Ordering Whole Numbers

Learn to put several whole numbers, up to millions, in ascending and descending order using digit counts and place value.

Elementary20 min18 problems
Practice Problems

Definition

Ordering whole numbers means putting three or more numbers in a row by size.
  • Ascending order goes from least to greatest: 88<975<4,82188 < 975 < 4{,}821
  • Descending order goes from greatest to least: 4,821>975>884{,}821 > 975 > 88
To order numbers, use the same place value method you know from comparing two numbers, just more than once:
  1. 1.Count the digits. A number with more digits is always larger.
  2. 2.Line up the place values of the numbers with the same number of digits, then compare digits from the left.
  3. 3.The first different digit decides. If the digits are equal, move one place to the right.
For example, order 36,45236{,}452, 35,64235{,}642 and 36,52436{,}524:
NumberTen thousandsThousandsHundredsTensOnes
36,45236452
35,64235642
36,52436524
The ten thousands are all 3. In the thousands, 5<65 < 6, so 35,64235{,}642 is the smallest. The other two differ in the hundreds: 4<54 < 5. Ascending order:
35,642<36,452<36,52435{,}642 < 36{,}452 < 36{,}524

Try it now

Which list is in ascending order (from least to greatest)?

Worked Examples

Order from least to greatest: 4,8214{,}821, 975975, 12,00412{,}004, 8888
1

Count the digits of each number

4,8214{,}821 has 4 digits, 975975 has 3, 12,00412{,}004 has 5 and 8888 has 2 → 2, 3, 4 and 5 digits
2

Fewer digits means a smaller number

Every number has a different digit count, so the digit count alone decides the order → 8888 is the smallest, 12,00412{,}004 the largest
3

Write the order with the < sign

From 2 digits up to 5 digits → 88<975<4,821<12,00488 < 975 < 4{,}821 < 12{,}004

Common Mistakes

Ordering by the first digit only, for example putting 912912 after 4,1054{,}105 because 9>49 > 4

Why it's wrong: The first digit only decides when the numbers have the same number of digits. 912912 has 3 digits, 4,1054{,}105 has 4.

Correct: Count the digits first: 912<4,105912 < 4{,}105. Only numbers with the same digit count are compared digit by digit.

Mixing up ascending and descending order

Why it's wrong: The question asked for least to greatest, but the list was written starting with the greatest number.

Correct: Ascending = going up, like climbing stairs: smallest first. Descending = going down: largest first. Check your list with the signs: all < for ascending, all > for descending.

Ignoring a zero inside a number, so that 6,0806{,}080 seems as large as 6,8006{,}800

Why it's wrong: The zero holds the hundreds place. 6,0806{,}080 has 0 hundreds, while 6,8006{,}800 has 8 hundreds.

Correct: Line up the place values and read the zero as a digit: 6,080<6,8006{,}080 < 6{,}800.

Starting with 0 when building the smallest number from digits, for example writing 0358903589

Why it's wrong: A whole number does not start with 0. 0358903589 is really the four-digit number 3,5893{,}589, not a five-digit number.

Correct: Put the smallest digit that is not 0 first, then the 0: the digits 5, 0, 8, 3, 9 make 30,58930{,}589.

Interactive Visual

Place Value Chart

Enter a number:

36,452
Ten Thousands
10,000
Thousands
1,000
Hundreds
100
Tens
10
Ones
1
36452

Expanded Form:

3 × 10,000 + 6 × 1,000 + 4 × 100 + 5 × 10 + 2 × 1

Click on any place value to highlight it and see its value.

to

Click on numbers to select them. Adjust the range to explore different values.

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which list is in ascending order (from least to greatest)?

Why It Matters

Ordering numbers helps you make sense of lists, tables and rankings:
  • Games and sports: A leaderboard puts scores in descending order, so the best score is on top.
  • Geography: Atlases rank mountains by height and cities by population.
  • Shopping: Sorting prices from least to greatest shows the cheapest offer first.
  • Data: Before you can find the middle value of a list (the median), the list must be in order.
If you can order numbers quickly, you can read any ranking and spot the biggest or smallest value at a glance.

Real World Applications

Town Populations

Atlases and city websites rank towns by how many people live there. When the populations start with the same digits, you need place value to rank them.

Example:

Berlin has about 3,782,0003{,}782{,}000 people, Madrid about 3,332,0003{,}332{,}000 and Rome about 2,755,0002{,}755{,}000. All have 7 digits, and the millions decide first: 2,755,000<3,332,000<3,782,0002{,}755{,}000 < 3{,}332{,}000 < 3{,}782{,}000.

1Try It Yourself

Three towns have these populations: Town A has 45,31245{,}312 people, Town B has 45,12345{,}123 and Town C has 45,23145{,}231.

Order the populations from least to greatest.

Step 1: Choose the right answer

Which order goes from least to greatest?

Game Leaderboards

Video games and sports tables show scores in descending order, so the winner is at the top.

Example:

Scores of 12,50012{,}500, 9,8009{,}800 and 12,05012{,}050 go on the leaderboard as 12,500>12,050>9,80012{,}500 > 12{,}050 > 9{,}800.

2Try It Yourself

Four players finished a game with these scores: Ana 98,45098{,}450, Ben 98,40598{,}405, Chloe 89,54089{,}540, David 98,54098{,}540.

Put the scores in descending order for the leaderboard.

Step 1: Choose the right answer

Which order goes from greatest to least?

Digit Card Games

In many classroom and board games you draw digit cards and try to build the largest or smallest possible number.

Example:

With the cards 2, 7 and 4 the largest number is 742742 (digits in descending order) and the smallest is 247247 (digits in ascending order).

3Try It Yourself

You draw the digit cards 6, 0, 2, 9 and 4. You must use every card exactly once.

How much larger is the largest five-digit number than the smallest five-digit number?

Step 1: Write the mathematical expression

Write the subtraction: largest number minus smallest number (no spaces or commas in the numbers):

Key Takeaways

  • 1Ascending order goes from least to greatest (a<b<ca < b < c), descending order from greatest to least (c>b>ac > b > a)
  • 2Count the digits first: a number with more digits is always larger
  • 3For numbers with the same digit count, line up the place values and compare digits from the left
  • 4The first place where the digits differ decides the order; if they are equal, move one place to the right
  • 5A zero inside a number holds its place: 6,080<6,8006{,}080 < 6{,}800
  • 6Largest number from digits: digits in descending order. Smallest: digits in ascending order, but never start with 0

Frequently Asked Questions

Think of stairs. You ascend (go up) when you climb, so ascending order starts small and gets bigger. You descend (go down) when you come back, so descending order starts with the largest number.
Think of stairs. You ascend (go up) when you climb, so ascending order starts small and gets bigger. You descend (go down) when you come back, so descending order starts with the largest number.
Put a sign between each pair of neighbors and check every sign. In 2,310<2,301<3,0122{,}310 < 2{,}301 < 3{,}012 the first sign is wrong, because 2,310>2,3012{,}310 > 2{,}301. Correct: 2,301<2,310<3,0122{,}301 < 2{,}310 < 3{,}012.
The separators group the digits in threes so the number is easier to read: 1,250,0001{,}250{,}000 is one million two hundred fifty thousand. Different countries use commas, dots or small spaces, but the number is the same. When you type an answer here, write only the digits, for example 1250000.
A zero at the front adds nothing: 0472 is just 472472, a three-digit number. To keep all the digits, start with the smallest digit that is not 0 and put the zero right after it: 2,0472{,}047.

Glossary

Ascending order
Numbers arranged from least to greatest, for example 12<45<30012 < 45 < 300
Descending order
Numbers arranged from greatest to least, for example 300>45>12300 > 45 > 12
Place value
The value of a digit based on its position (ones, tens, hundreds, thousands and so on)
Digit
One of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 used to write numbers
Leading digit
The first digit on the left of a number, in its largest place, for example 7 in 7,3067{,}306

Formula Card

Ascending order

a<b<c<da < b < c < d

Least to greatest: every sign is <

Descending order

d>c>b>ad > c > b > a

Greatest to least: every sign is >

Digit count rule

1,000>9991{,}000 > 999

More digits always means a larger whole number, so count the digits first

Today's challengeOne problem, three tries

More in This Topic

Introduction to Place Value

Elementary20 min

Ones, Tens, and Hundreds

Elementary15 min

Expanded Form

Elementary15 min

Comparing Whole Numbers

Elementary15 min