Two's Complement &
Binary Arithmetic

Negative Numbers · Binary Addition · Overflow

CSZoneCambridge IGCSE Computer Science 0478

Binary Addition

Adding Binary Numbers

Rules: 0+0=0 0+1=1 1+0=1 1+1=10 (carry 1)
1+1+1=11 (write 1, carry 1)

Example: 00110101 (53)
+ 00011010 (26)
---------
01001111 (79) ✓

Work right to left, carrying any excess into the next column

Overflow: when the result is too large to fit in the given number of bits — carry out of the leftmost column

Two's Complement

Representing Negative Numbers

Two's complement: the most significant bit (MSB) has a negative place value. For 8-bit: MSB is −128. To negate a number: invert all bits, then add 1.

Represent −37 in 8-bit two's complement:
1. Write +37: 00100101
2. Invert bits: 11011010
3. Add 1: 11011011 ← this is −37

Check: −128+64+16+8+2+1 = −128+91 = −37 ✓

Reading Two's Complement

Interpreting Negative Values

If the MSB is 1, the number is negative. The MSB contributes −128 (for 8-bit), all other bits are positive as normal

Range for 8-bit two's complement: −128 to +127

Example: 10100000 = −128+32 = −96

Overflow: adding two positives gives a negative result, or vice versa — the answer is wrong

Exam Practice

Have a go at this question

Cambridge IGCSE 0478 style

Show how the number −45 is represented using 8-bit two's complement. Show your working.

3 marks

+45 = 00101101 [1]. Invert: 11010010 [1]. Add 1: 11010011 [1].

Key Takeaways

What to Remember

Binary addition: 1+1=10, carry the 1; work right to left

Two's complement: MSB has negative value (−128 for 8-bit); to negate: invert + add 1

8-bit two's complement range: −128 to +127

Overflow: result is too large for the number of bits — carry out of the MSB