Large Numbers to a Million and Beyond

A big number uses the same ten digits you already know. Its digits are 0 through 9. Each digit's place tells you how much it is worth.

Start on the right

Name places from the right, starting with ones. Move left through tens, hundreds, thousands, ten thousands, hundred thousands, and millions. Starting on the right works for numbers of any length.

Look at 352,407. You can put each digit into a place value chart.

The digit in the ten thousands place is 5. Its value is 50,000. A place name tells you where it sits. A value tells you how much it adds to the number.

Read groups of three

Commas split large numbers into periods, or groups of three digits. Make the groups from the right. In 352,407, the thousands period is 352. The ones period is 407. In 1,352,407, a millions period sits to their left.

Each period repeats ones, tens, and hundreds of its own size. The thousands period holds thousands, ten thousands, and hundred thousands. Commas help you see the groups; they do not count as digits.

Give zero its place

Zero holds an empty place. In 352,407, there are no tens. The zero keeps the 4 in hundreds and the 7 in ones. Leaving it out would make 35,247, a different number.

Compare 305,000 and 35,000. The first has 3 hundred thousands. The second has 3 ten thousands. A zero may add no amount itself, but you still need it in the written number.

Reach a million

One million is 1,000,000: a 1 followed by six zeros. It has seven digits. The greatest six-digit number is 999,999. Add one and you reach a million.

Repeated digits can have different values. In 444,444, the values run from 400,000 down to 4.

Imagine packing counters into groups of ten. Ten single counters make one group of ten. Ten groups of ten make one hundred. The same pattern keeps going through every large-number place.

Follow the ten-times pattern

Each place is worth ten times the place just to its right.

• 10 hundreds make 1 thousand.
• 10 thousands make 1 ten thousand.
• 10 ten thousands make 1 hundred thousand.
• 10 hundred thousands make 1 million.

Going right reverses the change. A thousand is one tenth of a ten thousand. You can split one ten thousand into ten equal groups of one thousand.

In 770,000, the first 7 is worth 700,000. The next 7 is worth 70,000. Compare their values:

The left 7 is worth ten times as much. Both digits look the same, but their places give them different values.

Move more than one place

Two places to the left makes a value 100 times as large. That is ten times, then ten times again. Three places makes it 1,000 times as large. For example, 300,000 is 100 times 3,000.

Multiplying a whole number by 10 moves each digit one place left. A zero fills the empty ones place:

Multiplying by 100 moves every digit two places left:

When you divide 630,000 by 10, each digit moves one place right. The result is 63,000.

Count with different-sized units

You can rename a number by counting groups of a chosen size. There are 45 thousands in 45,000. There are also 450 hundreds. Smaller groups mean you need more groups for the same total.

If an example fund has $230,000 in $1,000 bundles, it has 230 bundles. Divide the total by the bundle size to find the count.

A million can be named as 10 hundred thousands, 100 ten thousands, or 1,000 thousands. Be careful with 100 thousands: that makes 100,000. You need ten times that amount to reach a million.

Counting in thousands follows the same idea as counting in tens. You add or subtract the same step each time. The places help you track what changes.

Find the step

The numbers 10, 100, 1,000, 10,000, 100,000, and 1,000,000 are powers of ten. Each is ten times the one before it. Here, the step tells you which place you add to or subtract from.

Count on by 10,000 from 254,318:

Only the ten thousands digit changes in these steps. The last four digits stay 4,318 because you keep adding whole ten thousands.

Cross into the next place

A place can show only digits 0 through 9. When you add one more to a 9, regroup ten of that place into the next place.

For 295,000 plus 10,000, the 9 ten thousands become 10 ten thousands. Those make 1 hundred thousand:

You can cross several places. From 990,000, one more 10,000 reaches 1,000,000. The digits after the comma become zeros.

Count backward across zeros

Counting back by 100 from 403,500 gives 403,400. To take 1,000 from 400,250, you must regroup across the zeros:

Think of 400,250 as 400 thousands and 250 ones. One fewer thousand leaves 399 thousands and 250 ones.

Complete a sequence

In 718,000; 728,000; ___; 748,000, find the gap between the first two numbers. It is 10,000. Add that step again to get 738,000. Check by adding once more to reach 748,000.

Read the step carefully. From 96,450, adding 10,000 gives 106,450. Adding 1,000 would give 97,450, which is a smaller jump.

Adding 100,000 to 900,000 reaches exactly 1,000,000. Starting at 905,600 reaches 1,005,600. Crossing into millions keeps any smaller amount you already had.

You can read a long number in small pieces. Each three-digit group works like a number below one thousand. Its period name tells you how large that piece is.

Read from left to right

Start with the leftmost period. Read its digits, then say its name. Do the same for each period after it. The ones period needs no period name.

For 352,407, read 352 as "three hundred fifty-two." Add "thousand." Then read 407 as "four hundred seven." Together, the name is "three hundred fifty-two thousand, four hundred seven."

Zero holds a place but is not spoken. Read 406,025 as "four hundred six thousand, twenty-five." In 2,000,300, the whole thousands period is zero. Skip it when speaking: "two million, three hundred."

Turn words into digits

Write one group for each period. After the first group, every group must have exactly three digits.

Take "two million, forty thousand, nine."

1. The millions group is 2.
2. The thousands group is forty, written 040.
3. The ones group is nine, written 009.
4. Put commas between the groups: 2,040,009.

The zeros show the empty places. Without them, the 4 and 9 would move into the wrong places.

"Six hundred thousand, thirty" is 600,030. The ones period is 030, even though you say only "thirty." To write "forty thousand, six," use 40 followed by 006: 40,006.

Keep the period names

The number 3,050,000 is "three million, fifty thousand." If you leave out "thousand," you name 3,000,050 instead. Check where each nonzero group sits before you speak.

In American math number names, leave out "and" inside whole numbers. Say "four hundred seven." Use a hyphen for names from twenty-one through ninety-nine, such as "sixty-four." Period names stay singular: "three hundred thousand."

Check the written groups

After every comma, count exactly three digits. You may also see a small space between periods: 1 000 000. It names the same number as 1,000,000.

A number can be written as digits or as parts added together. Splitting it helps you see what each place contributes. It can also make mental math easier.

Show each digit's value

Standard form uses the usual digits: 352,407. Expanded form adds the values of the digits:

There is no tens amount to add. You can leave out a zero part in expanded form. Keep that zero when writing standard form.

You can show how each part is built with multiplication:

Each digit is multiplied by the size of its place. These products add to the same total.

Build the number again

Start with $\mathrm{600,000}+\mathrm{4,000}+30+9$. Put each part in its place. There are 6 hundred thousands, no ten thousands, 4 thousands, no hundreds, 3 tens, and 9 ones. The result is 604,039.

Parts may arrive in a different order. For $8+\mathrm{70,000}+500$ + 200,000, arrange the parts by size first:

Writing 64,039 for the first example would lose the empty ten thousands place. A place value chart gives each part its own slot.

Choose a handy split

Partitioning means splitting a number into parts. The parts need not be single digit values. You can split 352,407 as $\mathrm{350,000}+\mathrm{2,407}$. You can also call it 352 thousands and 407 ones.

Check a split by adding its parts. Both must return the starting number. In $\mathrm{482,000}=\mathrm{400,000}$ + ___ + 2,000, the missing part is 80,000. The hundred thousands and thousands are already filled.

Regroup a crowded place

Sometimes parts give you more than nine of one place. For 12 thousands and 15 hundreds, first find their values:

The 15 hundreds make 1 thousand and 5 hundreds. Add that thousand to the 12 thousands. The result has 13 thousands and 5 hundreds.

A number line shows numbers in order and shows their distances apart. With large numbers, each jump can stand for thousands or more. Start with the labels, not with a guess about the jump size.

Count jumps between marks

Subtract two labeled values to find the distance. Then divide by the number of equal jumps between them.

A line runs from 200,000 to 300,000 in ten equal jumps. The total distance is 100,000, so each jump is:

The marks after 200,000 are 210,000, 220,000, 230,000, and 240,000. So the fourth mark after the start is 240,000.

Count spaces, not marks. Eleven marks have ten spaces between them. You can check a small example with three marks: there are only two jumps.

Use halfway as a landmark

Halfway from 400,000 to 500,000 is 450,000. The gap is 100,000, and half that gap is 50,000. Add 50,000 to the left label.

The same method works on a smaller section. Halfway from 60,000 to 70,000 is 65,000. Halfway from 0 to 1,000,000 is 500,000.

Place a number between marks

Suppose marks count by 10,000. To place 365,000, find 360,000 and 370,000. The number is 5,000 above the lower mark. That is half of the 10,000 gap, so put it halfway between them.

On a line from 300,000 to 400,000, 365,000 lies past halfway. Halfway is 350,000. It is closer to 400,000 than to 300,000.

Estimate an unlabeled point

If a point sits about one quarter of the way from 0 to 1,000,000, estimate 250,000. Four equal parts of one million are 250,000 each. Use "about" when the point's position is only an estimate.

Read both ends before labeling a point. A jump of 10,000 on one line may be 100,000 on another. Values grow as you move right. This also helps you find the two round numbers around a value when you round it.

Large numbers can look alike until you check the right place. Compare their biggest places first. One early difference can decide the whole comparison.

Count digits first

A whole number with more digits is greater. The number 102,345 has six digits. The number 98,765 has five. So 102,345 is greater, even though the smaller number starts with a 9.

Commas do not count as digits. For usual whole-number writing, the leftmost digit is nonzero. Its place tells you the number's size.

Compare matching places

When both numbers have the same number of digits, line them up by place. Matching commas can help. Start at the left and compare one pair of digits at a time.

Compare 538,214 and 538,941:

The first three places match. At hundreds, 2 is less than 9. That decides the comparison:

The tens and ones cannot make up that difference. The biggest amount in tens and ones together is 99. It cannot outweigh an extra hundred.

Write the symbol

The symbol < means "is less than." The symbol > means "is greater than." The open side faces the greater number. The point faces the smaller number.

Read your statement from left to right. "405,000 is greater than 45,000" becomes $\mathrm{405,000}>\mathrm{45,000}$. Use = when both amounts are the same.

Put different forms into digits

Before comparing "three hundred five thousand" with 350,000, write 305,000. Both numbers have six digits. Their hundred thousands match, but their ten thousands are 0 and 5. So 305,000 is less.

For $\mathrm{400,000}+\mathrm{6,000}$ + 20, build 406,020 first. Compare it with 460,020. Their ten thousands differ: 0 is less than 6. A number line would put 406,020 farther left.

Do not count how many large digits a number contains. Each digit matters through its place.

Ordering is comparing more than two numbers. Check which direction you need before you begin. Ascending means least to greatest. Descending means greatest to least.

Sort by size

Put these numbers in ascending order: 89,450; 1,002,300; 809,450; 98,450; 890,450.

Start with digit counts. The two five-digit numbers must come first. Compare their ten thousands digits: 8 is less than 9. That gives 89,450, then 98,450.

Next compare the six-digit numbers, 809,450 and 890,450. Both start with 8. At ten thousands, 0 is less than 9. So 809,450 comes first. The seven-digit number goes last.

The complete order is:

89,450; 98,450; 809,450; 890,450; 1,002,300.

For descending order, reverse that finished list. Check each pair of neighbors. Every number should be greater than the one after it.

Keep the places lined up

The first two digits alone can fool you. You might see 89 in 89,450 and 80 in 809,450. But those groups sit in different places. The six-digit number has hundred thousands; the five-digit number does not.

When several numbers begin alike, keep checking. For example, 642,305 comes before 642,350. Their first four digits match. The tens digit decides: 0 is less than 5.

Make a number from digit cards

Suppose your cards show 0, 3, 5, 7, 8, and 9. Use each card once. To make the greatest number, put the largest digits in the largest places: 987,530.

To make the least six-digit number, begin with the smallest nonzero digit, 3. Put 0 next, then the remaining digits in increasing order. You get 305,789. Starting with 0 would leave a five-digit number.

Find a number inside a gap

The number 455,000 lies between 450,000 and 460,000. Many other whole numbers do too. If you need the greatest whole number below 500,000, step back just one: 499,999.

For an example town list, populations of 45,820, 145,280, and 45,280 in descending order are 145,280; 45,820; 45,280. Keep each full number together when moving it in the list.

Rounding finds a nearby multiple of a chosen place. It helps you work with simpler numbers. If your number is already that multiple, it stays the same.

See the two choices

To round 4,362 to the nearest hundred, find the hundreds on each side: 4,300 and 4,400. The distances are 62 down and 38 up. Since 38 is smaller, round to 4,400.

Exactly halfway, round up. The number 4,350 is equally far from both hundreds, so it rounds to 4,400. The number 4,400 already is a whole hundred and stays 4,400.

Use the digit beside the place

You can make the same decision with digits.

1. Find the place you are rounding to.
2. Look at the digit immediately to its right.
3. For 0 through 4, keep the rounding digit. For 5 through 9, add one to it.
4. Change every digit to the right of the rounding place to zero.

For the nearest ten, look at ones. For the nearest hundred, look at tens. For the nearest thousand, look at hundreds.

Try the same starting number, 728,546, at three places:

The zero endings keep the number at the right size. To the nearest thousand, write 729,000. Writing 729 loses the thousands.

Round up across a 9

For 39,962 to the nearest hundred, the tens digit is 6. Round up. There are 399 whole hundreds before rounding; one more makes 400 hundreds. The result is 40,000.

Go straight from the original number

Do not round in a chain. For 4,449 to the nearest hundred, check the tens digit, 4. The answer is 4,400. If you rounded to tens first, you would get 4,450. Rounding that again would give the wrong hundred, 4,500.

Use "about" when reporting a rounded estimate. The sign ≈ means "is approximately equal to." An exact count and its rounded value usually differ.

The rounding rule stays the same for ten thousands, hundred thousands, and millions. The place you choose decides which digit you check. Larger steps leave less detail in the answer.

Round to ten thousands

For 458,213 to the nearest 10,000, the rounding digit is 5. Look one place right, at the thousands digit, 8. Round the 5 up to 6. Replace the smaller digits with zeros:

The original number lies between 450,000 and 460,000 and is nearer the upper end. Keeping 213 at the end would fail to round to a whole ten thousand.

Round to hundred thousands

Start again with 458,213. This time, the rounding digit is 4. Check the ten thousands digit, 5. That means round up to 500,000.

The two results differ because the rounding places differ. To the nearest 10,000 you keep more detail. Always find the named place before checking its neighbor.

Round to millions

For 1,458,213 to the nearest million, check the hundred thousands digit, 4. Keep the 1 in millions and replace all smaller digits with zeros. The result is 1,000,000.

The number 2,500,000 sits exactly halfway between 2,000,000 and 3,000,000. Halfway rounds up, so its nearest million is 3,000,000.

Carry across several places

Round 996,500 to the nearest 10,000. Check thousands: 6 means round up. The 99 ten thousands become 100 ten thousands:

The new digit at the left comes from regrouping. You have reached the next million.

When zero is nearest

A positive number can round to zero. To the nearest 100,000, compare 38,000 with 0 and 100,000. It is 38,000 from zero but 62,000 from 100,000. So it rounds to 0.

The number 72,000 is closer to 100,000 and rounds up. Zero is a valid rounded result when the chosen step is large compared with the number.

A rounded answer helps when an exact count is not needed. Choose a place that keeps useful detail. When a question names the rounding place, use that place.

Match the situation

A crowd of 48,763 people can be described as "about 49,000." That rounds to the nearest thousand. "Nearly 50,000" gives a rougher picture using ten thousands. If you need the exact number of numbered seats sold, keep the exact count.

Rounding is about closeness. It is not just cutting off smaller digits. For 67,900 to the nearest thousand, choose 68,000. Cutting off 900 would give 67,000, which is farther away.

Work backward from a rounded value

Suppose a whole-number count rounds to 50,000 to the nearest thousand. Find the halfway points on either side.

• Between 49,000 and 50,000, halfway is 49,500.
• Between 50,000 and 51,000, halfway is 50,500.

The first halfway point rounds up to 50,000. The second rounds up to 51,000. So the possible whole numbers run from 49,500 through 50,499.

For an attendance rounded to 30,000 to the nearest ten thousand, use the halfway points 25,000 and 35,000. The possible whole-number counts run from 25,000 through 34,999.

Estimate a sum

Round each number to a useful place before adding. For 389,512 + 207,904, use the nearest ten thousand:

The exact total is 597,416, close to 600,000. A result of 5,974,160 would be much too large. An estimate catches that size error, though a close estimate does not check every digit.

Estimate a difference

For 812,300 minus 296,800, use the nearest hundred thousand:

The exact difference is 515,500. Keep "about" with the estimate so a reader knows which kind of answer you are giving.

The places keep growing beyond one million. The same groups of three help you read the numbers. The same ten-times pattern tells you each digit's value.

Extend the place value chart

The millions period holds millions, ten millions, and hundred millions. After that comes the billions period.

Ten hundred millions make one billion. A billion is also one thousand millions. Check the groups: a thousand has three zeros, a million six, and a billion nine.

Read each period

For 47,305,260, read 47 as "forty-seven million." Read 305 as "three hundred five thousand." Finish with 260, "two hundred sixty."

The full name is "forty-seven million, three hundred five thousand, two hundred sixty." Empty places still matter. The number 4,005,000 is "four million, five thousand." Saying "four million, five" would name 4,000,005.

In 384,000,000, the 8 sits in ten millions. Its value is 80,000,000. Repeating a digit farther left gives it a larger value.

Compare and round as before

The number 12,450,000 has eight digits, while 9,980,000 has seven. So 12,450,000 is greater.

To round 47,305,260 to the nearest million, check hundred thousands, 3. Keep 47 millions and write 47,000,000. For the nearest ten million, check millions, 7. Round up to 50,000,000.

Check number words across languages

In English, billion means 1,000,000,000. French calls that amount "milliard," German "Milliarde," and Dutch "miljard." Their words "billion," "Billion," and "biljoen" mean a million million. Spanish "billón" also means a million million.

Similar-looking words can name different amounts. Looking at the digits lets you check the intended size.

A million and a billion are hard to picture as single piles. Smaller groups and familiar amounts help. A benchmark is an amount you can use to judge another amount.

Build a million from groups

Imagine 1,000 children each bringing 1,000 paper clips. The total is:

A thousand groups of a thousand make a million. A thousand groups of a million make a billion. Those extra three zeros mean 1,000 times as many.

Compare with time

One million seconds lasts about eleven and a half days. More exactly, it is 11 days, 13 hours, 46 minutes, and 40 seconds. Counting one number each second without stopping would take that long to reach one million.

One billion seconds lasts almost 32 years. The step from about eleven and a half days to almost 32 years helps you feel how much larger a billion is.

Build a tall stack

A pack of 500 sheets of printer paper is about 2 inches thick. A million sheets would need 2,000 packs:

At about 2 inches per pack, the stack is about 4,000 inches. That is roughly 330 feet, about the height of a 30-story building. Paper thickness and building heights vary, so this picture uses rounded sizes.

Check a number against its setting

A school often has hundreds of children. A town may have thousands or tens of thousands of people. The biggest cities have millions. The whole world has more than 8 billion people.

If you wrote 1,200,000 children for one ordinary school building, check your zeros. A count of 1,200 is a much more sensible size for a large school. Ask whether the unit and the size match what you are counting.

On a line from 0 to 1,000,000, the number 1,000 is very close to zero. You need 1,000 jumps of that size to reach the end.

Large numbers tell you about people, distances, money, and many other things. Read both the amount and its unit. Then decide whether you need an exact answer or an estimate.

Unpack digits mixed with words

"3 million" means 3,000,000. "450 thousand" means 450,000. "Half a million" is 500,000. A small decimal can stand for part of a million: "2.5 million" means two million plus half a million, or 2,500,000.

For money, "$1.2 million" means $1,200,000. The 0.2 million part is 200,000. It is not just two extra dollars.

Read and round a distance

NASA gives the Moon's average distance from Earth as 238,855 miles. Say "two hundred thirty-eight thousand, eight hundred fifty-five miles."

To the nearest thousand, check hundreds, 8: about 239,000 miles. To the nearest ten thousand, check thousands, 8: about 240,000 miles. Keep "miles" with both answers.

Compare example game crowds

An example stadium has crowds of 68,415, 70,092, and 69,870 at three games. The greatest is 70,092. In descending order, the counts are 70,092; 69,870; 68,415.

To find how many more attended the largest game than the smallest, subtract:

An estimate of 70,000 minus 68,000 gives about 2,000. The exact difference, 1,677 people, is close enough to fit that rough check.

For a quick total, use about 70,000 per game. Three games give about 210,000 visits. The exact sum is 208,377. These are visits across games; a person may attend more than once.

Count up to a target

An example school fund has $345,000 toward a $1,000,000 target. Count up in friendly steps. It needs $55,000 to reach $400,000, then $600,000 more to reach the target.

Write the answer with its meaning: "The fund needs $655,000 more." Check by adding that amount to the money already raised.