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Introduction to Whole Numbers

Identify counting numbers and 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: $1,2,3,4,5,\ldots$ and so on. These are called the 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.

Counting numbers and whole numbers can be visualized on a number line as shown in .

The point labeled $0$ is called the origin. The points are equally spaced to the right of $0$ and labeled with the counting numbers. When a number is paired with a point, it is called the 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 whole numbers.

We stopped at $5$ when listing the first few counting numbers and whole numbers. We could have written more numbers if they were needed to make the patterns clear.

Example

Try it.

Which of the following are ⓐ counting numbers? ⓑ whole numbers?

$0,\frac{1}{4},3,5.2,15,105$

Solution

ⓐ The counting numbers start at $1,$ so $0$ is not a counting number. The numbers $3,15,\ \text{and}\ 105$ are all counting numbers.
ⓑ Whole numbers are counting numbers and $0.$ The numbers $0,3,15,\ \text{and}\ 105$ are whole numbers.

The numbers $\frac{1}{4}$ and $5.2$ are neither counting numbers nor whole numbers. We will discuss these numbers later.

Model Whole Numbers

Our number system is called a place value system because the value of a digit depends on its position, or place, in a number. The number $537$ has a different value than the number $735.$ Even though they use the same digits, their value is different because of the different placement of the $7$ and the $5.$

Money gives us a familiar model of place value. Suppose a wallet contains three $\text{\$100}$ bills, seven $\text{\$10}$ bills, and four $\text{\$1}$ bills. The amounts are summarized in . How much money is in the wallet?

Find the total value of each kind of bill, and then add to find the total. The wallet contains $\text{\$374}.$

Base-10 blocks provide another way to model place value, as shown in . The blocks can be used to represent hundreds, tens, and ones. Notice that the tens rod is made up of $10$ ones, and the hundreds square is made of $10$ tens, or $100$ ones.

shows the number $138$ modeled with $\text{base-10}$ blocks.

Digit	Place value	Number	Value	Total value
$1$	hundreds	$1$	$100$	$100\$
$3$	tens	$3$	$10$	$30\$
$8$	ones	$8$	$1$	$+\ 8\$
				$\text{Sum = }138\$

Example

Try it.

Use place value notation to find the value of the number modeled by the $\text{base-10}$ blocks shown.

Solution

There are $2$ hundreds squares, which is $200.$

There is $1$ tens rod, which is $10.$

There are $5$ ones blocks, which is $5.$

Digit	Place value	Number	Value	Total value
$2$	hundreds	$2$	$100$	$200\$
$1$	tens	$1$	$10$	$10\$
$5$	ones	$5$	$1$	$+\ 5\$
				$215\$

The $\text{base-10}$ blocks model the number $215.$

Identify the Place Value of a Digit

By looking at money and $\text{base-10}$ 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 periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods.

Just as with the $\text{base-10}$ 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 $5,278,194$ is written in a place value chart.

The digit $5$ is in the millions place. Its value is $5,000,000.$
The digit $2$ is in the hundred thousands place. Its value is $200,000.$
The digit $7$ is in the ten thousands place. Its value is $70,000.$
The digit $8$ is in the thousands place. Its value is $8,000.$
The digit $1$ is in the hundreds place. Its value is $100.$
The digit $9$ is in the tens place. Its value is $90.$
The digit $4$ is in the ones place. Its value is $4.$

Example

Try it.

In the number $63,407,218;$ find the place value of each of the following digits:

ⓐ $7$
ⓑ $0$
ⓒ $1$
ⓓ $6$
ⓔ $3$

Solution

Write the number in a place value chart, starting at the right.

ⓐ The $7$ is in the thousands place.
ⓑ The $0$ is in the ten thousands place.
ⓒ The $1$ is in the tens place.
ⓓ The $6$ is in the ten millions place.
ⓔ The $3$ is in the millions place.

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.

So the number $37,519,248$ is written thirty-seven million, five hundred nineteen thousand, two hundred forty-eight.

Notice that the word and is not used when naming a whole number.

Example

Try it.

Name the number $8,165,432,098,710$ in words.

Solution

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

Putting all of the words together, we write $8,165,432,098,710$ as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

Example

Try it.

A student conducted research and found that the number of mobile phone users in the United States during one month in $2014$ was $327,577,529.$ Name that number in words.

Solution

Identify the periods associated with the number.

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.

Condensed — the full section is in OpenStax Prealgebra 2e.

Use Place Value to Write Whole Numbers

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

Example

Try it.

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

Solution

ⓐ 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.

Put the numbers together, including the commas. The number is $53,401,742.$

ⓑ 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.

The number is $9,246,073,189.$

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.

Example

Try it.

A state budget was about $\text{\$77}$ billion. Write the budget in standard form.

Solution

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.

So the budget was about $\text{\$77,000,000,000.}$

Round Whole Numbers

In $2013,$ the U.S. Census Bureau reported the population of the state of New York as $19,651,127$ people. It might be enough to say that the population is approximately $20$ million. The word approximately means that $20$ million is not the exact population, but is close to the exact value.

The process of approximating a number is called rounding. Numbers are rounded to a specific place value depending on how much accuracy is needed. $20$ million was achieved by rounding to the millions place. Had we rounded to the one hundred thousands place, we would have $19,700,000$ as a result. Had we rounded to the ten thousands place, we would have $19,650,000$ 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 $76$ to the nearest ten. Is $76$ closer to $70$ or $80$ on the number line?

Now consider the number $72.$ Find $72$ in .

How do we round $75$ to the nearest ten. Find $75$ in .

So that everyone rounds the same way in cases like this, mathematicians have agreed to round to the higher number, $80.$ So, $75$ rounded to the nearest ten is $80.$

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 $5,$ round down. If it is greater than or equal to $5,$ round up.

Example

Try it.

Round $843$ to the nearest ten.

Solution

Locate the tens place.
Underline the digit to the right of the tens place.
Since 3 is less than 5, do not change the digit in the tens place.
Replace all digits to the right of the tens place with zeros.
	Rounding 843 to the nearest ten gives 840.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Name a whole number in words.
1. 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.
2. Use commas in the number to separate the periods.
Use place value to write a whole number.
1. Identify the words that indicate periods. (Remember the ones period is never named.)
2. Draw three blanks to indicate the number of places needed in each period.
3. Name the number in each period and place the digits in the correct place value position.
Round a whole number to a specific place value.
1. 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.).
2. Underline the digit to the right of the given place value.
3. 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.
4. Replace all digits to the right of the given place value with zeros.

Introduction to Whole Numbers

Identify Counting Numbers and Whole Numbers

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

Try it.

$0,\frac{2}{3},5,8.1,125$

Solution

ⓐ 5, 125
ⓑ 0, 5, 125

Try it.

$0,\frac{7}{10},3,20.5,300$

Try it.

$0,\frac{4}{9},3.9,50,221$

Solution

ⓐ 50, 221
ⓑ 0, 50, 221

Try it.

$0,\frac{3}{5},10,303,422.6$

Model Whole Numbers

In the following exercises, use place value notation to find the value of the number modeled by the $\text{base-10}$ blocks.

Try it.

Solution

561

Try it.

Solution

407

Try it.

Identify the Place Value of a Digit

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

Try it.

$579,601$

ⓐ 9
ⓑ 6
ⓒ 0
ⓓ 7
ⓔ 5

Solution

ⓐ thousands
ⓑ hundreds
ⓒ tens
ⓓ ten thousands
ⓔ hundred thousands

Try it.

$398,127$

ⓐ 9
ⓑ 3
ⓒ 2
ⓓ 8
ⓔ 7

Try it.

$56,804,379$

ⓐ 8
ⓑ 6
ⓒ 4
ⓓ 7
ⓔ 0

Solution

ⓐ hundred thousands
ⓑ millions
ⓒ thousands
ⓓ tens
ⓔ ten thousands

Try it.

$78,320,465$

ⓐ 8
ⓑ 4
ⓒ 2
ⓓ 6
ⓔ 7

Use Place Value to Name Whole Numbers

In the following exercises, name each number in words.

Try it.

$1,078$

Solution

One thousand, seventy-eight

Try it.

$5,902$

Try it.

$364,510$

Solution

Three hundred sixty-four thousand, five hundred ten

Try it.

$146,023$

Try it.

$5,846,103$

Solution

Five million, eight hundred forty-six thousand, one hundred three

Try it.

$1,458,398$

Try it.

$37,889,005$

Solution

Thirty seven million, eight hundred eighty-nine thousand, five

Try it.

$62,008,465$

Try it.

The height of Mount Rainier is $14,410$ feet.

Solution

Fourteen thousand, four hundred ten

Try it.

The height of Mount Adams is $12,276$ feet.

Try it.

Seventy years is $613,200$ hours.

Solution

Six hundred thirteen thousand, two hundred

Try it.

One year is $525,600$ minutes.

Try it.

The U.S. Census estimate of the population of Miami-Dade county was $2,617,176.$

Solution

Two million, six hundred seventeen thousand, one hundred seventy-six

Try it.

The population of Chicago was $2,718,782.$

Try it.

There are projected to be $23,867,000$ college and university students in the US in five years.

Solution

Twenty three million, eight hundred sixty-seven thousand

Try it.

About twelve years ago there were $20,665,415$ registered automobiles in California.

Try it.

The population of China is expected to reach $1,377,583,156$ in $2016.$

Solution

One billion, three hundred seventy-seven million, five hundred eighty-three thousand, one hundred fifty-six

Try it.

The population of India is estimated at $1,267,401,849$ as of July $1,2014.$

Use Place Value to Write Whole Numbers

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

Try it.

four hundred twelve

Solution

412

Try it.

two hundred fifty-three

Try it.

thirty-five thousand, nine hundred seventy-five

Solution

35,975

Try it.

sixty-one thousand, four hundred fifteen

Try it.

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

Solution

11,044,167

Try it.

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

Try it.

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

Solution

3,226,512,017

Try it.

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

Try it.

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

Solution

7,173,000,000

Try it.

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

Try it.

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

Solution

39,000,000,000,000

Try it.

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

Round Whole Numbers

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

Try it.

Round to the nearest ten:

ⓐ $386$
ⓑ $2,931$

Solution

ⓐ 390
ⓑ 2,930

Try it.

Round to the nearest ten:

ⓐ $792$
ⓑ $5,647$

Try it.

Round to the nearest hundred:

ⓐ $13,748$
ⓑ $391,794$

Solution

ⓐ 13,700
ⓑ 391,800

Try it.

Round to the nearest hundred:

ⓐ $28,166$
ⓑ $481,628$

Try it.

Round to the nearest ten:

ⓐ $1,492$
ⓑ $1,497$

Solution

ⓐ 1,490
ⓑ 1,500

Try it.

Round to the nearest thousand:

ⓐ $2,391$
ⓑ $2,795$

Try it.

Round to the nearest hundred:

ⓐ $63,994$
ⓑ $63,949$

Solution

ⓐ $64,000$
ⓑ $63,900$

Try it.

Round to the nearest thousand:

ⓐ $163,584$
ⓑ $163,246$

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Which of the following are ⓐ counting numbers? ⓑ whole numbers?
$0,\frac{1}{4},3,5.2,15,105$
Жауап
- ⓐ The counting numbers start at $1,$ so $0$ is not a counting number. The numbers $3,15,\ \text{and}\ 105$ are all counting numbers.
- ⓑ Whole numbers are counting numbers and $0.$ The numbers $0,3,15,\ \text{and}\ 105$ are whole numbers.
The numbers $\frac{1}{4}$ and $5.2$ are neither counting numbers nor whole numbers. We will discuss these numbers later.
Which of the following are ⓐ counting numbers ⓑ whole numbers?
$0,\frac{2}{3},2,9,11.8,241,376$
Жауап
- ⓐ 2, 9, 241, 376
- ⓑ 0, 2, 9, 241, 376
Which of the following are ⓐ counting numbers ⓑ whole numbers?
$0,\frac{5}{3},7,8.8,13,201$
Жауап
- ⓐ 7, 13, 201
- ⓑ 0, 7, 13, 201
Use place value notation to find the value of the number modeled by the $\text{base-10}$ blocks shown.

Жауап

There are $2$ hundreds squares, which is $200.$
There is $1$ tens rod, which is $10.$
There are $5$ ones blocks, which is $5.$

Digit Place value Number Value Total value
$2$ hundreds $2$ $100$ $200\$
$1$ tens $1$ $10$ $10\$
$5$ ones $5$ $1$ $+\ 5\$
$215\$
The $\text{base-10}$ blocks model the number $215.$
Use place value notation to find the value of the number modeled by the $\text{base-10}$ blocks shown.

Жауап

176
Use place value notation to find the value of the number modeled by the $\text{base-10}$ blocks shown.

Жауап

237
In the number $63,407,218;$ find the place value of each of the following digits:
1. ⓐ $7$
2. ⓑ $0$
3. ⓒ $1$
4. ⓓ $6$
5. ⓔ $3$
Жауап
Write the number in a place value chart, starting at the right.
- ⓐ The $7$ is in the thousands place.
- ⓑ The $0$ is in the ten thousands place.
- ⓒ The $1$ is in the tens place.
- ⓓ The $6$ is in the ten millions place.
- ⓔ The $3$ is in the millions place.
For each number, find the place value of digits listed: $27,493,615$
1. ⓐ $2$
2. ⓑ $1$
3. ⓒ $4$
4. ⓓ $7$
5. ⓔ $5$
Жауап
- ⓐ ten millions
- ⓑ tens
- ⓒ hundred thousands
- ⓓ millions
- ⓔ ones
For each number, find the place value of digits listed: $519,711,641,328$
1. ⓐ $9$
2. ⓑ $4$
3. ⓒ $2$
4. ⓓ $6$
5. ⓔ $7$
Жауап
- ⓐ billions
- ⓑ ten thousands
- ⓒ tens
- ⓓ hundred thousands
- ⓔ hundred millions

Name the number $8,165,432,098,710$ in words.

Жауап

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

Putting all of the words together, we write $8,165,432,098,710$ as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

Name each number in words: $9,258,137,904,061$

Жауап

nine trillion, two hundred fifty-eight billion, one hundred thirty-seven million, nine hundred four thousand, sixty-one
Name each number in words: $17,864,325,619,004$

Жауап

seventeen trillion, eight hundred sixty-four billion, three hundred twenty-five million, six hundred nineteen thousand, four
A student conducted research and found that the number of mobile phone users in the United States during one month in $2014$ was $327,577,529.$ Name that number in words.

Жауап

Identify the periods associated with the number.

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.
The population in a country is $316,128,839.$ Name that number.

Жауап

three hundred sixteen million, one hundred twenty-eight thousand, eight hundred thirty-nine
One year is $31,536,000$ seconds. Name that number.

Жауап

thirty-one million, five hundred thirty-six thousand
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
Жауап

ⓐ 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.

Put the numbers together, including the commas. The number is $53,401,742.$
ⓑ 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.

The number is $9,246,073,189.$
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.
Write each number in standard form:
fifty-three million, eight hundred nine thousand, fifty-one.

Жауап

53,809,051
Write each number in standard form:
two billion, twenty-two million, seven hundred fourteen thousand, four hundred sixty-six.

Жауап

2,022,714,466
A state budget was about $\text{\$77}$ billion. Write the budget in standard form.

Жауап

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.

So the budget was about $\text{\$77,000,000,000.}$
Write each number in standard form:
The closest distance from Earth to Mars is about $34$ million miles.

Жауап

34,000,000 miles
Write each number in standard form:
The total weight of an aircraft carrier is $204$ million pounds.

Жауап

204,000,000 pounds

Round $843$ to the nearest ten.

Жауап

Locate the tens place.
Underline the digit to the right of the tens place.
Since 3 is less than 5, do not change the digit in the tens place.
Replace all digits to the right of the tens place with zeros.
	Rounding 843 to the nearest ten gives 840.

Round to the nearest ten: $157.$

Жауап

160
Round to the nearest ten: $884.$

Жауап

880

Round each number to the nearest hundred:

ⓐ $23,658$
ⓑ $3,978$

Жауап

ⓐ
Locate the hundreds place.
The digit to the right of the hundreds place is 5. Underline the digit to the right of the hundreds place.
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.	So 23,658 rounded to the nearest hundred is 23,700.

ⓑ
Locate the hundreds place.
Underline the digit to the right of the hundreds place.
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.	So 3,978 rounded to the nearest hundred is 4,000.

Round to the nearest hundred: $17,852.$

Жауап

17,900
Round to the nearest hundred: $4,951.$

Жауап

5,000

Round each number to the nearest thousand:

ⓐ $147,032$
ⓑ $29,504$

Жауап

ⓐ
Locate the thousands place. Underline the digit to the right of the thousands place.
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.
We then replace all digits to the right of the thousands pace with zeros.	So 147,032 rounded to the nearest thousand is 147,000.

ⓑ
Locate the thousands place.
Underline the digit to the right of the thousands place.
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.	So 29,504 rounded to the nearest thousand is 30,000.

Notice that in part ⓑ , when we add $1$ thousand to the $9$ thousands, the total is $10$ thousands. We regroup this as $1$ ten thousand and $0$ thousands. We add the $1$ ten thousand to the $2$ ten thousands and put a $0$ in the thousands place.

Round to the nearest thousand: $63,921.$

Жауап

64,000
Round to the nearest thousand: $156,437.$

Жауап

156,000
$0,\frac{2}{3},5,8.1,125$
Жауап
1. ⓐ 5, 125
2. ⓑ 0, 5, 125
$0,\frac{7}{10},3,20.5,300$
$0,\frac{4}{9},3.9,50,221$
Жауап
1. ⓐ 50, 221
2. ⓑ 0, 50, 221
$0,\frac{3}{5},10,303,422.6$
$579,601$
1. ⓐ 9
2. ⓑ 6
3. ⓒ 0
4. ⓓ 7
5. ⓔ 5
Жауап
1. ⓐ thousands
2. ⓑ hundreds
3. ⓒ tens
4. ⓓ ten thousands
5. ⓔ hundred thousands
$56,804,379$
1. ⓐ 8
2. ⓑ 6
3. ⓒ 4
4. ⓓ 7
5. ⓔ 0
Жауап
1. ⓐ hundred thousands
2. ⓑ millions
3. ⓒ thousands
4. ⓓ tens
5. ⓔ ten thousands
$5,846,103$

Жауап

Five million, eight hundred forty-six thousand, one hundred three
$1,458,398$

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Introduction to Whole Numbers

Identify counting numbers and whole numbers
Model whole numbers
Identify the place value of a digit
Use place value to name whole numbers
Use place value to write whole numbers
Round whole numbers
The digit
The digit

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Өзүңүздүн аракетиңизди көрүңүз

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Кээ бирлери Arithmetic

Order of operations Fractions Percentages Powers and roots

Digit	Place value	Number	Value	Total value
\(1\)	hundreds	\(1\)	\(100\)	\(100\\)
\(3\)	tens	\(3\)	\(10\)	\(30\\)
\(8\)	ones	\(8\)	\(1\)	\(+\ 8\\)
				\(\text{Sum = }138\\)

Digit	Place value	Number	Value	Total value
\(2\)	hundreds	\(2\)	\(100\)	\(200\\)
\(1\)	tens	\(1\)	\(10\)	\(10\\)
\(5\)	ones	\(5\)	\(1\)	\(+\ 5\\)
				\(215\\)

Digit	Place value	Number	Value	Total value
\(2\)	hundreds	\(2\)	\(100\)	\(200\\)
\(1\)	tens	\(1\)	\(10\)	\(10\\)
\(5\)	ones	\(5\)	\(1\)	\(+\ 5\\)
				\(215\\)