + 50 + 6
    b. 1,000 + 900 + 30 + 2
13. a. 2,000 + 600 + 5
    b. 9,000 + 20 + 4
14. a. 5,000 + 300+1
    b. 7,000 + 80 + 8
15. a. 40,000 + 900 + 90
    b. 10,000 + 50 + 3
16. a. 60,000 + 700 + 80
    b. 20,000 + 100 + 4

For Problems 17 to 24, write the numbers in their (i) standard form and (ii) expanded form.

17. a. Five hundred seventy
    b. Eight hundred three
18. a. One thousand, five
    b. Seven thousand, twenty
19. a. Eighty thousand, six hundred thirty
    b. Seventy-five thousand, twenty-five
20. a. Sixty-five thousand, two hundred forty-four
    b. Eight hundred thirty-three thousand, six hundred forty-one
21. a. Twelve million, four hundred fifty-two thousand, eight hundred thirty-two
    b. Thirty-two million, six hundred eighty-four thousand, two hundred fifty-six
22. a. Two billion, one thousand
    b. One billion, twenty-five thousand
23. a. One-eighth of a million
    b. One-quarter of a million
24. a. Half of a billion
    b. One-tenth of a billion

For Problems 25 and 26, plot the numbers on a number line.

25. a. 14, 19, 15, 7
    b. 12, 8, 17, 5
26. a. 18, 9, 6, 11
    b. 4, 10, 7, 16

For Problems 27 and 28, place the correct sign ‘>’ or ‘<’ in the space between the numbers.

27. a. 7 <> 15   b. 19 <> 14   c. 0 <> 5   d. 19 <> 0


28. a. 12 <> 17   b. 8 <> 5   c. 17 <> 0   d. 0 <> 8

For Problems 29 and 30, express the relationship between the numbers using the statements (i) “less than” and (ii) “greater than”.

29. a. 6 < 9 b. 18 > 11 c. 5 < 11 d. 11 > 0
30. a. 4 < 7 b. 16 > 7 c. 10 < 16 d. 0 < 4

For Problems 31 to 34, arrange the numbers in order from least to greatest.

31. a. 87; 108; 99; 103; 96
    b. 159; 141; 108; 139; 167
32. a. 58; 129; 147; 49; 68
    b. 836; 820; 805; 873; 875
33. a. 2,067; 2,040; 2,638; 2,533
    b. 79,487; 79,534; 79,468; 78,812
34. a. 2,668; 2,630; 2,579; 2,759
    b. 68,336; 69,999; 69,067; 68,942

For Problems 35 and 36, create the (i) least and (ii) greatest possible numbers using all the given digits.

35. a. 9, 2, 5
    b. 7, 9, 1, 8
    c. 3, 5, 4, 8
36. a. 6, 1, 7
    b. 9, 4, 8, 5
    c. 4, 7, 2, 6, 5

For Problems 37 and 38, round the numbers to (i) nearest ten, (ii) nearest hundred, and (iii) nearest thousand.

37. Table	Number	Nearest Ten	Nearest Hundred	Nearest Thousant
    a.	524	Blank Cell	Blank Cell	Blank Cell
    b.	1,645	Blank Cell	Blank Cell	Blank Cell
    c.	53,562	Blank Cell	Blank Cell	Blank Cell
    d.	235,358	Blank Cell	Blank Cell	Blank Cell




38. Table	Number	Nearest Ten	Nearest Hundred	Nearest Thousant
    a.	895	Blank Cell	Blank Cell	Blank Cell
    b.	9,157	Blank Cell	Blank Cell	Blank Cell
    c.	25,972	Blank Cell	Blank Cell	Blank Cell
    d.	139,835	Blank Cell	Blank Cell	Blank Cell

For Problems 39 and 40, round the numbers to (i) nearest ten thousand, (ii) nearest hundred thousand, and (iii) nearest million.

39. Table	Number	Nearest Ten Thousand	Nearest Ten Thousand	Nearest Million
    a.	875,555	Blank Cell	Blank Cell	Blank Cell
    b.	1,656,565	Blank Cell	Blank Cell	Blank Cell
    c.	3,368,850	Blank Cell	Blank Cell	Blank Cell
    d.	Blank Cell	Blank Cell	Blank Cell




40. Table	Number	Nearest Ten Thousand	Nearest Ten Thousand	Nearest Million
    a.	759,850	Blank Cell	Blank Cell	Blank Cell
    b.	3,254,599	Blank Cell	Blank Cell	Blank Cell
    c.	7,555,450	Blank Cell	Blank Cell	Blank Cell
    d.	2,959,680	Blank Cell	Blank Cell	Blank Cell

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