Decimal 49 converts to the 8421 BCD code 0100 1001. Unlike standard base-2 binary conversion which divides the entire number by powers of 2, the formula for Binary Coded Decimal (BCD) isolates each base-10 digit and maps it individually to a 4-bit binary equivalent using the 8-4-2-1 weighting assumption. For the value 49: the tens digit (4) evaluates as (0×8) + (1×4) + (0×2) + (0×1) yielding 0100. The ones digit (9) evaluates as (1×8) + (0×4) + (0×2) + (1×1) yielding 1001. Concatenating them produces the final BCD string: 0100 1001.
| Decimal | Tens Nibble | Ones Nibble | Full BCD Code |
|---|---|---|---|
| 39 | 0011 | 1001 | 0011 1001 |
| 44 | 0100 | 0100 | 0100 0100 |
| 49 | 0100 | 1001 | 0100 1001 |
| 54 | 0101 | 0100 | 0101 0100 |
| 59 | 0101 | 1001 | 0101 1001 |
The 8421 BCD Conversion Reference Table
The foundational assumption that fixes a BCD answer is the 8421 weighting system. Each 4-bit nibble represents a single decimal digit from 0 to 9. Because 4 bits can technically count up to 15 (in pure hex/binary), the states from 10 to 15 are strictly forbidden in standard BCD. Below is the complete data-dense mapping for a single BCD digit, including the hex equivalent and the standard 7-segment display outputs (where 'a' through 'g' represent the LED segments).
| Decimal | BCD (8421) | Hex | 7-Segment (a-b-c-d-e-f-g) | State |
|---|---|---|---|---|
| 0 | 0000 | 0x0 | 1-1-1-1-1-1-0 | Valid |
| 1 | 0001 | 0x1 | 0-1-1-0-0-0-0 | Valid |
| 2 | 0010 | 0x2 | 1-1-0-1-1-0-1 | Valid |
| 3 | 0011 | 0x3 | 1-1-1-1-0-0-1 | Valid |
| 4 | 0100 | 0x4 | 0-1-1-0-0-1-1 | Valid |
| 5 | 0101 | 0x5 | 1-0-1-1-0-1-1 | Valid |
| 6 | 0110 | 0x6 | 1-0-1-1-1-1-1 | Valid |
| 7 | 0111 | 0x7 | 1-1-1-0-0-0-0 | Valid |
| 8 | 1000 | 0x8 | 1-1-1-1-1-1-1 | Valid |
| 9 | 1001 | 0x9 | 1-1-1-1-0-1-1 | Valid |
| 10 | 1010 | 0xA | Blank / Invalid | Invalid |
| 11 | 1011 | 0xB | Blank / Invalid | Invalid |
| 12 | 1100 | 0xC | Blank / Invalid | Invalid |
| 13 | 1101 | 0xD | Blank / Invalid | Invalid |
| 14 | 1110 | 0xE | Blank / Invalid | Invalid |
| 15 | 1111 | 0xF | Blank / Invalid | Invalid |
For a deeper look at how these digital logic states map to physical circuits, the All About Circuits digital textbook provides excellent schematic examples of BCD adders and encoders.
Physical Implementation: Logic Families and Voltage Thresholds
In AC power math, conversions shift wildly depending on whether you are calculating for 120V single-phase, 230V single-phase, or 400V 3-phase systems. In digital logic, the mathematical conversion of BCD is absolute, but the physical hardware implementation shifts drastically based on logic family voltage thresholds. You cannot universally apply a 5V TTL output to a 3.3V CMOS input without risking silicon damage or floating logic states.
74LS Series (TTL)
VCC: 4.75V to 5.25V
Example IC: 74LS147 (10-line to 4-line priority encoder)
Thresholds: $V_{IH}$ (Logic High) requires minimum 2.0V. $V_{IL}$ (Logic Low) maxes at 0.8V.
Use Case: Legacy 5V breadboard designs. Draws higher static current.
74HC Series (CMOS)
VCC: 2.0V to 6.0V
Example IC: SN74HC4511 (BCD-to-7-Segment Latch/Decoder)
Thresholds: Scales with VCC. At 5V, $V_{IH}$ is ~3.15V. At 3.3V, $V_{IH}$ is ~2.1V.
Use Case: Modern microcontroller interfaces (ESP32/Arduino) running at 3.3V or 5V.
CD4000 Series (CMOS)
VCC: 3.0V to 15.0V
Example IC: CD4028BE (BCD to Decimal Decoder)
Thresholds: Highly tolerant. At 12V, $V_{IH}$ is ~9.0V.
Use Case: High-voltage industrial logic or 12V automotive dash clusters.
When selecting a BCD converter IC, always match the logic family to your microcontroller's GPIO voltage. Driving a 74LS147 with a 3.3V ESP32 GPIO might fail to register a logic HIGH, as the ESP32's 3.3V output falls dangerously close to the TTL $V_{IH}$ minimum threshold when accounting for trace voltage drop. Opt for the 74HC or HCT families for 3.3V compatibility. Texas Instruments provides comprehensive logic circuit family guides to help match voltage translation requirements.
Handling Invalid States: When BCD Conversion is Meaningless
When is a BCD conversion mathematically meaningless? The conversion fails when the binary input falls into the six invalid states (1010 through 1111, or hex A through F). Because standard 8421 BCD only maps to base-10 digits (0-9), feeding a binary 1100 (12) into a BCD decoder yields no valid decimal equivalent. This is the digital logic equivalent of trying to calculate 3-phase power with an unknown power factor—the inputs exist, but the output framework cannot resolve them.
In hardware design, you must account for these invalid states to prevent erratic display behavior or bus contention. Dedicated BCD-to-7-segment ICs like the SN74HC4511 handle this via a dedicated Blanking Input (BI/RBO pin). If the IC detects an input greater than 1001 (9), it automatically forces all 7-segment outputs LOW, turning the display completely blank rather than showing a garbled, non-standard glyph.
Frequently Asked Questions
How do I convert a 3-digit decimal like 125 to BCD?
Apply the isolation formula to all three digits. 1 becomes 0001, 2 becomes 0010, and 5 becomes 0101. The full 12-bit BCD string is 0001 0010 0101. Note that in pure binary, 125 is 01111101 (8 bits), which highlights why BCD is considered less memory-efficient but significantly easier to decode for human-readable displays.
What is the difference between BCD and Excess-3 (XS-3) code?
Standard 8421 BCD maps directly to binary weights. Excess-3 is a self-complementing code where you add 3 (binary 0011) to the standard BCD value. For example, decimal 4 in standard BCD is 0100, but in XS-3 it is 0111 (4+3). XS-3 was historically used in older arithmetic logic units (ALUs) to simplify subtraction operations via 9's complement math.
Which IC converts BCD to decimal outputs (1-of-10)?
If you need to convert a 4-bit BCD input into 10 individual active-HIGH decimal output lines (useful for driving relays or discrete LEDs), use the CD4028BE or 74HC4511 (though the 4511 is for 7-segment, the 74HC154 or 74LS42 are dedicated 4-line to 10-line BCD decoders). The 74LS42 specifically features open-collector outputs, allowing you to switch higher voltage loads directly from the logic IC.
