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== 1.1#h(0.6em)Introduction to Whole Numbers


=== Identify Counting Numbers and Whole Numbers

Learning algebra is similar to learning a language. You start with a basic vocabulary and then add to it as you go along. You need to practice often until the vocabulary becomes easy to you. The more you use the vocabulary, the more familiar it becomes.

Algebra uses numbers and symbols to represent words and ideas. Let’s look at the numbers first. The most basic numbers used in algebra are those we use to count objects: #math.equation(block: false, alt: "1 , 2 , 3 , 4 , 5 , and so on")[$1 , 2 , 3 , 4 , 5 , …$] and so on. These are called the #strong[counting numbers]. The notation “…” is called an ellipsis, which is another way to show “and so on”, or that the pattern continues endlessly. Counting numbers are also called natural numbers.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Counting Numbers]

The counting numbers start with #math.equation(block: false, alt: "1")[$1$] and continue.

#math.equation(block: true, alt: "1 , 2 , 3 , 4 , 5…")[$1 , 2 , 3 , 4 , "5…"$]
]

Counting numbers and whole numbers can be visualized on a #strong[number line] as shown.

#figure(figph[An image of a number line from 0 to 6 in increments of one. An arrow above the number line pointing to the right with the label “larger”. An arrow pointing to the left with the label “smaller”.], alt: "An image of a number line from 0 to 6 in increments of one. An arrow above the number line pointing to the right with the label “larger”. An arrow pointing to the left with the label “smaller”.", caption: [The numbers on the number line increase from left to right, and decrease from right to left.])

The point labeled #math.equation(block: false, alt: "0")[$0$] is called the #strong[origin]. The points are equally spaced to the right of #math.equation(block: false, alt: "0")[$0$] and labeled with the counting numbers. When a number is paired with a point, it is called the #strong[coordinate] of the point.

The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the #strong[whole numbers].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Whole Numbers]

The whole numbers are the counting numbers and zero.

#math.equation(block: true, alt: "0 , 1 , 2 , 3 , 4 , 5…")[$0 , 1 , 2 , 3 , 4 , "5…"$]
]

We stopped at #math.equation(block: false, alt: "5")[$5$] when listing the first few #strong[counting numbers] and #strong[whole numbers]. We could have written more numbers if they were needed to make the patterns clear.

#examplebox("Example 1")[][
Which of the following are ⓐ counting numbers? ⓑ whole numbers?

#math.equation(block: true, alt: "0 , the fraction 1 over 4 , 3 , 5.2 , 15 , 105")[$0 , frac(1, 4) , 3 , 5.2 , 15 , 105$]

#solutionbox[
- ⓐ The counting numbers start at #math.equation(block: false, alt: "1 ,")[$1 ,$] so #math.equation(block: false, alt: "0")[$0$] is not a counting number. The numbers #math.equation(block: false, alt: "3 , 15 , and 105")[$3 , 15 , #h(0.2em) "and" #h(0.2em) 105$] are all counting numbers.
- ⓑ Whole numbers are counting numbers and #math.equation(block: false, alt: "0 .")[$0 .$] The numbers #math.equation(block: false, alt: "0 , 3 , 15 , and 105")[$0 , 3 , 15 , #h(0.2em) "and" #h(0.2em) 105$] are whole numbers.

The numbers #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] and #math.equation(block: false, alt: "5.2")[$5.2$] are neither counting numbers nor whole numbers. We will discuss these numbers later.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Which of the following are ⓐ counting numbers ⓑ whole numbers?

#math.equation(block: true, alt: "0 , the fraction 2 over 3 , 2 , 9 , 11.8 , 241 , 376")[$0 , frac(2, 3) , 2 , 9 , 11.8 , 241 , 376$]

#solutionbox[
- ⓐ 2, 9, 241, 376
- ⓑ 0, 2, 9, 241, 376
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Which of the following are ⓐ counting numbers ⓑ whole numbers?

#math.equation(block: true, alt: "0 , the fraction 5 over 3 , 7 , 8.8 , 13 , 201")[$0 , frac(5, 3) , 7 , 8.8 , 13 , 201$]

#solutionbox[
- ⓐ 7, 13, 201
- ⓑ 0, 7, 13, 201
]
]


=== Model Whole Numbers

Our number system is called a #strong[place value system] because the value of a digit depends on its position, or place, in a number. The number #math.equation(block: false, alt: "537")[$537$] has a different value than the number #math.equation(block: false, alt: "735 .")[$735 .$] Even though they use the same digits, their value is different because of the different placement of the #math.equation(block: false, alt: "7")[$7$] and the #math.equation(block: false, alt: "5 .")[$5 .$]

Money gives us a familiar model of place value. Suppose a wallet contains three #math.equation(block: false, alt: "$100")[$"$100"$] bills, seven #math.equation(block: false, alt: "$10")[$"$10"$] bills, and four #math.equation(block: false, alt: "$1")[$"$1"$] bills. The amounts are summarized in . How much money is in the wallet?

#figure(figph[An image of three stacks of American currency. First stack from left to right is a stack of 3 \$100 bills, with label “Three \$100 bills, 3 times \$100 equals \$300”. Second stack from left to right is a stack of 7 \$10 bills, with label “Seven \$10 bills, 7 times \$10 equals \$70”. Third stack from left to right is a stack of 4 \$1 bills, with label “Four \$1 bills, 4 times \$1 equals \$4”.], alt: "An image of three stacks of American currency. First stack from left to right is a stack of 3 $100 bills, with label “Three $100 bills, 3 times $100 equals $300”. Second stack from left to right is a stack of 7 $10 bills, with label “Seven $10 bills, 7 times $10 equals $70”. Third stack from left to right is a stack of 4 $1 bills, with label “Four $1 bills, 4 times $1 equals $4”.", caption: none)

Find the total value of each kind of bill, and then add to find the total. The wallet contains #math.equation(block: false, alt: "$374 .")[$"$374" .$]

#figure(figph[An image of “\$300 + \$70 +\$4” where the “3” in “\$300”, the “7” in “\$70”, and the “4” in “\$4” are all in red instead of black like the rest of the expression. Below this expression there is the value “\$374”. An arrow points from the red “3” in the expression to the “3” in “\$374”, an arrow points to the red “7” in the expression to the “7” in “\$374”, and an arrow points from the red “4” in the expression to the “4” in “\$374”.], alt: "An image of “$300 + $70 +$4” where the “3” in “$300”, the “7” in “$70”, and the “4” in “$4” are all in red instead of black like the rest of the expression. Below this expression there is the value “$374”. An arrow points from the red “3” in the expression to the “3” in “$374”, an arrow points to the red “7” in the expression to the “7” in “$374”, and an arrow points from the red “4” in the expression to the “4” in “$374”.", caption: none)

Base-10 blocks provide another way to model place value, as shown. The blocks can be used to represent hundreds, tens, and ones. Notice that the tens rod is made up of #math.equation(block: false, alt: "10")[$10$] ones, and the hundreds square is made of #math.equation(block: false, alt: "10")[$10$] tens, or #math.equation(block: false, alt: "100")[$100$] ones.

#figure(figph[An image with three items. The first item is a single block with the label “A single block represents 1”. The second item is a horizontal rod consisting of 10 blocks, with the label “A rod represents 10”. The third item is a square consisting of 100 blocks, with the label “A square represents 100”. The square is 10 blocks tall and 10 blocks wide.], alt: "An image with three items. The first item is a single block with the label “A single block represents 1”. The second item is a horizontal rod consisting of 10 blocks, with the label “A rod represents 10”. The third item is a square consisting of 100 blocks, with the label “A square represents 100”. The square is 10 blocks tall and 10 blocks wide.", caption: none)

shows the number #math.equation(block: false, alt: "138")[$138$] modeled with #math.equation(block: false, alt: "base-10")[$"base-10"$] blocks.

#figure(figph[An image consisting of three items. The first item is a square of 100 blocks, 10 blocks wide and 10 blocks tall, with the label “1 hundred”. The second item is 3 horizontal rods containing 10 blocks each, with the label “3 tens”. The third item is 8 individual blocks with the label “8 ones”.], alt: "An image consisting of three items. The first item is a square of 100 blocks, 10 blocks wide and 10 blocks tall, with the label “1 hundred”. The second item is 3 horizontal rods containing 10 blocks each, with the label “3 tens”. The third item is 8 individual blocks with the label “8 ones”.", caption: [We use place value notation to show the value of the number #math.equation(block: false, alt: "138 .")[$138 .$]])

#figure(figph[An image of “100 + 30 +8” where the “1” in “100”, the “3” in “30”, and the “8” are all in red instead of black like the rest of the expression. Below this expression there is the value “138”. An arrow points from the red “1” in the expression to the “1” in “138”, an arrow points to the red “3” in the expression to the “3” in “138”, and an arrow points from the red “8” in the expression to the “8” in 138.], alt: "An image of “100 + 30 +8” where the “1” in “100”, the “3” in “30”, and the “8” are all in red instead of black like the rest of the expression. Below this expression there is the value “138”. An arrow points from the red “1” in the expression to the “1” in “138”, an arrow points to the red “3” in the expression to the “3” in “138”, and an arrow points from the red “8” in the expression to the “8” in 138.", caption: none)

#figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([Digit], [Place value], [Number], [Value], [Total value]),
  [#math.equation(block: false, alt: "1")[$1$]], [hundreds], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "100")[$100 #h(1em)$]],
  [#math.equation(block: false, alt: "3")[$3$]], [tens], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "30")[$30 #h(1em)$]],
  [#math.equation(block: false, alt: "8")[$8$]], [ones], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "plus 8")[$+ #h(0.5em) 8 #h(1em)$]],
  [], [], [], [], [#math.equation(block: false, alt: "Sum = 138")[$"Sum = " 138 #h(1em)$]],
))

#examplebox("Example 2")[][
Use place value notation to find the value of the number modeled by the #math.equation(block: false, alt: "base-10")[$"base-10"$] blocks shown.

#figure(figph[An image consisting of three items. The first item is two squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is one horizontal rod containing 10 blocks. The third item is 5 individual blocks.], alt: "An image consisting of three items. The first item is two squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is one horizontal rod containing 10 blocks. The third item is 5 individual blocks.", caption: none)

#solutionbox[
There are #math.equation(block: false, alt: "2")[$2$] hundreds squares, which is #math.equation(block: false, alt: "200 .")[$200 .$]

There is #math.equation(block: false, alt: "1")[$1$] tens rod, which is #math.equation(block: false, alt: "10 .")[$10 .$]

There are #math.equation(block: false, alt: "5")[$5$] ones blocks, which is #math.equation(block: false, alt: "5 .")[$5 .$] #linebreak() 
#figure(figph[An image of “200 + 10 + 5” where the “2” in “200”, the “1” in “10”, and the “5” are all in red instead of black like the rest of the expression. Below this expression there is the value “215”. An arrow points from the red “2” in the expression to the “2” in “215”, an arrow points to the red “1” in the expression to the “1” in “215”, and an arrow points from the red “5” in the expression to the “5” in 215.], alt: "An image of “200 + 10 + 5” where the “2” in “200”, the “1” in “10”, and the “5” are all in red instead of black like the rest of the expression. Below this expression there is the value “215”. An arrow points from the red “2” in the expression to the “2” in “215”, an arrow points to the red “1” in the expression to the “1” in “215”, and an arrow points from the red “5” in the expression to the “5” in 215.", caption: none)

#figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([Digit], [Place value], [Number], [Value], [Total value]),
  [#math.equation(block: false, alt: "2")[$2$]], [hundreds], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "200")[$200 #h(1em)$]],
  [#math.equation(block: false, alt: "1")[$1$]], [tens], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "10")[$10 #h(1em)$]],
  [#math.equation(block: false, alt: "5")[$5$]], [ones], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "plus 5")[$+ #h(0.5em) 5 #h(1em)$]],
  [], [], [], [], [#math.equation(block: false, alt: "215")[$215 #h(1em)$]],
))

The #math.equation(block: false, alt: "base-10")[$"base-10"$] blocks model the number #math.equation(block: false, alt: "215 .")[$215 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use place value notation to find the value of the number modeled by the #math.equation(block: false, alt: "base-10")[$"base-10"$] blocks shown.

#figure(figph[An image consisting of three items. The first item is a square of 100, 10 blocks wide and 10 blocks tall. The second item is 7 horizontal rods containing 10 blocks each. The third item is 6 individual blocks.], alt: "An image consisting of three items. The first item is a square of 100, 10 blocks wide and 10 blocks tall. The second item is 7 horizontal rods containing 10 blocks each. The third item is 6 individual blocks.", caption: none)

#solutionbox[
176
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use place value notation to find the value of the number modeled by the #math.equation(block: false, alt: "base-10")[$"base-10"$] blocks shown.

#figure(figph[An image consisting of three items. The first item is two squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is three horizontal rods containing 10 blocks each. The third item is 7 individual blocks.], alt: "An image consisting of three items. The first item is two squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is three horizontal rods containing 10 blocks each. The third item is 7 individual blocks.", caption: none)

#solutionbox[
237
]
]


=== Identify the Place Value of a Digit

By looking at money and #math.equation(block: false, alt: "base-10")[$"base-10"$] blocks, we saw that each place in a number has a different value. A place value chart is a useful way to summarize this information. The place values are separated into groups of three, called periods. The #strong[periods] are #emph[ones, thousands, millions, billions, trillions], and so on. In a written number, commas separate the periods.

Just as with the #math.equation(block: false, alt: "base-10")[$"base-10"$] blocks, where the value of the tens rod is ten times the value of the ones block and the value of the hundreds square is ten times the tens rod, the value of each place in the place-value chart is ten times the value of the place to the right of it.

shows how the number #math.equation(block: false, alt: "5,278,194")[$5,278,194$] is written in a place value chart.

#figure(figph[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'.], alt: "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'.", caption: none)

- The digit #math.equation(block: false, alt: "5")[$5$] is in the millions place. Its value is #math.equation(block: false, alt: "5,000,000 .")[$5,000,000 .$]
- The digit #math.equation(block: false, alt: "2")[$2$] is in the hundred thousands place. Its value is #math.equation(block: false, alt: "200,000 .")[$200,000 .$]
- The digit #math.equation(block: false, alt: "7")[$7$] is in the ten thousands place. Its value is #math.equation(block: false, alt: "70,000 .")[$70,000 .$]
- The digit #math.equation(block: false, alt: "8")[$8$] is in the thousands place. Its value is #math.equation(block: false, alt: "8,000 .")[$8,000 .$]
- The digit #math.equation(block: false, alt: "1")[$1$] is in the hundreds place. Its value is #math.equation(block: false, alt: "100 .")[$100 .$]
- The digit #math.equation(block: false, alt: "9")[$9$] is in the tens place. Its value is #math.equation(block: false, alt: "90 .")[$90 .$]
- The digit #math.equation(block: false, alt: "4")[$4$] is in the ones place. Its value is #math.equation(block: false, alt: "4 .")[$4 .$]

#examplebox("Example 3")[][
In the number #math.equation(block: false, alt: "63,407,218 ;")[$63,407,218 ;$] find the place value of each of the following digits:

+ ⓐ #math.equation(block: false, alt: "7")[$7$]
+ ⓑ #math.equation(block: false, alt: "0")[$0$]
+ ⓒ #math.equation(block: false, alt: "1")[$1$]
+ ⓓ #math.equation(block: false, alt: "6")[$6$]
+ ⓔ #math.equation(block: false, alt: "3")[$3$]

#solutionbox[
Write the number in a place value chart, starting at the right. #linebreak() 
#figure(figph[A figure titled “Place Values” with fifteen columns and 2 rows, with the colums broken down into five groups of three. The first row has the values “Hundred trillions”, “Ten trillions”, “trillions”, “hundred billions”, “ten billions”, “billions”, “hundred millions”, “ten millions”, “millions”, “hundred thuosands”, “ten thousands”, “thousands”, “hundreds”, “tens”, and “ones”. The first 7 values in the second row are blank. Starting with eighth column, the values are “6”, “3”, “4”, “0”, “7”, “2”, “1” and “8”. The first group is labeled “trillions” and contains the first row values of “Hundred trillions”, “ten trillions”, and “trillions”. The second group is labeled “billions” and contains the first row values of “Hundred billions”, “ten billions”, and “billions”. The third group is labeled “millions” and contains the first row values of “Hundred millions”, “ten millions”, and “millions”. The fourth group is labeled “thousands” and contains the first row values of “Hundred thousands”, “ten thousands”, and “thousands”. The fifth group is labeled “ones” and contains the first row values of “Hundreds”, “tens”, and “ones”.], alt: "A figure titled “Place Values” with fifteen columns and 2 rows, with the colums broken down into five groups of three. The first row has the values “Hundred trillions”, “Ten trillions”, “trillions”, “hundred billions”, “ten billions”, “billions”, “hundred millions”, “ten millions”, “millions”, “hundred thuosands”, “ten thousands”, “thousands”, “hundreds”, “tens”, and “ones”. The first 7 values in the second row are blank. Starting with eighth column, the values are “6”, “3”, “4”, “0”, “7”, “2”, “1” and “8”. The first group is labeled “trillions” and contains the first row values of “Hundred trillions”, “ten trillions”, and “trillions”. The second group is labeled “billions” and contains the first row values of “Hundred billions”, “ten billions”, and “billions”. The third group is labeled “millions” and contains the first row values of “Hundred millions”, “ten millions”, and “millions”. The fourth group is labeled “thousands” and contains the first row values of “Hundred thousands”, “ten thousands”, and “thousands”. The fifth group is labeled “ones” and contains the first row values of “Hundreds”, “tens”, and “ones”.", caption: none)

- ⓐ The #math.equation(block: false, alt: "7")[$7$] is in the thousands place.
- ⓑ The #math.equation(block: false, alt: "0")[$0$] is in the ten thousands place.
- ⓒ The #math.equation(block: false, alt: "1")[$1$] is in the tens place.
- ⓓ The #math.equation(block: false, alt: "6")[$6$] is in the ten millions place.
- ⓔ The #math.equation(block: false, alt: "3")[$3$] is in the millions place.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For each number, find the place value of digits listed: #math.equation(block: false, alt: "27,493,615")[$27,493,615$]

+ ⓐ #math.equation(block: false, alt: "2")[$2$]
+ ⓑ #math.equation(block: false, alt: "1")[$1$]
+ ⓒ #math.equation(block: false, alt: "4")[$4$]
+ ⓓ #math.equation(block: false, alt: "7")[$7$]
+ ⓔ #math.equation(block: false, alt: "5")[$5$]

#solutionbox[
- ⓐ ten millions
- ⓑ tens
- ⓒ hundred thousands
- ⓓ millions
- ⓔ ones
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For each number, find the place value of digits listed: #math.equation(block: false, alt: "519,711,641,328")[$519,711,641,328$]

+ ⓐ #math.equation(block: false, alt: "9")[$9$]
+ ⓑ #math.equation(block: false, alt: "4")[$4$]
+ ⓒ #math.equation(block: false, alt: "2")[$2$]
+ ⓓ #math.equation(block: false, alt: "6")[$6$]
+ ⓔ #math.equation(block: false, alt: "7")[$7$]

#solutionbox[
- ⓐ billions
- ⓑ ten thousands
- ⓒ tens
- ⓓ hundred thousands
- ⓔ hundred millions
]
]


=== Use Place Value to Name Whole Numbers

When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period followed by the name of the period without the ‘s’ at the end. Start with the digit at the left, which has the largest place value. The commas separate the periods, so wherever there is a comma in the number, write a comma between the words. The ones period, which has the smallest place value, is not named.

#figure(figph[An image with three values separated by commas. The first value is “37” and has the label “millions”. The second value is “519” and has the label thousands. The third value is “248” and has the label ones. Underneath, the value “37” has an arrow pointing to “Thirty-seven million”, the value “519” has an arrow pointing to “Five hundred nineteen thousand”, and the value “248” has an arrow pointing to “Two hundred forty-eight”.], alt: "An image with three values separated by commas. The first value is “37” and has the label “millions”. The second value is “519” and has the label thousands. The third value is “248” and has the label ones. Underneath, the value “37” has an arrow pointing to “Thirty-seven million”, the value “519” has an arrow pointing to “Five hundred nineteen thousand”, and the value “248” has an arrow pointing to “Two hundred forty-eight”.", caption: none)

So the number #math.equation(block: false, alt: "37,519,248")[$37,519,248$] is written thirty-seven million, five hundred nineteen thousand, two hundred forty-eight.

Notice that the word #emph[and] is not used when naming a whole number.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Name a whole number in words.]

+ 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.
+ Use commas in the number to separate the periods.
]

#examplebox("Example 4")[][
Name the number #math.equation(block: false, alt: "8,165,432,098,710")[$8,165,432,098,710$] in words.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Begin with the leftmost digit, which is 8. It is in the trillions place.], [eight trillion]),
  [The next period to the right is billions.], [one hundred sixty-five billion],
  [The next period to the right is millions.], [four hundred thirty-two million],
  [The next period to the right is thousands.], [ninety-eight thousand],
  [The rightmost period shows the ones.], [seven hundred ten],
))

#figure(figph[An image with five values separated by commas. The first value is “8” and has the label “trillions”. The second value is “165” and has the label “bilions”. The third value is “432” and has the label “millions”. The fourth value is “098” and has the label “thousands”. The fifth value is “710” and has the label “ones”. Underneath, the value “8” has an arrow pointing to “Eight trillion”, the value “165” has an arrow pointing to “One hundred sixty-five billion”, the value “432” has an arrow pointing to “Four hundred thirty-two million”, the value “098” has an arrow pointing to “Ninety-eight thousand”, and the value “710” has an arrow pointing to “seven hundred ten”.], alt: "An image with five values separated by commas. The first value is “8” and has the label “trillions”. The second value is “165” and has the label “bilions”. The third value is “432” and has the label “millions”. The fourth value is “098” and has the label “thousands”. The fifth value is “710” and has the label “ones”. Underneath, the value “8” has an arrow pointing to “Eight trillion”, the value “165” has an arrow pointing to “One hundred sixty-five billion”, the value “432” has an arrow pointing to “Four hundred thirty-two million”, the value “098” has an arrow pointing to “Ninety-eight thousand”, and the value “710” has an arrow pointing to “seven hundred ten”.", caption: none)

Putting all of the words together, we write #math.equation(block: false, alt: "8,165,432,098,710")[$8,165,432,098,710$] as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Name each number in words: #math.equation(block: false, alt: "9,258,137,904,061")[$9,258,137,904,061$]

#solutionbox[
nine trillion, two hundred fifty-eight billion, one hundred thirty-seven million, nine hundred four thousand, sixty-one
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Name each number in words: #math.equation(block: false, alt: "17,864,325,619,004")[$17,864,325,619,004$]

#solutionbox[
seventeen trillion, eight hundred sixty-four billion, three hundred twenty-five million, six hundred nineteen thousand, four
]
]

#examplebox("Example 5")[][
A student conducted research and found that the number of mobile phone users in the United States during one month in #math.equation(block: false, alt: "2014")[$2014$] was #math.equation(block: false, alt: "327,577,529 .")[$327,577,529 .$] Name that number in words.

#solutionbox[
Identify the periods associated with the number. #linebreak() 
#figure(figph[An image with three values separated by commas. The first value is “327” and has the label “millions”. The second value is “577” and has the label “thousands”. The third value is “529” and has the label “ones”.], alt: "An image with three values separated by commas. The first value is “327” and has the label “millions”. The second value is “577” and has the label “thousands”. The third value is “529” and has the label “ones”.", caption: none)

Name the number in each period, followed by the period name. Put the commas in to separate the periods.

Millions period: three hundred twenty-seven million

Thousands period: five hundred seventy-seven thousand

Ones period: five hundred twenty-nine

So the number of mobile phone users in the Unites States during the month of April was three hundred twenty-seven million, five hundred seventy-seven thousand, five hundred twenty-nine.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The population in a country is #math.equation(block: false, alt: "316,128,839 .")[$316,128,839 .$] Name that number.

#solutionbox[
three hundred sixteen million, one hundred twenty-eight thousand, eight hundred thirty-nine
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
One year is #math.equation(block: false, alt: "31,536,000")[$31,536,000$] seconds. Name that number.

#solutionbox[
thirty-one million, five hundred thirty-six thousand
]
]


=== Use Place Value to Write Whole Numbers

We will now reverse the process and write a number given in words as digits.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Use place value to write a whole number.]

+ Identify the words that indicate periods. (Remember the ones period is never named.)
+ Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.
+ Name the number in each period and place the digits in the correct place value position.
]

#examplebox("Example 6")[][
Write the following numbers using digits.

- ⓐ fifty-three million, four hundred one thousand, seven hundred forty-two
- ⓑ nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine

#solutionbox[
ⓐ Identify the words that indicate periods.

Except for the first period, all other periods must have three places. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.

Then write the digits in each period. #linebreak() 
#figure(figph[An image with three blocks of text pointing to numerical values. The first block of text is “fifty-three million”, has the label “millions”, and points to value 53. The second block of text is “four hundred one thousand”, has the label “thousands”, and points to value 401. The third block of text is “seven hundred forty-two”, has the label “ones”, and points to value 742.], alt: "An image with three blocks of text pointing to numerical values. The first block of text is “fifty-three million”, has the label “millions”, and points to value 53. The second block of text is “four hundred one thousand”, has the label “thousands”, and points to value 401. The third block of text is “seven hundred forty-two”, has the label “ones”, and points to value 742.", caption: none)

Put the numbers together, including the commas. The number is #math.equation(block: false, alt: "53,401,742 .")[$53,401,742 .$]

ⓑ Identify the words that indicate periods.

Except for the first period, all other periods must have three places. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.

Then write the digits in each period. #linebreak() 
#figure(figph[An image with four blocks of text pointing to numerical values. The first block of text is “nine billion”, has the label “billions”, and points to value 9. The second block of text is “two hundred forty-six million”, has the label “millions”, and points to value 246. The third block of text is “seventy-three thousand”, has the label “thousands”, and points to value 742. The fourth block of text is “one hundred eighty-nine”, has the label “ones”, and points to the value 189.], alt: "An image with four blocks of text pointing to numerical values. The first block of text is “nine billion”, has the label “billions”, and points to value 9. The second block of text is “two hundred forty-six million”, has the label “millions”, and points to value 246. The third block of text is “seventy-three thousand”, has the label “thousands”, and points to value 742. The fourth block of text is “one hundred eighty-nine”, has the label “ones”, and points to the value 189.", caption: none)

The number is #math.equation(block: false, alt: "9,246,073,189 .")[$9,246,073,189 .$]

Notice that in part ⓑ , a zero was needed as a place-holder in the hundred thousands place. Be sure to write zeros as needed to make sure that each period, except possibly the first, has three places.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each number in standard form:

fifty-three million, eight hundred nine thousand, fifty-one.

#solutionbox[
53,809,051
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each number in standard form:

two billion, twenty-two million, seven hundred fourteen thousand, four hundred sixty-six.

#solutionbox[
2,022,714,466
]
]

#examplebox("Example 7")[][
A state budget was about #math.equation(block: false, alt: "$77")[$"$77"$] billion. Write the budget in standard form.

#solutionbox[
Identify the periods. In this case, only two digits are given and they are in the billions period. To write the entire number, write zeros for all of the other periods. #linebreak() 
#figure(figph[An image with four blocks of text pointing to numerical values. The first block of text is “77 billion”, has the label “billions”, and points to value “77”. The second block of text is null, has the label “millions”, and points to value “000”. The third block of text is null, has the label “thousands”, and points to value “000”. The fourth block of text is null, has the label “ones”, and points to the value “000”.], alt: "An image with four blocks of text pointing to numerical values. The first block of text is “77 billion”, has the label “billions”, and points to value “77”. The second block of text is null, has the label “millions”, and points to value “000”. The third block of text is null, has the label “thousands”, and points to value “000”. The fourth block of text is null, has the label “ones”, and points to the value “000”.", caption: none)

So the budget was about #math.equation(block: false, alt: "$77,000,000,000.")[$"$77,000,000,000."$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each number in standard form:

The closest distance from Earth to Mars is about #math.equation(block: false, alt: "34")[$34$] million miles.

#solutionbox[
34,000,000 miles
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each number in standard form:

The total weight of an aircraft carrier is #math.equation(block: false, alt: "204")[$204$] million pounds.

#solutionbox[
204,000,000 pounds
]
]


=== Round Whole Numbers

In #math.equation(block: false, alt: "2013 ,")[$2013 ,$] the U.S. Census Bureau reported the population of the state of New York as #math.equation(block: false, alt: "19,651,127")[$19,651,127$] people. It might be enough to say that the population is approximately #math.equation(block: false, alt: "20")[$20$] million. The word #emph[approximately] means that #math.equation(block: false, alt: "20")[$20$] million is not the exact population, but is close to the exact value.

The process of approximating a number is called #strong[rounding]. Numbers are rounded to a specific place value depending on how much accuracy is needed. #math.equation(block: false, alt: "20")[$20$] million was achieved by rounding to the millions place. Had we rounded to the one hundred thousands place, we would have #math.equation(block: false, alt: "19,700,000")[$19,700,000$] as a result. Had we rounded to the ten thousands place, we would have #math.equation(block: false, alt: "19,650,000")[$19,650,000$] as a result, and so on. The place value to which we round to depends on how we need to use the number.

Using the number line can help you visualize and understand the rounding process. Look at the number line in . Suppose we want to round the number #math.equation(block: false, alt: "76")[$76$] to the nearest ten. Is #math.equation(block: false, alt: "76")[$76$] closer to #math.equation(block: false, alt: "70")[$70$] or #math.equation(block: false, alt: "80")[$80$] on the number line?

#figure(figph[An image of a number line from 70 to 80 with increments of one. All the numbers on the number line are black except for 70 and 80 which are red. There is an orange dot at the value “76” on the number line.], alt: "An image of a number line from 70 to 80 with increments of one. All the numbers on the number line are black except for 70 and 80 which are red. There is an orange dot at the value “76” on the number line.", caption: [We can see that #math.equation(block: false, alt: "76")[$76$] is closer to #math.equation(block: false, alt: "80")[$80$] than to #math.equation(block: false, alt: "70 .")[$70 .$] So #math.equation(block: false, alt: "76")[$76$] rounded to the nearest ten is #math.equation(block: false, alt: "80 .")[$80 .$]])

Now consider the number #math.equation(block: false, alt: "72 .")[$72 .$] Find #math.equation(block: false, alt: "72")[$72$] in .

#figure(figph[An image of a number line from 70 to 80 with increments of one. All the numbers on the number line are black except for 70 and 80 which are red. There is an orange dot at the value “72” on the number line.], alt: "An image of a number line from 70 to 80 with increments of one. All the numbers on the number line are black except for 70 and 80 which are red. There is an orange dot at the value “72” on the number line.", caption: [We can see that #math.equation(block: false, alt: "72")[$72$] is closer to #math.equation(block: false, alt: "70 ,")[$70 ,$] so #math.equation(block: false, alt: "72")[$72$] rounded to the nearest ten is #math.equation(block: false, alt: "70 .")[$70 .$]])

How do we round #math.equation(block: false, alt: "75")[$75$] to the nearest ten. Find #math.equation(block: false, alt: "75")[$75$] in .

#figure(figph[An image of a number line from 70 to 80 with increments of one. All the numbers on the number line are black except for 70 and 80 which are red. There is an orange dot at the value “75” on the number line.], alt: "An image of a number line from 70 to 80 with increments of one. All the numbers on the number line are black except for 70 and 80 which are red. There is an orange dot at the value “75” on the number line.", caption: [The number #math.equation(block: false, alt: "75")[$75$] is exactly midway between #math.equation(block: false, alt: "70")[$70$] and #math.equation(block: false, alt: "80 .")[$80 .$]])

So that everyone rounds the same way in cases like this, mathematicians have agreed to round to the higher number, #math.equation(block: false, alt: "80 .")[$80 .$] So, #math.equation(block: false, alt: "75")[$75$] rounded to the nearest ten is #math.equation(block: false, alt: "80 .")[$80 .$]

Now that we have looked at this process on the number line, we can introduce a more general procedure. To round a number to a specific place, look at the number to the right of that place. If the number is less than #math.equation(block: false, alt: "5 ,")[$5 ,$] round down. If it is greater than or equal to #math.equation(block: false, alt: "5 ,")[$5 ,$] round up.

So, for example, to round #math.equation(block: false, alt: "76")[$76$] to the nearest ten, we look at the digit in the ones place.

#figure(figph[An image of value “76”. The text “tens place” is in blue and points to number 7 in “76”. The text “is greater than 5” is in red and points to the number 6 in “76”.], alt: "An image of value “76”. The text “tens place” is in blue and points to number 7 in “76”. The text “is greater than 5” is in red and points to the number 6 in “76”.", caption: none)

The digit in the ones place is a #math.equation(block: false, alt: "6 .")[$6 .$] Because #math.equation(block: false, alt: "6")[$6$] is greater than or equal to #math.equation(block: false, alt: "5 ,")[$5 ,$] we increase the digit in the tens place by one. So the #math.equation(block: false, alt: "7")[$7$] in the tens place becomes an #math.equation(block: false, alt: "8 .")[$8 .$] Now, replace any digits to the right of the #math.equation(block: false, alt: "8")[$8$] with zeros. So, #math.equation(block: false, alt: "76")[$76$] rounds to #math.equation(block: false, alt: "80 .")[$80 .$]

#figure(figph[An image of the value “76”. The “6” in “76” is crossed out and has an arrow pointing to it which says “replace with 0”. The “7” has an arrow pointing to it that says “add 1”. Under the value “76” is the value “80”.], alt: "An image of the value “76”. The “6” in “76” is crossed out and has an arrow pointing to it which says “replace with 0”. The “7” has an arrow pointing to it that says “add 1”. Under the value “76” is the value “80”.", caption: none)

Let’s look again at #strong[rounding] #math.equation(block: false, alt: "72")[$72$] to the nearest #math.equation(block: false, alt: "10 .")[$10 .$] Again, we look to the ones place.

#figure(figph[An image of value “72”. The text “tens place” is in blue and points to number 7 in “72”. The text “is less than 5” is in red and points to the number 2 in “72”.], alt: "An image of value “72”. The text “tens place” is in blue and points to number 7 in “72”. The text “is less than 5” is in red and points to the number 2 in “72”.", caption: none)

The digit in the ones place is #math.equation(block: false, alt: "2 .")[$2 .$] Because #math.equation(block: false, alt: "2")[$2$] is less than #math.equation(block: false, alt: "5 ,")[$5 ,$] we keep the digit in the tens place the same and replace the digits to the right of it with zero. So #math.equation(block: false, alt: "72")[$72$] rounded to the nearest ten is #math.equation(block: false, alt: "70 .")[$70 .$]

#figure(figph[An image of the value “72”. The “2” in “72” is crossed out and has an arrow pointing to it which says “replace with 0”. The “7” has an arrow pointing to it that says “do not add 1”. Under the value “72” is the value “70”.], alt: "An image of the value “72”. The “2” in “72” is crossed out and has an arrow pointing to it which says “replace with 0”. The “7” has an arrow pointing to it that says “do not add 1”. Under the value “72” is the value “70”.", caption: none)

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Round a whole number to a specific place value.]

+ Locate the given place value. All digits to the left of that place value do not change unless the given place value is a 9, in which case it may. (See Step 3.)
+ Underline the digit to the right of the given place value.
+ Determine if this digit is greater than or equal to #math.equation(block: false, alt: "5 .")[$5 .$]
- Yes—add #math.equation(block: false, alt: "1")[$1$] 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.
- No—do not change the digit in the given place value.
+ Replace all digits to the right of the given place value with zeros.
]

#examplebox("Example 8")[][
Round #math.equation(block: false, alt: "843")[$843$] to the nearest ten.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Locate the tens place.], [#figure(figph[The number 843 with the label “tens place” pointed at the 4 in 843.], alt: "The number 843 with the label “tens place” pointed at the 4 in 843.", caption: none)]),
  [Underline the digit to the right of the tens place.], [#figure(figph[The number 843 with the 3 underlined.], alt: "The number 843 with the 3 underlined.", caption: none)],
  [Since 3 is less than 5, do not change the digit in the tens place.], [#figure(figph[The number 843 with the 3 underlined.], alt: "The number 843 with the 3 underlined.", caption: none)],
  [Replace all digits to the right of the tens place with zeros.], [#figure(figph[The number 840 with the 0 underlined.], alt: "The number 840 with the 0 underlined.", caption: none)],
  [], [Rounding 843 to the nearest ten gives 840.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round to the nearest ten: #math.equation(block: false, alt: "157 .")[$157 .$]

#solutionbox[
160
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round to the nearest ten: #math.equation(block: false, alt: "884 .")[$884 .$]

#solutionbox[
880
]
]

#examplebox("Example 9")[][
Round each number to the nearest hundred:

+ ⓐ #math.equation(block: false, alt: "23,658")[$23,658$]
+ ⓑ #math.equation(block: false, alt: "3,978")[$3,978$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [Locate the hundreds place.], [#figure(figph[An arrow points from 'hundreds place' to the digit 6 in the number 23,658, illustrating the concept of place value.], alt: "An arrow points from 'hundreds place' to the digit 6 in the number 23,658, illustrating the concept of place value.", caption: none)],
  [The digit to the right of the hundreds place is 5. Underline the digit to the right of the hundreds place.], [#figure(figph[The number 23,658 is written with a horizontal line drawn under the digit 5.], alt: "The number 23,658 is written with a horizontal line drawn under the digit 5.", caption: none)],
  [Since 5 is greater than or equal to 5, round up by adding 1 to the digit in the hundreds place. Then replace all digits to the right of the hundreds place with zeros.], [#figure(figph[The number 23,658 is written with an arrow pointing to the digit 6 and the instruction to add 1, and an arrow pointing to the digits 58 with the instructions to replace with zeros. The final product is 23,700.], alt: "The number 23,658 is written with an arrow pointing to the digit 6 and the instruction to add 1, and an arrow pointing to the digits 58 with the instructions to replace with zeros. The final product is 23,700.", caption: none)

 #linebreak() So 23,658 rounded to the nearest hundred is 23,700.],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [Locate the hundreds place.], [#figure(figph[An arrow points to the '9' in the number 3,978, with the words 'hundreds place' written above, illustrating the hundreds digit in a four-digit number.], alt: "An arrow points to the '9' in the number 3,978, with the words 'hundreds place' written above, illustrating the hundreds digit in a four-digit number.", caption: none)],
  [Underline the digit to the right of the hundreds place.], [#figure(figph[The number 3,978 is written with a horizontal line drawn underneath the digit 7.], alt: "The number 3,978 is written with a horizontal line drawn underneath the digit 7.", caption: none)],
  [The digit to the right of the hundreds place is 7. Since 7 is greater than or equal to 5, round up by adding 1 to the 9. Then place all digits to the right of the hundreds place with zeros.], [#figure(figph[An illustration of rounding 3,978 to the nearest thousand. The hundreds digit 9 is underlined, indicating that 1 should be added to the thousands place (3+1=4) and the remaining digits replaced with zeros, resulting in 4,000.], alt: "An illustration of rounding 3,978 to the nearest thousand. The hundreds digit 9 is underlined, indicating that 1 should be added to the thousands place (3+1=4) and the remaining digits replaced with zeros, resulting in 4,000.", caption: none)

 #linebreak() So 3,978 rounded to the nearest hundred is 4,000.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round to the nearest hundred: #math.equation(block: false, alt: "17,852 .")[$17,852 .$]

#solutionbox[
17,900
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round to the nearest hundred: #math.equation(block: false, alt: "4,951 .")[$4,951 .$]

#solutionbox[
5,000
]
]

#examplebox("Example 10")[][
Round each number to the nearest thousand:

+ ⓐ #math.equation(block: false, alt: "147,032")[$147,032$]
+ ⓑ #math.equation(block: false, alt: "29,504")[$29,504$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [Locate the thousands place. Underline the digit to the right of the thousands place.], [#figure(figph[The number 147,032 is written with an arrow pointing to the digit 7, with a label indicating that this is the 'thousands place.'], alt: "The number 147,032 is written with an arrow pointing to the digit 7, with a label indicating that this is the 'thousands place.'", caption: none)],
  [The digit to the right of the thousands place is 0. Since 0 is less than 5, we do not change the digit in the thousands place.], [#figure(figph[The number 147,032 is displayed in bold black text on a white background, with the digit '0' in the hundreds place underlined.], alt: "The number 147,032 is displayed in bold black text on a white background, with the digit '0' in the hundreds place underlined.", caption: none)],
  [We then replace all digits to the right of the thousands pace with zeros.], [#figure(figph[The number 147,000 is written.], alt: "The number 147,000 is written.", caption: none)

 #linebreak() So 147,032 rounded to the nearest thousand is 147,000.],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [Locate the thousands place.], [#figure(figph[The number 29,504 is written with an arrow pointing to the digit 9 and a label indicating that this number is in the thousands place.], alt: "The number 29,504 is written with an arrow pointing to the digit 9 and a label indicating that this number is in the thousands place.", caption: none)],
  [Underline the digit to the right of the thousands place.], [#figure(figph[The number 29,504 is written with a horizontal line drawn underneath the digit 5.], alt: "The number 29,504 is written with a horizontal line drawn underneath the digit 5.", caption: none)],
  [The digit to the right of the thousands place is 5. Since 5 is greater than or equal to 5, round up by adding 1 to the 9. Then replace all digits to the right of the thousands place with zeros.], [#figure(figph[An image demonstrating how to round the number 29,504 up to the nearest ten thousand, resulting in 30,000, by adding 1 to the thousands place (9+1=10), writing 0, adding 1 to the ten thousands place, and replacing the last three digits with zeros.], alt: "An image demonstrating how to round the number 29,504 up to the nearest ten thousand, resulting in 30,000, by adding 1 to the thousands place (9+1=10), writing 0, adding 1 to the ten thousands place, and replacing the last three digits with zeros.", caption: none)

 #linebreak() So 29,504 rounded to the nearest thousand is 30,000.],
))

Notice that in part ⓑ , when we add #math.equation(block: false, alt: "1")[$1$] thousand to the #math.equation(block: false, alt: "9")[$9$] thousands, the total is #math.equation(block: false, alt: "10")[$10$] thousands. We regroup this as #math.equation(block: false, alt: "1")[$1$] ten thousand and #math.equation(block: false, alt: "0")[$0$] thousands. We add the #math.equation(block: false, alt: "1")[$1$] ten thousand to the #math.equation(block: false, alt: "2")[$2$] ten thousands and put a #math.equation(block: false, alt: "0")[$0$] in the thousands place.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round to the nearest thousand: #math.equation(block: false, alt: "63,921 .")[$63,921 .$]

#solutionbox[
64,000
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round to the nearest thousand: #math.equation(block: false, alt: "156,437 .")[$156,437 .$]

#solutionbox[
156,000
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://www.openstax.org/l/24detplaceval")[Determine Place Value]
- #link("https://www.openstax.org/l/24numdigword")[Write a Whole Number in Digits from Words]
]


=== Key Concepts

#figure(figph[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'.], alt: "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'.", caption: none)

- #emph[Name a whole number in words.]
+ 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.
+ Use commas in the number to separate the periods.
- #emph[Use place value to write a whole number.]
+ Identify the words that indicate periods. (Remember the ones period is never named.)
+ Draw three blanks to indicate the number of places needed in each period.
+ Name the number in each period and place the digits in the correct place value position.
- #emph[Round a whole number to a specific place value.]
+ Locate the given place value. All digits to the left of that place value do not change unless the given place value is a 9, in which case it may. (See Step 3.).
+ Underline the digit to the right of the given place value.
+ 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.
+ Replace all digits to the right of the given place value with zeros.


==== Practice Makes Perfect

#strong[Identify Counting Numbers and Whole Numbers]

In the following exercises, determine which of the following numbers are ⓐ counting numbers ⓑ whole numbers.

#math.equation(block: true, alt: "0 , the fraction 2 over 3 , 5 , 8.1 , 125")[$0 , frac(2, 3) , 5 , 8.1 , 125$]

#solutionbox[
+ ⓐ 5, 125
+ ⓑ 0, 5, 125
]

#math.equation(block: true, alt: "0 , the fraction 7 over 10 , 3 , 20.5 , 300")[$0 , frac(7, 10) , 3 , 20.5 , 300$]

#math.equation(block: true, alt: "0 , the fraction 4 over 9 , 3.9 , 50 , 221")[$0 , frac(4, 9) , 3.9 , 50 , 221$]

#solutionbox[
+ ⓐ 50, 221
+ ⓑ 0, 50, 221
]

#math.equation(block: true, alt: "0 , the fraction 3 over 5 , 10 , 303 , 422.6")[$0 , frac(3, 5) , 10 , 303 , 422.6$]

#strong[Model Whole Numbers]

In the following exercises, use place value notation to find the value of the number modeled by the #math.equation(block: false, alt: "base-10")[$"base-10"$] blocks.

#figure(figph[An image consisting of three items. The first item is five squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is six horizontal rods containing 10 blocks each. The third item is 1 individual block.], alt: "An image consisting of three items. The first item is five squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is six horizontal rods containing 10 blocks each. The third item is 1 individual block.", caption: none)

#solutionbox[
561
]

#figure(figph[An image consisting of three items. The first item is three squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is eight horizontal rods containing 10 blocks each. The third item is 4 individual blocks.], alt: "An image consisting of three items. The first item is three squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is eight horizontal rods containing 10 blocks each. The third item is 4 individual blocks.", caption: none)

#figure(figph[An image consisting of two items. The first item is four squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is 7 individual blocks.], alt: "An image consisting of two items. The first item is four squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is 7 individual blocks.", caption: none)

#solutionbox[
407
]

#figure(figph[An image consisting of two items. The first item is six squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is 2 horizontal rods with 10 blocks each.], alt: "An image consisting of two items. The first item is six squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is 2 horizontal rods with 10 blocks each.", caption: none)

#strong[Identify the Place Value of a Digit]

In the following exercises, find the place value of the given digits.

#math.equation(block: true, alt: "579,601")[$579,601$]

+ ⓐ 9
+ ⓑ 6
+ ⓒ 0
+ ⓓ 7
+ ⓔ 5

#solutionbox[
+ ⓐ thousands
+ ⓑ hundreds
+ ⓒ tens
+ ⓓ ten thousands
+ ⓔ hundred thousands
]

#math.equation(block: true, alt: "398,127")[$398,127$]

+ ⓐ 9
+ ⓑ 3
+ ⓒ 2
+ ⓓ 8
+ ⓔ 7

#math.equation(block: true, alt: "56,804,379")[$56,804,379$]

+ ⓐ 8
+ ⓑ 6
+ ⓒ 4
+ ⓓ 7
+ ⓔ 0

#solutionbox[
+ ⓐ hundred thousands
+ ⓑ millions
+ ⓒ thousands
+ ⓓ tens
+ ⓔ ten thousands
]

#math.equation(block: true, alt: "78,320,465")[$78,320,465$]

+ ⓐ 8
+ ⓑ 4
+ ⓒ 2
+ ⓓ 6
+ ⓔ 7

#strong[Use Place Value to Name Whole Numbers]

In the following exercises, name each number in words.

#math.equation(block: true, alt: "1,078")[$1,078$]

#solutionbox[
One thousand, seventy-eight
]

#math.equation(block: true, alt: "5,902")[$5,902$]

#math.equation(block: true, alt: "364,510")[$364,510$]

#solutionbox[
Three hundred sixty-four thousand, five hundred ten
]

#math.equation(block: true, alt: "146,023")[$146,023$]

#math.equation(block: true, alt: "5,846,103")[$5,846,103$]

#solutionbox[
Five million, eight hundred forty-six thousand, one hundred three
]

#math.equation(block: true, alt: "1,458,398")[$1,458,398$]

#math.equation(block: true, alt: "37,889,005")[$37,889,005$]

#solutionbox[
Thirty seven million, eight hundred eighty-nine thousand, five
]

#math.equation(block: true, alt: "62,008,465")[$62,008,465$]

The height of Mount Rainier is #math.equation(block: false, alt: "14,410")[$14,410$] feet.

#solutionbox[
Fourteen thousand, four hundred ten
]

The height of Mount Adams is #math.equation(block: false, alt: "12,276")[$12,276$] feet.

Seventy years is #math.equation(block: false, alt: "613,200")[$613,200$] hours.

#solutionbox[
Six hundred thirteen thousand, two hundred
]

One year is #math.equation(block: false, alt: "525,600")[$525,600$] minutes.

The U.S. Census estimate of the population of Miami-Dade county was #math.equation(block: false, alt: "2,617,176 .")[$2,617,176 .$]

#solutionbox[
Two million, six hundred seventeen thousand, one hundred seventy-six
]

The population of Chicago was #math.equation(block: false, alt: "2,718,782 .")[$2,718,782 .$]

There are projected to be #math.equation(block: false, alt: "23,867,000")[$23,867,000$] college and university students in the US in five years.

#solutionbox[
Twenty three million, eight hundred sixty-seven thousand
]

About twelve years ago there were #math.equation(block: false, alt: "20,665,415")[$20,665,415$] registered automobiles in California.

The population of China is expected to reach #math.equation(block: false, alt: "1,377,583,156")[$1,377,583,156$] in #math.equation(block: false, alt: "2016 .")[$2016 .$]

#solutionbox[
One billion, three hundred seventy-seven million, five hundred eighty-three thousand, one hundred fifty-six
]

The population of India is estimated at #math.equation(block: false, alt: "1,267,401,849")[$1,267,401,849$] as of July #math.equation(block: false, alt: "1 , 2014 .")[$1 , 2014 .$]

#strong[Use Place Value to Write Whole Numbers]

In the following exercises, write each number as a whole number using digits.

four hundred twelve

#solutionbox[
412
]

two hundred fifty-three

thirty-five thousand, nine hundred seventy-five

#solutionbox[
35,975
]

sixty-one thousand, four hundred fifteen

eleven million, forty-four thousand, one hundred sixty-seven

#solutionbox[
11,044,167
]

eighteen million, one hundred two thousand, seven hundred eighty-three

three billion, two hundred twenty-six million, five hundred twelve thousand, seventeen

#solutionbox[
3,226,512,017
]

eleven billion, four hundred seventy-one million, thirty-six thousand, one hundred six

The population of the world was estimated to be seven billion, one hundred seventy-three million people.

#solutionbox[
7,173,000,000
]

The age of the solar system is estimated to be four billion, five hundred sixty-eight million years.

Lake Tahoe has a capacity of thirty-nine trillion gallons of water.

#solutionbox[
39,000,000,000,000
]

The federal government budget was three trillion, five hundred billion dollars.

#strong[Round Whole Numbers]

In the following exercises, round to the indicated place value.

Round to the nearest ten:

+ ⓐ #math.equation(block: false, alt: "386")[$386$]
+ ⓑ #math.equation(block: false, alt: "2,931")[$2,931$]

#solutionbox[
+ ⓐ 390
+ ⓑ 2,930
]

Round to the nearest ten:

+ ⓐ #math.equation(block: false, alt: "792")[$792$]
+ ⓑ #math.equation(block: false, alt: "5,647")[$5,647$]

Round to the nearest hundred:

+ ⓐ #math.equation(block: false, alt: "13,748")[$13,748$]
+ ⓑ #math.equation(block: false, alt: "391,794")[$391,794$]

#solutionbox[
+ ⓐ 13,700
+ ⓑ 391,800
]

Round to the nearest hundred:

+ ⓐ #math.equation(block: false, alt: "28,166")[$28,166$]
+ ⓑ #math.equation(block: false, alt: "481,628")[$481,628$]

Round to the nearest ten:

+ ⓐ #math.equation(block: false, alt: "1,492")[$1,492$]
+ ⓑ #math.equation(block: false, alt: "1,497")[$1,497$]

#solutionbox[
+ ⓐ 1,490
+ ⓑ 1,500
]

Round to the nearest thousand:

+ ⓐ #math.equation(block: false, alt: "2,391")[$2,391$]
+ ⓑ #math.equation(block: false, alt: "2,795")[$2,795$]

Round to the nearest hundred:

+ ⓐ #math.equation(block: false, alt: "63,994")[$63,994$]
+ ⓑ #math.equation(block: false, alt: "63,949")[$63,949$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "64,000")[$64,000$]
+ ⓑ #math.equation(block: false, alt: "63,900")[$63,900$]
]

Round to the nearest thousand:

+ ⓐ #math.equation(block: false, alt: "163,584")[$163,584$]
+ ⓑ #math.equation(block: false, alt: "163,246")[$163,246$]


==== Everyday Math

#strong[Writing a Check] Jorge bought a car for #math.equation(block: false, alt: "$24,493 .")[$"$24,493" .$] He paid for the car with a check. Write the purchase price in words.

#solutionbox[
Twenty four thousand, four hundred ninety-three dollars
]

#strong[Writing a Check] Marissa’s kitchen remodeling cost #math.equation(block: false, alt: "$18,549 .")[$"$18,549" .$] She wrote a check to the contractor. Write the amount paid in words.

#strong[Buying a Car] Jorge bought a car for #math.equation(block: false, alt: "$24,493 .")[$"$24,493" .$] Round the price to the nearest:

+ ⓐ ten dollars
+ ⓑ hundred dollars
+ ⓒ thousand dollars
+ ⓓ ten-thousand dollars

#solutionbox[
+ ⓐ \$24,490
+ ⓑ \$24,500
+ ⓒ \$24,000
+ ⓓ \$20,000
]

#strong[Remodeling a Kitchen] Marissa’s kitchen remodeling cost #math.equation(block: false, alt: "$18,549 .")[$"$18,549" .$] Round the cost to the nearest:

+ ⓐ ten dollars
+ ⓑ hundred dollars
+ ⓒ thousand dollars
+ ⓓ ten-thousand dollars

#strong[Population] The population of China was #math.equation(block: false, alt: "1,355,692,544")[$1,355,692,544$] in #math.equation(block: false, alt: "2014 .")[$2014 .$] Round the population to the nearest:

+ ⓐ billion people
+ ⓑ hundred-million people
+ ⓒ million people

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "1,000,000,000")[$1,000,000,000$]
+ ⓑ #math.equation(block: false, alt: "1,400,000,000")[$1,400,000,000$]
+ ⓒ #math.equation(block: false, alt: "1,356,000,000")[$1,356,000,000$]
]

#strong[Astronomy] The average distance between Earth and the sun is #math.equation(block: false, alt: "149,597,888")[$149,597,888$] kilometers. Round the distance to the nearest:

+ ⓐ hundred-million kilometers
+ ⓑ ten-million kilometers
+ ⓒ million kilometers


==== Writing Exercises

In your own words, explain the difference between the counting numbers and the whole numbers.

#solutionbox[
Answers may vary. The whole numbers are the counting numbers with the inclusion of zero.
]

Give an example from your everyday life where it helps to round numbers.


==== Self Check

ⓐ After completing the exercises, use this checklist to
evaluate your mastery of the objectives of this section.

#figure(figph[A self-assessment chart for students to rate their understanding of whole numbers, place value, and rounding skills with options: Confidently, With some help, or No-I don't get it!.], alt: "A self-assessment chart for students to rate their understanding of whole numbers, place value, and rounding skills with options: Confidently, With some help, or No-I don't get it!.", caption: none)

ⓑ If most of your checks were...

…confidently. Congratulations! You have achieved the
objectives in this section. Reflect on the study skills you
used so that you can continue to use them. What did you do
to become confident of your ability to do these things? Be
specific.

…with some help. This must be addressed quickly because
topics you do not master become potholes in your road to
success. In math, every topic builds upon previous work.
It is important to make sure you have a strong foundation
before you move on. Whom can you ask for help? Your
fellow classmates and instructor are good resources. Is
there a place on campus where math tutors are available?
Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must
not ignore it. You should get help right away or you will
quickly be overwhelmed. See your instructor as soon as you
can to discuss your situation. Together you can come up
with a plan to get you the help you need.