Chapter 1: Binary Basics

Binary: What’s it all mean?

A set of bits, like 10101100 has no inherent meaning. This is one reason we have data types in programming languages like Java and Python — so there are cues about how people want bits to be interpreted.

Take a look at the C Programming Language’s Data Types and indicate at least two that could perfectly fit the binary 10101100 (at least two are directly relevant to the reading). Indicate the meaning of 10101100 for each.

The Unsigned Number Line

In class we looked at a diagram of the segment of the integer number line that can be used for unsigned, 3-bit binary:

4-bits

Sketch out the number line for 4-bit unsigned numbers.

$n$-bits

Consider how two $n$-bit number lines can be combined together to create a $(n+1)$-bit number line. Explain the pattern and how to take two 3-bit lines to build the 4-bit line.

The Signed Number Line

In class we drew the signed number line as a single, long line. In some ways it may help to think of “bending” that line so that half is above the bend and half is below the bend:

Draw the 3-bit and 4-bit number lines with the bend. Label the binary and decimal values of each point on the lines assuming it’s using two’s complement numbers.

Going negative:

Signs of negativity

One way to identify a negative 2’s complement number if via the leading (leftmost) bit. Explain how it indicates negative values and what it means in terms of the “bent number line”.

$-1$

Consider the binary representation of negative one (-1) in both examples of signed number lines. How can you describe the representation of -1 for an arbitrary $n$-bit, 2’s complement value?

Addition

As we discussed in class, we can think of addition as moving to the right on the flat number line. How would you describe the motion caused by addition on this “bent” depiction of the number line? And what happens when “adding one” to the point in the top right? And to the bottom left?

Bitwise Inversion & the bent number line

Inverting a single bit is flipping it to the complementary value: a 0 becomes a 1 and a 1 becomes a zero. Bitwise inversion of an $n$-bit number flips each individual bit. When working with equations we represent bit-wise inversion of a multi-bit value $A$ by placing a bar over it: $\bar{A}$, or in plain text with a slash in front of it to indicate the bar: /A.

Describe the “motion” that results when inverting any number on the bent number line. That is, if you start at $A$, where is $\bar{A}$? (Be sure to consider the top left and bottom right values)

Mathematical Negation

We will be emulating subtraction by adding the mathematical negation of a value. For example, we will transform $4-5$ into $4 + (-5)$, which can be thought of as $4 + (-1 \times 5)$. Another example: $4 - (-5)$ would be thought of as $4 - (-1 \times -5)$. Consequently, we need to be able to handle the concept of “$-1 \times$”.

Moving from $A$ to $-1 \times A$

Using the “bent number line” describe how to get from value $A$ to the value $-1 \times A$

Formal Equation for Mathematical Negation

What is the equation that can be used to mathematically negate a value?

Chapter 2: Gates

Tees

Although you may not find Bill’s shirt today to be a witty take on Shakespeare, he feels that it’s eventually true, no matter what. Explain what he means by this.

Simulation of the Tee

One person in your group should download and install the simulator used in class, JLS: https://github.com/bsiever/JLS/releases.

• Intel Macs & Linux: The macOS version does not currently support Intel-based Macs. The JAR version will will run on Intel Macs (or PCs or Linux), but requires Java’s JRE 6.0 or later to be installed.
• Apple Silicon / macOS: Different versions of macOS may require different approaches to allow you to open the file after unzipping it.
  • In many cases doing a “right-click” and selecting open will suffice.
  • You may need to adjust security settings to give it permission to run. See: https://support.apple.com/en-us/102445

Also download this starter file: studio_2a_211.jls

Open the file in JLS and look it over. Note that:

1. Throughout the class “ports” will be a key part of breaking complex circuits into manageable parts. Moreover, JLS requires “ports” to interact with circuits. The only two elements in the circuit are two ports.
2. One port is highlighted because it’s being “watched”. Discuss what this means.

Now:

1. Add in a Signal Generator () and, using it, a description of a test case for this circuit. The description can be just 3 lines… or even just 10 characters on one line!
2. Simulate the circuit (Simulator > Show Simulator Window and then the Start button). Does it clearly show that the circuit is “eventually true”?
3. Update the test case to show that it’s “eventually true, no matter what”. I.e., that it becomes true for any initial condition that could be given.

Simple Circuits

Now download this starter file: studio_2a_22.jls (it’s empty, but please use the starter file anyway).

1. Write out the full truth table for Figure 2.25 of the text: $Y = \bar{B} \cdot \bar{C} + A \cdot \bar{B}$ (or, in text: Y=/B*/C + A*/B).
   1. Be sure to go in a “counting order”, with the bits in the order $A$, $B$, $C$. Initially they should all be zero (false), then $C$ should be come a one (001), then $B$ should become a 1 and C should be a zero (010), etc.
   2. How many rows are needed for the table? Why? (There are three inputs. Consider the 3-bit number line)
2. Create a circuit for the equation using JLS and the provided starter file.
3. Create a simulation that cycles through all possible values of $A$, $B$, and $C$. Be sure your simulation accounts for the computation time (propagation delay) needed for the circuit. Again, use a “Counting Order”, with the bits in the order $A$, $B$, $C$.
4. How does the simulation compare to your truth table? Explain any instants in time where it does not precisely match the table.
5. What is the propagation delay (total time that may be needed) for your circuit?

Looking It Up

As has been mentioned, the concept of a “Look-Up Table” (LUT) is vital in digital logic. Explain how one can use your truth table to “lookup” the output needed.

Challenge: Simple Selection

Another (empty) starter file: studio_2a_23.jls

We’ll soon be using Multiplexers in larger circuits. Multiplexers can also have some unique behaviors that correspond to LUTs. Can you use a multiplexer () to implement the above equation? You may need to use other parts, like a bundle of wires or constants (hover over parts in JLS for the help text). You should be able to use the same test-case you used before.