STEP-BY-STEP LESSON · §2.5
Application: Number Systems and Circuits for Addition
Understand binary place values and carries in addition.
Before you begin
Application: Digital Logic Circuits
Use 0 and 1 as two logical values in the model. Gates perform specified operations.
Read a compound circuit from its inputs: evaluate each node after its inputs.
r=p∧q; out=¬r
Symbols
- ₂
- binary notation
- ⊕
- exclusive OR
Definitions and notation for this topic
Binary numerals and circuit values
(b_k…b_0)₂; 0/1
In a binary numeral each digit is 0 or 1 and the subscript 2 specifies place-value base two. In digital logic, 0 and 1 can instead encode false and true.
Distinguish arithmetic addition, 1+1=(10)₂, from OR. A binary numeral’s subscript is a base marker, not a sequence index.
Separate glossary example
(1011)₂=8+2+1=11. In a circuit, 1∧0=0 and 1∨1=1.
Exclusive OR / XOR
⊕
True exactly when one of the two statements is true.
Unlike inclusive OR, XOR is false when both parts are true.
Separate glossary example
T ⊕ T = F
Step by step
Step 1 / 6
Positions have weights
Binary digits 0 and 1 multiply powers of two. From the right, place values are 1,2,4,8.
101₂=1·4+0·2+1·1=5
The numeral system has base two.
Worked example
Compute 101₂+011₂ and verify in decimal.
- Rightmost column: 1+1=10₂, so write 0 and carry 1.
- Middle column: 0+1+carry 1=10₂, so write 0 and carry 1.
- Left column: 1+0+carry 1=10₂. Write 0 and place the final 1 in a new column: 1000₂.
- 101₂=5 and 011₂=3; 1000₂=8 agrees with 5+3.
Optional self-check
What is 110₂?
Show answer
4+2+0=6.