Numbers with Different Digit Counts

Order from least to greatest: $4{,}821$, $975$, $12{,}004$, $88$

Count the digits of each number: $4{,}821$ has 4 digits, $975$ has 3, $12{,}004$ has 5 and $88$ has 2 = 2, 3, 4 and 5 digits

Fewer digits means a smaller number: Every number has a different digit count, so the digit count alone decides the order = $88$ is the smallest, $12{,}004$ the largest

Write the order with the < sign: From 2 digits up to 5 digits = $88 < 975 < 4{,}821 < 12{,}004$

Answer: $88 < 975 < 4{,}821 < 12{,}004$

Same Digit Count: Line Up the Place Values

Order from greatest to least: $36{,}452$, $36{,}524$, $35{,}642$, $36{,}425$

Count the digits: All four numbers have 5 digits = Compare place by place

Compare the ten thousands and thousands: All have 3 ten thousands. Thousands: 6, 6, 5, 6 = $35{,}642$ is the smallest

Compare the hundreds of the other three: $36{,}452$ has 4, $36{,}524$ has 5, $36{,}425$ has 4 = $36{,}524$ is the largest

Compare the tens of the two numbers left: $36{,}452$ has 5 tens, $36{,}425$ has 2 tens, and $5 > 2$ = $36{,}452 > 36{,}425$

Write the order with the > sign: Greatest first, least last = $36{,}524 > 36{,}452 > 36{,}425 > 35{,}642$

Answer: $36{,}524 > 36{,}452 > 36{,}425 > 35{,}642$

Zeros Inside Numbers

Order from least to greatest: $7{,}060$, $7{,}600$, $7{,}006$, $7{,}606$

Check the digit count and the thousands: All have 4 digits and 7 thousands = Look at the hundreds

Compare the hundreds: $7{,}060$ and $7{,}006$ have 0 hundreds, $7{,}600$ and $7{,}606$ have 6 hundreds = The two numbers with 0 hundreds come first

Order each pair: $7{,}006$ has 0 tens, $7{,}060$ has 6 tens. $7{,}600$ has 0 ones, $7{,}606$ has 6 ones = $7{,}006 < 7{,}060$ and $7{,}600 < 7{,}606$

Write the whole order: Pair with 0 hundreds, then pair with 6 hundreds = $7{,}006 < 7{,}060 < 7{,}600 < 7{,}606$

Answer: $7{,}006 < 7{,}060 < 7{,}600 < 7{,}606$. A zero holds its place: it means none of that place value, not that the digit is missing.

Ranking Mountains by Height

Order these mountains from highest to lowest: Kilimanjaro $5{,}895$ m, Matterhorn $4{,}478$ m, Elbrus $5{,}642$ m, Denali $6{,}190$ m.

Count the digits: All four heights have 4 digits = Compare the thousands

Compare the thousands: Denali 6, Kilimanjaro 5, Elbrus 5, Matterhorn 4 = Denali is highest, Matterhorn lowest

Compare the hundreds of the two with 5 thousands: Kilimanjaro has 8 hundreds, Elbrus has 6 hundreds = $5{,}895 > 5{,}642$

Write the ranking: Highest first = $6{,}190 > 5{,}895 > 5{,}642 > 4{,}478$

Answer: Denali ($6{,}190$ m), Kilimanjaro ($5{,}895$ m), Elbrus ($5{,}642$ m), Matterhorn ($4{,}478$ m)

Largest and Smallest Number from Digits

Use each of the digits 5, 0, 8, 3 and 1 exactly once. Make the largest and the smallest five-digit number.

Largest: put the digits in descending order: The biggest digit goes to the biggest place: 8, 5, 3, 1, 0 = $85{,}310$

Smallest: put the digits in ascending order: 0, 1, 3, 5, 8 would start with 0, which gives only a four-digit number = 0 cannot be first

Start with the smallest digit that is not 0: 1 goes first, then the rest in ascending order: 0, 3, 5, 8 = $10{,}358$

Check: Both numbers have 5 digits and use every digit once = $85{,}310 > 10{,}358$

Answer: Largest: $85{,}310$. Smallest: $10{,}358$.

Mistake: Ordering by the first digit only, for example putting $912$ after $4{,}105$ because $9 > 4$

Why: The first digit only decides when the numbers have the same number of digits. $912$ has 3 digits, $4{,}105$ has 4.

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

Mistake: Mixing up ascending and descending order

Why: 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.

Mistake: Ignoring a zero inside a number, so that $6{,}080$ seems as large as $6{,}800$

Why: The zero holds the hundreds place. $6{,}080$ has 0 hundreds, while $6{,}800$ has 8 hundreds.

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

Mistake: Starting with 0 when building the smallest number from digits, for example writing $03589$

Why: A whole number does not start with 0. $03589$ is really the four-digit number $3{,}589$, 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 $30{,}589$.

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.

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

Game Leaderboards

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

Scores of $12{,}500$, $9{,}800$ and $12{,}050$ go on the leaderboard as $12{,}500 > 12{,}050 > 9{,}800$.

Digit Card Games

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

With the cards 2, 7 and 4 the largest number is $742$ (digits in descending order) and the smallest is $247$ (digits in ascending order).

Q: How do I remember which is ascending and which is descending?

A: 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.

Q: How can I check that my order is right?

A: Put a sign between each pair of neighbors and check every sign. In $2{,}310 < 2{,}301 < 3{,}012$ the first sign is wrong, because $2{,}310 > 2{,}301$. Correct: $2{,}301 < 2{,}310 < 3{,}012$.

Q: Why are big numbers written with commas, dots or spaces?

A: The separators group the digits in threes so the number is easier to read: $1{,}250{,}000$ 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.

Q: Why can 0 not be the first digit of the smallest number?

A: A zero at the front adds nothing: 0472 is just $472$, 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: $2{,}047$.