To compare two big numbers, use what you know about reading large numbers and place value. The greatest place matters most.
Key idea
First count the digits: a whole number with more digits is greater. If they have the same number of digits, compare from the left, place by place. The first place where the digits differ decides which number is greater.
Why it works
One ten thousand is more than any amount you can make with only thousands, hundreds, tens and ones (the most is $9{,}999$). So a bigger digit in a higher place outweighs everything to its right. That is why you stop at the first difference.
| Place | 45,312 | 45,231 | Same? |
|---|---|---|---|
| ten thousands | 4 | 4 | yes |
| thousands | 5 | 5 | yes |
| hundreds | 3 | 2 | no: 3 > 2 |
Worked example: Same number of digits
Compare $45{,}312$ and $45{,}231$.
- Both have 5 digits, so start at the left.
- Ten thousands: 4 and 4. Thousands: 5 and 5. Still the same.
- Hundreds: 3 and 2. Since $3 > 2$, the first number is greater. The tens and ones do not matter now.
Answer: $45{,}312 > 45{,}231$
Worked example: Different number of digits
Compare $9{,}876$ and $10{,}234$.
- $9{,}876$ has 4 digits; $10{,}234$ has 5.
- The 5-digit number has a ten thousand, which is more than all of $9{,}876$.
Answer: $9{,}876 < 10{,}234$
Common mistake
Comparing the last digits, or deciding by the first digit alone when the numbers have different lengths ($9{,}876$ starts with 9 but is less than $10{,}234$). Count digits first, then compare from the left.
The symbol opens toward the greater number: $45{,}312 > 45{,}231$ reads "is greater than"; $9{,}876 < 10{,}234$ reads "is less than."