• Table of Contents

<aside> <img src="/icons/map-pin_gray.svg" alt="/icons/map-pin_gray.svg" width="40px" /> Most of the topics covered in Week 3 are a review of what we have learned in COE328 — more specifically Ch 5 if you would like to refer back to old notes.

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Number Representation

A number is a mathematical concept. There are many ways to represent a number.

• As humans, we use decimal (base 10), which are composed of digits 0 to 9.
• Computers use binary (base 2), which only use digits 0 and 1.

Numbers that are only positive are called unsigned and numbers that can also be negative are called signed.

Figure 3.1 Formats for representation of integers.

In the binary system, the sign of a number is denoted by the left-most bit.

Unsigned Numbers

Binary addition is performed in the same way as decimal addition except that the values of individual digits can be only 0 or 1.

This circuit which implements the addition of only two $1$-bits is called a half-adder. It does not take the carry from previous sum — hence the name half-adder.

• The sum $s$ is the $\textrm{XOR}$ function.
• The carry $c$ is the $\textrm{AND}$ function.

Figure 3.2 Half-adder.

To add more than $1$-bits, we introduce full-adders. It takes the carry from previous sum to perform addition for the next bit.

Figure 3.3 Full-adder.

One can expect that a full-adder can be constructed using half-adders.