Learn everything about 9’s and 10’s complement in digital logic and decimal arithmetic, including their definition, methods of calculation, subtraction applications, and real-world examples for computing and digital systems.
Introduction to 9’s and 10’s Complement
In digital systems and computer arithmetic, complements are used to simplify subtraction and signed number representation. While 1’s and 2’s complements are widely used in binary arithmetic, the 9’s and 10’s complements are the decimal counterparts, essential for decimal arithmetic in computing, calculators, and digital circuits.
Understanding 9’s and 10’s complements helps students and engineers perform subtraction using addition, reducing hardware complexity and computation time.
What is a Complement in Decimal Systems?
A complement of a decimal number is a value that, when added to the original number, results in a base power of 10, typically 10^n, where n is the number of digits.
- 9’s Complement: One less than the next power of 10 (all digits replaced by 9 minus that digit).
- 10’s Complement: One more than the 9’s complement, equal to 10^n – number.
Purpose: Complements allow subtraction to be performed as addition in decimal arithmetic, similar to binary systems.
∴ Concept of MINUEND and SUBTRAHEND
→ 9`s Complement & 10`s complement:
If the number is binary, then we use 1’s complement and 2’s complement. But in case, when the number is a decimal number, we will use the 9’s and 10’s complement.
• 9`s Complement:
The 9’s complement is used to find the subtraction of the decimal numbers.
• The 9’s complement of a number is calculated by subtracting each digit of the number by 9.
• Subtraction using 9’s complement
• Step 1: Find 9`s complement of subtrahend
• Step 2: Add the 9`s complement with the minuend.
Note: If there is carry, add the carry to the result to get the final result, else find the 9`s complement of the reuslt and put negative sign to MSB.
Case 1: When the subtrahend is smaller than the minuend.
Case 2: When the subtrahend is greater than the minuend.
⁘ 10’s complement
The 10’s complement is also used to find the subtraction of the decimal numbers.
• The 10’s complement of a number is calculated by subtracting each digit by 9 and then adding 1 to the result. Simply, by adding 1 to its 9’s complement we can get its 10’s complement value.
• Subtraction using 10’s complement
• Step 1: Find 10`s complement of subtrahend
• Step 2: Add 10`s complement with minuend
Note: if there is any carry discard it, else again find 10`s complement of result and put negative sign to MSB.
Case 1: When the subtrahend is smaller than the minuend.
Case 2: When the subtrahend is greater than the minuend.
Advantages of Using 9’s and 10’s Complement
- Simplifies subtraction in decimal arithmetic.
- Eliminates the need for borrowing in manual calculation.
- Essential in digital calculators, arithmetic units, and BCD arithmetic.
- Reduces hardware complexity in digital circuits performing decimal arithmetic.
Applications of 9’s and 10’s Complement
- Digital Computers:
- Efficient subtraction in binary-coded decimal (BCD) systems.
- Calculators:
- Used for fast subtraction using addition.
- Arithmetic Logic Units (ALU):
- Enables subtraction by addition in embedded systems.
- Error Detection:
- Forms part of checksum calculations and validation.
- Learning & Education:
- Helps students understand complement arithmetic in decimal systems.
Summary Table: 9’s vs 10’s Complement
| Feature | 9’s Complement | 10’s Complement |
|---|---|---|
| Definition | Subtract each digit from 9 | 9’s complement + 1 |
| Purpose | Intermediate step for subtraction | Direct subtraction via addition |
| Method | Digit inversion | Digit inversion + 1 |
| Example | 275 → 724 | 275 → 725 |
| Advantage | Simple to calculate | Direct subtraction without extra steps |
Conclusion
9’s and 10’s complements are foundational concepts in decimal arithmetic and digital logic design. They allow efficient subtraction by converting it into addition, simplifying calculations in computers, calculators, and digital circuits. Mastery of these methods is essential for students, engineers, and professionals working with binary-coded decimal systems and arithmetic units.
Call to Action:
Practice calculating 9’s and 10’s complements for multi-digit numbers and implement subtraction problems to strengthen your decimal arithmetic skills in digital logic.
Frequently Asked Questions (FAQ)
1. What is the difference between 9’s and 10’s complement?
9’s complement is obtained by subtracting each digit from 9; 10’s complement is 9’s complement + 1.
2. Why is 10’s complement preferred in computers?
Because it allows subtraction using addition without manually handling carry and simplifies hardware implementation.
3. How do you perform subtraction using 9’s complement?
Add the 9’s complement of the subtrahend to the minuend, then add 1, and ignore overflow beyond the number of digits.
4. Are complements used in binary arithmetic?
Yes, binary arithmetic uses 1’s and 2’s complements to perform subtraction by addition.
5. Where are 9’s and 10’s complements applied?
They are applied in calculators, BCD arithmetic, digital computers, ALUs, and teaching arithmetic operations.
