--- title: "BCD decoder; 1's and 2's complements" subject: "Digital Logic" chapter: 4 key: "semester_3/digital_logic:bcd-decoder-complements" source: https://www.bctengineer.com/q/semester_3/digital_logic/bcd-decoder-complements site: BCT Engineering --- # BCD decoder; 1's and 2's complements Digital Logic · Chapter 4: Data Processing Circuits - Asked in: 2075 Asoj, 3+3 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/digital_logic/bcd-decoder-complements - Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-4 ## Question Realize the circuit diagram for BCD decoder. Explain 1's and 2's complements with examples? ## Short answer BCD decoder activates one of ten outputs: D0=A'B'C'D', D1=A'B'C'D, D2=B'CD', D3=B'CD, D4=BC'D', D5=BC'D, D6=BCD', D7=BCD, D8=AD', D9=AD. 1's complement inverts all bits; 2's complement adds 1 to the 1's complement. ## Answer #### BCD decoder (BCD-to-decimal, 4-to-10) A **BCD decoder** takes a 4-bit BCD code $ABCD$ ($A$ = MSB) and activates one of ten outputs $D_0$–$D_9$ (IC 7442). Codes 1010–1111 never occur, so they are don't-cares and allow partial decoding. | Digit | ABCD | Full decoding | Simplified (with don't-cares) | |---|---|---|---| | 0 | 0000 | $A'B'C'D'$ | $A'B'C'D'$ | | 1 | 0001 | $A'B'C'D$ | $A'B'C'D$ | | 2 | 0010 | $A'B'CD'$ | $B'CD'$ | | 3 | 0011 | $A'B'CD$ | $B'CD$ | | 4 | 0100 | $A'BC'D'$ | $BC'D'$ | | 5 | 0101 | $A'BC'D$ | $BC'D$ | | 6 | 0110 | $A'BCD'$ | $BCD'$ | | 7 | 0111 | $A'BCD$ | $BCD$ | | 8 | 1000 | $AB'C'D'$ | $AD'$ | | 9 | 1001 | $AB'C'D$ | $AD$ | ```text A B C D A' B' C' D' (inverters) | | | | +--+--+--+--[AND]--- D0 = A'B'C'D' +--+--+--+--[AND]--- D1 = A'B'C'D ... (one AND gate per output) +--------+--[AND]--- D8 = AD' +--------+--[AND]--- D9 = AD ``` Ten AND gates and four inverters form the decoder. #### 1's and 2's complements **1's complement:** invert every bit (0 → 1, 1 → 0). Example: 1's complement of $10110$ is $01001$. **2's complement:** 1's complement + 1. Example: 2's complement of $10110$ is $01001 + 1 = 01010$. They represent negative numbers and turn subtraction into addition. Example $9 - 5$ in 4 bits: ```text 1's complement method 2's complement method 1001 (9) 1001 (9) + 1010 (1's comp of 0101) + 1011 (2's comp of 0101) ------ ------ 1 0011 1 0100 +1 end-around carry carry discarded ------ 0100 = 4 0100 = 4 ``` In 1's complement the end carry is added back; in 2's complement it is discarded. 2's complement has a single zero and is used in computers. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why can the BCD decoder outputs D8 and D9 be simplified to AD' and AD? A. Codes 1010-1111 never occur, so they are don't-cares B. A is always 1 in BCD C. B and C are always equal D. Inputs are inverted internally Answer: A. Since inputs 10-15 never occur, B and C need not be checked once A = 1. (Key point: Don't-care simplification) 2. How many outputs does a BCD (4-to-10) decoder have, and what does it activate? A. 16 outputs, two at a time B. 10 outputs, one for each decimal digit C. 4 outputs D. 7 outputs for the display Answer: B. A BCD-to-decimal decoder (IC 7442) activates one of ten lines for the digit 0-9. (Key point: Decoder size) 3. What is the 2's complement of 10110? A. 01001 B. 01010 C. 01011 D. 10111 Answer: B. The 1's complement is 01001; adding 1 gives 01010. (Key point: 2's complement example) 4. How is the end carry treated in 1's versus 2's complement subtraction? A. Both discard it B. 1's complement discards it; 2's adds it back C. 1's complement adds it back (end-around); 2's complement discards it D. Both add it back Answer: C. In 1's complement the end carry is added to the result; in 2's complement it is simply discarded. 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