1. Unsigned arithmetic

Unsigned arithmetic

Unsigned addition

We follow the following 4 (bit addition) rules:

• $1+0=1$
• $0+1=1$
• $0+0=0$
• $1+1=10$

Overflow

Take, a $3$-bit value, say $111$. If we perform $111+001$, we end up at $000$ since the addition has overflowed past the MSB.

$$\begin{array}{r}111\\ +\underline{001}\\ 1000\end{array}$$

Hence, this $1$ value is discarded and we are left with $000$.

A question may specify a $n$-bit system.
**Always** pick `overflow` if you end with a carry-out, such as above.

Unsigned subtraction

We follow the following 4 (bit subtraction) rules:

• $0-0=0$
• $1-0=1$
• $1-1=0$
• $0-1=\text{borrow}$

Before doing unsigned subtraction, always check the magnitudes of the operands.
For example, $001 - 101$, the second operand is larger than the first, hence since we are working with unsigned binary, this operation is invalid.

Unsigned multiplication

To multiply two binary numbers together:

1. Select the number with the least $1$ digits.
2. Split the number into parts. $101\Rightarrow 100\text{ and }001$
3. Multiply each part to the other number.
4. Sum resulting parts.

For example, take $01111\times 10010$.

• The number we first take is $10010$.
• Split it into $10000$ and $00010$.
• Take the other number $01111$ and multiply by the specified power of $2$:
  1. $01111\times 10000=01111<<4=011110000$
  2. $01111\times 00010=01111<<1=000011110$
• Sum the resulting parts: $$\begin{array}{r}011110000\\ +\underline{000011110}\\ 0100001110\end{array}$$ Hence the final answer is $0100001110$.

Unsigned division

Computer systems may opt for short or long division algorithms. Here is an example using long division:

$$\begin{array}{l}001001_2\\ 101\overline{)101101}\\ -\underline{101}\\ 0101\\ -\underline{101}\\ 0\end{array}$$

Each comparison should be justified as you progress through each column:

$$\begin{array}{rl}1& <101_2\\ 10& <101_2\\ 101& \ge 101_2\\ 1& <101_2\\ 10& <101_2\\ 101& \ge 101_2\end{array}$$