Binary Arithmetic

This lesson covers binary addition, binary subtraction using two's complement, overflow, and bit shifting (logical and arithmetic shifts). These are core skills for the OCR H446 specification.

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Binary Addition

Binary addition follows the same rules as denary addition, but with only two digits (0 and 1).

Addition Rules

A	B	Sum	Carry
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

When both bits are 1, the sum is 0 and the carry is 1 (just like 1+1=2 in denary, which is "10" in binary).

If there is also a carry-in of 1: 1 + 1 + 1 = 1 with carry 1.

Worked Example: 01101011 + 00110101

  Carry:  1 1 1 1 1 1
          0 1 1 0 1 0 1 1   (107)
        + 0 0 1 1 0 1 0 1   ( 53)
        ─────────────────
          1 0 1 0 0 0 0 0   (160)

Verification: 107 + 53 = 160. Correct.

Worked Example: 11111111 + 00000001

  Carry: 1 1 1 1 1 1 1 1
          1 1 1 1 1 1 1 1   (255)
        + 0 0 0 0 0 0 0 1   (  1)
        ─────────────────
        1 0 0 0 0 0 0 0 0   (256)

This result requires 9 bits. If we are limited to 8 bits, this is an overflow.

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Overflow

Overflow occurs when the result of a calculation exceeds the maximum value that can be stored in the available number of bits.

Bits	Unsigned Range	Max Value
8	0 to 255	255
16	0 to 65,535	65,535
32	0 to 4,294,967,295	~4.29 billion

For unsigned 8-bit numbers, overflow occurs if the result exceeds 255 (i.e., if there is a carry out of the most significant bit).

For signed two's complement numbers, overflow occurs when adding two positive numbers gives a negative result, or adding two negative numbers gives a positive result.

Exam Tip: When performing binary addition in an exam, always check whether the result fits within the given number of bits. If a carry is produced beyond the most significant bit, state that overflow has occurred and explain the consequence.

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Binary Subtraction

Binary subtraction can be performed by adding the two's complement of the number being subtracted.

A - B = A + (-B)

To find the two's complement (negative) of B:

1. Flip all bits (one's complement / bitwise NOT)
2. Add 1

Worked Example: 10110100 - 01001011

Step 1: Find two's complement of 01001011

• Flip: 10110100 -> 10110101 (flip 01001011 -> 10110100, then add 1 -> 10110101)

Actually let us work step by step:

• Original B: 01001011
• Flip all bits: 10110100
• Add 1: 10110101

Step 2: Add A + (-B)

          1 0 1 1 0 1 0 0   (180)
        + 1 0 1 1 0 1 0 1   (-75 in two's complement)
        ─────────────────
        1 0 1 1 0 1 0 0 1

Discard the carry (9th bit): result = 01101001 = 105

Verification: 180 - 75 = 105. Correct.

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Bit Shifting