Question

How to convert hex to BCD and vice versa?

Answer 1

Could you clarify whether you want a detailed step-by-step process for converting hex to BCD and BCD to hex, or just a summary of the methods?

Answer 2

Hexadecimal (hex) and Binary-Coded Decimal (BCD) are two different number systems used in computing and electronics. Converting between them involves a few steps, and understanding these conversions can be quite useful. Here’s a detailed explanation of how to convert between hex and BCD:

### 1. Hexadecimal to BCD Conversion

**Hexadecimal (Hex) System:**
- Uses base 16.
- Digits range from 0 to 9 and A to F (where A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15).

**Binary-Coded Decimal (BCD) System:**
- Each decimal digit (0-9) is represented by its 4-bit binary equivalent.

**Steps to Convert Hex to BCD:**

1. **Convert Hex to Decimal:**
   - Convert the hex number to its decimal equivalent. For example, for hex `2F`, calculate:
     \[
     2F_{16} = (2 \times 16^1) + (15 \times 16^0) = 32 + 15 = 47_{10}
     \]

2. **Convert Decimal to BCD:**
   - Break the decimal number into individual decimal digits.
   - Convert each decimal digit to its 4-bit binary equivalent.

   For example, converting 47 to BCD:
   - Decimal 47 breaks into digits 4 and 7.
   - Decimal 4 in BCD is `0100`.
   - Decimal 7 in BCD is `0111`.

   So, the BCD representation of 47 is `0100 0111`.

### 2. BCD to Hexadecimal Conversion

**Steps to Convert BCD to Hex:**

1. **Convert BCD to Decimal:**
   - Combine the 4-bit binary groups from BCD to form decimal digits.
   - Convert the combined decimal digits to a decimal number.

   For example, BCD `0100 0111`:
   - `0100` is 4 and `0111` is 7.
   - Combine them to form decimal 47.

2. **Convert Decimal to Hex:**
   - Convert the decimal number obtained from the BCD conversion to hex.

   For decimal 47:
   - Convert 47 to hex:
     \[
     47_{10} = 2F_{16}
     \]

   So, BCD `0100 0111` converts to hex `2F`.

### Summary of Conversions

1. **Hex to BCD:**
   - Convert hex to decimal.
   - Convert decimal to BCD by representing each decimal digit with its 4-bit binary equivalent.

2. **BCD to Hex:**
   - Convert BCD to decimal by combining and converting each 4-bit group.
   - Convert decimal to hex.

### Examples:

**Example 1: Convert Hex `1A` to BCD**

1. Convert hex `1A` to decimal:
   \[
   1A_{16} = (1 \times 16^1) + (10 \times 16^0) = 16 + 10 = 26_{10}
   \]

2. Convert decimal 26 to BCD:
   - Decimal 26 breaks into 2 and 6.
   - Decimal 2 in BCD is `0010`.
   - Decimal 6 in BCD is `0110`.

   So, hex `1A` is `0010 0110` in BCD.

**Example 2: Convert BCD `0001 1010` to Hex**

1. Convert BCD `0001 1010` to decimal:
   - `0001` is 1 and `1010` is 10.
   - Combine them to form decimal 1A (26 in decimal).

2. Convert decimal 26 to hex:
   - Decimal 26 is `1A` in hex.

So, BCD `0001 1010` is hex `1A`.

Understanding these steps and practicing with different numbers will help you become proficient in converting between hex and BCD.