1.1 Understanding Whole Numbers
1.2 Arithmetic Operations with Whole Numbers
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.
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.
10^{9} | 10^{8} | 10^{7} | 10^{6} | 10^{5} | 10^{4} | 10^{3} | 10^{2} | 10^{1} | 10^{0} |
1,000,000,000 | 100,000,000 | 10,000,000 | 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 |
Billions | Hundred millions |
Ten millions |
Millions | Hundred thousands |
Ten thousands |
Thousands | Hundreds | Tens | Ones |
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.
3 | 8 | 6 | 7 | 2 | 5 | 5 | 1 | 2 | 9 |
Billions | Hundred millions |
Ten millions |
Millions | Hundred thousands |
Ten thousands |
Thousands | Hundreds | Tens | Ones |
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
What is the place value of the digit 5 in each of the following numbers and what amount does it represent?
$2,543
$75,342
$6,521,890
$915,203,847
$2,543
Place value of the digit 5: Hundreds.
Amount it represents: $500
$75,342
Place value of the digit 5: Thousands.
Amount it represents: $5,000
$6,521,890
Place value of the digit 5: Hundred thousands.
Amount it represents: $500,000
Place value of the digit 5: Millions.
Amount it represents: $5,000,000
In the number 5,320,948, identify the digit that occupies the following place values:
Hundred thousands
Ten thousands
Thousands
Tens
Hundreds
Millions
5,320,948 3 Hundred thousands
5,320,948 2 Ten thousands
5,320,948 0 Thousands
5,320,348 4 Tens
5,320,948 9 Hundreds
5,320,948 5 Millions
Write the following numbers in expanded form:
698
8,564
49,005
521,076
9,865,323
43,583,621
698
600 + 90 + 8
8,564
8,000 + 500 + 60 + 4
49,005
40,000 + 9,000 + 5
521,076
500,000 + 20,000 + 1,000 + 70 + 6
9,865,323
9,000,000 + 800,000 + 60,000 + 5,000 + 300 + 20 + 3
43,583,621
40,000,000 + 3,000,000 + 500,000 + 80,000 + 3,000 + 600 + 20 + 1
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.
Follow these steps to write large numbers in word form:
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.
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:
Write the following numbers in word form:
7 4 3
5 , 0 0 6
1 5 , 0 1 7
8 0 0 , 6 2 9
6 , 7 8 3 , 2 5 1
5 2 , 6 3 0 , 0 4 2
7 4 3
Seven hundred forty-three
5 , 0 0 6
Five thousand, six
1 5 , 0 1 7
Fifteen thousand, seventeen
8 0 0 , 6 2 9
Eight hundred thousand, six hundred twenty-nine
6 , 7 8 3 , 2 5 1
Six million, seven hundred eighty-three thousand, two hundred fifty-one
5 2 , 6 3 0 , 0 4 2
Fifty-two million, six hundred thirty thousand, forty-two
Write the following numbers in standard form:
Two hundred five
Six thousand, four
Thirty-five thousand, eight hundred twenty-five
Eight hundred thousand, five
Two million, three hundred forty-two thousand, six hundred seventeen
Half of a million
One-quarter of a billion
Two hundred five
205
Six thousand, four
6,004
Thirty-five thousand, eight hundred twenty-five
35,825
Eight hundred thousand, five
800,005
Two million, three hundred forty-two thousand, six hundred seventeen
2,342,617
Half of a million
\(\displaystyle{\frac{1,000,000}{2} = 500,000}\)
One-quarter of a billion
\(\displaystyle{\frac{1,000,000,000}{4} = 250,000,000}\)
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:
Plot the following numbers on a number line and place the correct sign of inequality, ‘>’ or ‘<’, in the space between the numbers.
7 11
7 5
11 5
5 12
3 5
12 11
7 < 11
7 > 5
11 > 5
5 < 12
3 < 5
12 > 11
Write statements using the words “greater than” or “less than” for the following expressions:
24 > 22
36 < 39
9 > 0
0 < 5
24 > 22
24 is greater than 22, or 22 is less than 24.
36 < 39
36 is less than 39, or 39 is greater than 36.
9 > 0
9 is greater than 0, or 0 is less than 9.
0 < 5
0 is less than 5, or 5 is greater than 0.
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 m^{2}. However, rounding the measurements to the nearest ten can provide a quick estimate.
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.
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 m^{2}.
Round the following numbers to the indicated place value using a number line:
624 to the nearest ten (multiple of 10).
150 to the nearest hundred (multiple of 100).
1,962 to the nearest hundred (multiple of 100).
We can visualize these numbers on a number line to determine the nearest number to round to:
624 to the nearest ten (multiple of 10)
624 is closer to 620 than it is to 630
Therefore, 624 rounded to the nearest ten is 620.
150 to the nearest hundred (multiple of 100)
150 is exactly midway between 100 and 200. By convention, if a number is exactly in the middle, we round up.
Therefore, 150 rounded to the nearest hundred is 200.
1,962 to the nearest hundred (multiple of 100)
1,962 is closer to 2,000 than it is to 1,900.
Therefore, 1,962 rounded to the nearest hundred is 2,000.
Follow these steps to round whole numbers:
Round the following numbers to the indicated place value using a number line:
$568 to the nearest $10.
$795 to the nearest $10.
$5,643 to the nearest $100.
$19,958 to the nearest $100.
For Problems 1 to 4, write (i) the place value of the underlined digit and (ii) the value it represents.
For Problems 5 to 10, write the numbers in their (i) expanded form and (ii) word form.
For Problems 11 to 16, write the numbers in their (i) standard form and (ii) word form.
For Problems 17 to 24, write the numbers in their (i) standard form and (ii) expanded form.
For Problems 25 and 26, plot the numbers on a number line.
For Problems 27 and 28, place the correct sign ‘>’ or ‘<’ in the space between the numbers.
For Problems 29 and 30, express the relationship between the numbers using the statements (i) “less than” and (ii) “greater than”.
For Problems 31 to 34, arrange the numbers in order from least to greatest.
For Problems 35 and 36, create the (i) least and (ii) greatest possible numbers using all the given digits.
For Problems 37 and 38, round the numbers to (i) nearest ten, (ii) nearest hundred, and (iii) nearest thousand.
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 |
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.
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 |
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 |