Binary numbers and adders: counting in binary, the half adder (sum and carry), the full adder with a carry in, and adding multi-bit numbers.

What this dot point is asking

Eduqas wants you to count in binary and convert to decimal, build a half adder (sum and carry) from logic gates, extend it to a full adder that includes a carry in, and explain how full adders are chained to add multi-bit binary numbers. This shows how the gates of the previous topics perform real arithmetic, the basis of every processor.

The answer

Counting in binary

The half adder

The full adder

Adding multi-bit numbers

Examples in context

Adders are where logic gates become arithmetic: the full adder is the building block of the arithmetic logic unit inside every microprocessor and microcontroller. The half adder shows the XOR and AND gates doing useful work, and chaining full adders into a ripple-carry adder demonstrates how a simple block scales to add large numbers. The binary counting introduced here is also exactly what the counters and seven-segment displays of the sequential module rely on.

Try this

Q1. Convert the binary number $1101_2$ to decimal. [1 mark]

• Cue. $8 + 4 + 0 + 1 = 13$.

Q2. State the sum and carry of a half adder for inputs $A = 1$, $B = 1$. [2 marks]

• Cue. Sum $= 0$, carry $= 1$ (since $1 + 1 = 10_2$).

Q3. State why columns after the first in a multi-bit adder need full adders. [1 mark]

• Cue. They may receive a carry in from the previous column, which only a full adder can include.