Binary addition and binary–decimal conversion

Binary Numbers

• **Binary** is a base-2 numeral system using only two digits: **0** and **1**.
• Each binary digit is called a **bit**.
• In binary, each position represents a **power of 2**, starting from 2⁰ on the right.
• Binary is used by computers because it is easy to implement with electronic circuits.

Converting Binary to Decimal

• To convert binary to decimal, write the binary number with place values (powers of 2) above each bit.
• Multiply each bit by its place value (0 or 1 times the power of 2).
• Add all the products together to get the decimal value.
• Example: 1011₂ = 1×8 + 0×4 + 1×2 + 1×1 = 11₁₀.

Converting Decimal to Binary

• One method is **division by 2**: repeatedly divide the decimal number by 2, recording the remainder.
• Read the remainders from **bottom to top** to get the binary equivalent.
• Example: 13₁₀ → 13÷2=6 r1, 6÷2=3 r0, 3÷2=1 r1, 1÷2=0 r1 → 1101₂.
• Another method is to subtract the largest power of 2 that fits, and mark that bit as 1.

Binary Addition

• Binary addition follows rules similar to decimal, but with only 0 and 1.
• Rules: 0+0=0, 0+1=1, 1+0=1, 1+1=0 with a **carry** of 1.
• When adding multiple bits, include any carry from the previous column.
• Example: 1011₂ + 1101₂ = 11000₂.

Carrying in Binary Addition

• A **carry** occurs when the sum in a column is 2 or more.
• In binary, 1+1=2, so write 0 and carry 1 to the next column.
• If a carry is added to an existing 1, the sum becomes 2 again, so write 0 and carry 1 again.
• Carries can propagate through multiple columns.

Overflow

• **Overflow** happens when the result of binary addition is too large to fit in the fixed number of bits.
• For example, adding two 4-bit numbers may produce a 5-bit result.
• If only 4 bits are allowed, the extra bit is lost, causing an incorrect result.
• Overflow can be detected when there is a carry out of the most significant bit.

Example: Binary Addition with Carries

• Add 0111₂ (7) and 0101₂ (5) using 4 bits.
• Column 1 (2⁰): 1+1=0 carry 1.
• Column 2 (2¹): 1+0+carry1 = 0 carry 1.
• Column 3 (2²): 1+1+carry1 = 1 carry 1.
• Column 4 (2³): 0+0+carry1 = 1, no carry out.
• Result: 1100₂ (12). No overflow.

Example: Overflow in Binary Addition

• Add 1001₂ (9) and 0111₂ (7) using 4 bits.
• Column 1: 1+1=0 carry 1.
• Column 2: 0+1+carry1 = 0 carry 1.
• Column 3: 0+1+carry1 = 0 carry 1.
• Column 4: 1+0+carry1 = 0 carry 1.
• There is a carry out of the most significant bit, so overflow occurs; the 4-bit result 0000₂ is incorrect (should be 16).

Practical Applications

• Binary addition is fundamental to computer arithmetic.
• Overflow detection is important in programming and hardware design.
• Understanding binary–decimal conversion helps in debugging and data representation.
• These skills are essential for working with low-level computer systems.

Binary to Decimal Conversion Place Values

Binary Addition with Carries

Overflow Example