Ten's Complement — Definition, Formula & Examples
Ten's complement is the number you must add to a given number to reach the next power of 10. For a single digit like 3, the ten's complement is 7 because 3 + 7 = 10.
For a positive integer with digits, the ten's complement of is defined as . This produces the value that, when added to , yields exactly .
Key Formula
Where:
- = The original positive integer
- = The number of digits in n
How It Works
To find the ten's complement, first count the number of digits in your number. Then subtract the number from the corresponding power of 10. For a 1-digit number, subtract from 10. For a 2-digit number, subtract from 100. For a 3-digit number, subtract from 1000, and so on. A quick shortcut for multi-digit numbers: subtract each digit from 9 except the last digit, which you subtract from 10.
Worked Example
Problem: Find the ten's complement of 374.
Step 1: Count the digits. The number 374 has 3 digits, so the target power of 10 is 1000.
Step 2: Subtract the number from 1000.
Step 3: Verify: add the result back to the original number.
Answer: The ten's complement of 374 is 626.
Why It Matters
Ten's complements are the foundation of how computers perform subtraction in decimal arithmetic — instead of subtracting, the processor adds the complement. In everyday mental math, knowing complements of 10 speeds up addition and making change (for example, instantly knowing that $6.26 is the change from $10.00 on a $3.74 purchase).
Common Mistakes
Mistake: Using the wrong power of 10 — for example, subtracting a 3-digit number from 100 instead of 1000.
Correction: Always match the power of 10 to the number of digits. A 3-digit number is subtracted from 10³ = 1000, a 2-digit number from 10² = 100, and so on.