Last Updated on July 5, 2023 by Mayank Dham

Binary-coded decimal (BCD) is a numerical representation widely used in digital systems to encode decimal numbers. BCD allows for direct conversion between binary and decimal representations, making it particularly useful for applications that involve arithmetic operations on decimal numbers. One crucial component in BCD arithmetic is the BCD adder, a circuit that performs addition operations on BCD numbers.

In this article, we delve into the world of BCD adders and BCD addition, providing a comprehensive guide to understanding their concepts and working principles. We explore the significance of BCD encoding, the need for BCD addition, and the design of BCD adder circuits. Whether you are a student, an electronics enthusiast, or a professional working with digital systems, gaining a deep understanding of BCD adder and BCD addition is crucial.

## What is a BCD Adder?

A BCD adder, also known as a Binary-Coded Decimal adder, is a digital circuit that performs addition operations on Binary-Coded Decimal numbers. BCD is a numerical representation that uses a four-bit binary code to represent each decimal digit from 0 to 9. BCD encoding allows for direct conversion between binary and decimal representations, making it useful for arithmetic operations on decimal numbers.

The purpose of a BCD adder is to add two BCD numbers together and produce a BCD result. It follows specific rules to ensure accurate decimal results. The BCD adder circuit typically consists of multiple stages, each representing one decimal digit, and utilizes binary addition circuits combined with BCD-specific rules.

## BCD Addition of Given Decimal Number

BCD addition of a given decimal number involves performing addition operations on the individual BCD digits of the number. Here’s a step-by-step guide to understanding how to perform BCD addition:

**Step 1: Convert the decimal number into BCD representation:**

- Take each decimal digit and convert it into its BCD equivalent, which is a four-bit binary code.
- For example, the decimal number 456 would be represented as 0100 0101 0110 in BCD.

**Step 2: Align the BCD numbers for addition:**

Ensure that the BCD numbers to be added have the same number of digits.

If necessary, pad the shorter BCD number with leading zeros to match the length of the longer BCD number.

**Step 3: Perform binary addition on the BCD digits:**

- Start from the rightmost digit and add the corresponding BCD digits of the two numbers.
- If the sum of the BCD digits is less than or equal to 9 (0000 to 1001 in binary), it represents a valid BCD digit.
- If the sum is greater than 9 (1010 to 1111 in binary), it indicates a carry-out, and a correction is needed.

**Step 4: Handle carry-out and correction:**

```
When a carry-out occurs, it needs to be propagated to the next higher-order digit.
Add the carry-out to the next higher-order digit's BCD digit and continue the process until all digits have been processed.
```

**Step 5: Obtain the final BCD result:**

`Once all the BCD digits have been processed, the resulting BCD numbers represent the decimal sum of the original BCD numbers.`

## Example of BCD Adder and BCD Addition of Given Decimal Number

Let’s consider an example to illustrate BCD addition of a given decimal number. We will perform the BCD addition of the decimal numbers 37 and 59.

**Step 1: Convert the decimal numbers into BCD representation:**

```
37 in BCD: 0011 0111
59 in BCD: 0101 1001
```

**Step 2: Align the BCD numbers for addition:**

`Both BCD numbers have the same number of digits, so no alignment is needed.`

**Step 3: Perform binary addition on the BCD digits:**

```
Start from the rightmost digit and add the corresponding BCD digits:
1 + 1 = 10 (carry-out)
1 + 0 = 01
1 + 1 = 10 (carry-out)
0 + 1 = 01
0 + 0 = 00
1 + 1 = 10 (carry-out)
1 + 0 = 01
```

**Step 4: Handle carry-out and correction:**

```
Propagate the carry-outs to the next higher-order digit:
1 (carry-out) + 0 = 01
1 (carry-out) + 0 = 01
0 (carry-out) + 0 = 00
```

**Step 5: Obtain the final BCD result:**

`The final BCD result is: 0101 0100, which represents the decimal number 94.`

Therefore, the BCD addition of the decimal numbers 37 and 59 is 94 in decimal representation. By following the steps of BCD addition, we can accurately perform arithmetic operations on BCD numbers, ensuring proper conversion between binary and decimal representations.

## Approach for BCD Addition of Given Decimal Number

The goal is to generate a BCD number from the sum of the two numbers A and B that are given. Here are the actions:

Find the sum of the two supplied numbers, A and B (say num).

Convert up to 4 bits of each digit in the number num into a binary representation.

Print the result after concatenating each digit’s binary representation.

**Code Implementation**

#include <bits/stdc++.h> using namespace std; string BCDAddition(int A, int B) { // Store the summation of A and B // in form of string string s = to_string(A + B); int l = s.length(); // To store the final result string ans; string str; // Forming BCD using Bitset for (int i = 0; i < l; i++) { // Find the binary representation // of the current characters str = bitset<4>(s[i]).to_string(); ans.append(str); } // Stripping off leading zeroes. const auto loc1 = ans.find('1'); // Return string ans if (loc1 != string::npos) { return ans.substr(loc1); } return "0"; } // Driver Code int main() { // Given Numbers int A = 12, B = 20; // Function Call cout << BCDAddition(A, B); return 0; }

class PrepBytes{ // Function to perform BCD Addition static String BCDAddition(int A, int B) { // Store the summation of A and B // in form of string String s = String.valueOf(A + B); int l = s.length(); // Forming BCD using Bitset String temp[] = { "0000", "0001", "0010", "0011", "0100", "0101", "0110", "0111", "1000", "1001" }; String ans = ""; for(int i = 0; i < l; i++) { // Find the binary representation // of the current characters String t = temp[s.charAt(i) - '0']; ans = ans + String.valueOf(t); } // Stripping off leading zeroes. int loc1 = 0; while (loc1 < l && ans.charAt(loc1) != '1') { loc1++; } // Return string ans return ans.substring(loc1); } // Driver code public static void main(String[] args) { // Given Numbers int A = 12; int B = 20; // Function Call System.out.println(BCDAddition(A, B)); } }

def BCDAddition(A, B): # Store the summation of A and B # in form of string s = str(A + B) l = len(s) # Forming BCD using Bitset temp = [ "0000", "0001", "0010", "0011", "0100", "0101", "0110", "0111", "1000", "1001" ] ans = "" for i in range(l): # Find the binary representation # of the current characters t = temp[ord(s[i]) - ord('0')] ans = ans + str(t) # Stripping off leading zeroes. loc1 = ans.find('1') # Return string ans return ans[loc1:] # Driver Code # Given Numbers A = 12 B = 20 # Function Call print(BCDAddition(A, B))

**Output**

110010

**Time Complexity:** The time complexity for BCD addition of Given Decimal Numbers will be O(log10(A+B)).

**Conclusion**

In conclusion, understanding BCD adders and BCD addition is crucial for working with decimal numbers in digital systems. BCD adders are digital circuits designed to perform addition operations on Binary-Coded Decimal numbers, which use a four-bit binary code to represent each decimal digit. By following specific rules and utilizing binary addition circuits, BCD adders ensure accurate and reliable computation of decimal values.

Performing BCD addition of a given decimal number involves converting the decimal digits into BCD representation, aligning the BCD numbers for addition, performing binary addition on the BCD digits, handling carry-outs and corrections, and obtaining the final BCD result. By following these steps, we can accurately add decimal numbers in their BCD representations, maintaining the BCD format and obtaining the correct decimal sum.

## FAQ on BCD Adder and BCD Addition of Given Decimal Number

**Q1. What is a BCD adder?**

A BCD adder is a digital circuit that performs addition operations on Binary-Coded Decimal numbers. It follows specific rules and combines binary addition circuits to ensure accurate computation of decimal values.

**Q2 .Why is BCD addition important?**

BCD addition is important because it allows for accurate arithmetic operations on decimal numbers within digital systems. It enables the conversion between binary and decimal representations and is commonly used in applications such as calculators and numerical displays.

**Q3. How do I perform BCD addition of a given decimal number?**

To perform BCD addition of a given decimal number, convert the decimal digits into BCD representation, align the BCD numbers for addition, perform binary addition on the BCD digits, handle carry-outs and corrections, and obtain the final BCD result.

**Q4. What are the benefits of using BCD adders?**

BCD adders allow for efficient and reliable arithmetic operations on decimal numbers. They maintain the BCD format and ensure accurate computation of decimal values, which is essential for applications that require precise handling of decimal arithmetic.

**Q5. Can BCD addition be performed using software or programming languages?**

Yes, BCD addition can be implemented using software or programming languages. By understanding the principles of BCD addition and using appropriate algorithms, you can perform BCD addition in software applications, enabling decimal arithmetic in digital systems.