Lecture Notes 02:
Bits, Binary, and Number representation

Outline

This class we'll discuss:

• Review: Evolution of computing
• The minimum amount of knowledge
• Information atoms
• What can we do with bits?
• Number Systems
• Binary
• Binary arithmetic
• Binary logic

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Evolution of computing system

An example problem: calculate the trajectory of a missile.

• Paper, pencil & brain: several days to a few weeks
• Gears & levers: several hours to a few days (including set-up)
• Switches, relays, shafts, and clutches: a few hours (including set-up)
• Voltages & logic circuits (ENIAC): 15 seconds (once set-up, which could take hours or days)
• Voltages & logic circuits (MacBook Pro): several \(ns\) to a few \(\mu s\) (study your prefixes)

So why move from humans to machines? ... Automation!

And why move from gears to electricity? ... SPEED!
(and size).

And why move from analog to digital? ... Precision!
(and compact representation).



Let's talk about how this electricity-based machine stores and uses data.

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The Minimum Amount of Knowledge

Part 1: The Medium

Activity 1 :[2 minutes] : What is the size of the most accurate map of a region?

We want to chose a good way to represent reality.

Step 1: choose the medium.

Moving gears takes a lot of energy and is slow.
Also, analog machines suffer from non-ideal effects (inaccuracy, loss, leaks, friction, etc).

What we want is a medium that moves fast and is energy efficient.
(and allows us to perform error correction!)

Electricity is that medium.
However, we need to be able to satisfy some requirements!

Part 2: The Requirements

• Represent and Save information
• Solve mathematical and logical expressions
• Accept input
• Make decisions according to input or stored information
• Produce output

Part 3: Questions about information

Lets start with: representing and saving information

When thinking about representing and saving information, we'd like to answer the following questions:

1. What types of information should we be able to save?
2. How should we structure this information in order to save it efficiently?
3. What is the amount of information we can (want to) save?
4. What is the minimum amount possible?

Activity 2 :[2 minutes] : Let's start with the fourth question:

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Information atoms: bits

A yes/no piece of data is the minimum you can know about a thing. It simply answers the question:
"is there or not something there?""

Anything that has two parts or two sides is called binary, so this piece of information is called a binary datum (a single piece of data), which could be represented with any two symbols:

• No/Yes
• False/True
• OFF/ON
• \( \bullet\) / \(\boldsymbol{-}\)
• 0/1

The term bit is derived from binary digit, a logical state (relation between a proposition and the truth) with one of two values.

It is common to represent these with \( 0 \) and \( 1 \), but as you can see below, we can use another approach (OFF/ON) with the same results.

Multiple bits

We've seen that a bit can represent something being there or not.

Activity 3 :[2 minutes] : how do we represent many things being there or not?


With this in mind, how do we save information using electricity?

Fast Water and Pipes: an analogy for an electric computer

To explain how we do this, we'll use an analogy with water and pipes, which contains:

• a water reservoir
• valves
• water circuits

Activity 4 :[2 minutes] : how do we represent information (a bit) using water?




Now the question is, what can we represent with that information representation?
We'll come back to the water analogy, but first, let's think about information in the abstract.

Activity 5 :[2 minutes] : can we represent the following information using bits?

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Number representation

There are many different ways of representing numbers:

Activity 6 :[2 minutes] : why would we want to "represent" a number?
From early representation, we've used different sounds (grunts \( \rightarrow \) words) to refer to numbers, and different symbols so that we can write them out.

Tally Marks

One of the easiest representations is using tally marks, like any of these:


Why could that be an issue?

Grouping Symbols

These employ several different symbols, each with a different value.
Numbers are constructed by placing the "largest" value symbols possible until the full amount is represented.

More advanced approaches used the position of the symbol to vary the value slightly.
One exampe of these is the Roman numerals:


Why could that be an issue?

Positional Number System

The genius of this approach is that you can reuse any number of symbols and give greater significance to their position when grouped together.

Decimal

Note that decimal is a positional system with the symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9;

The position change indicates "ten times as many as before" ("or a tenth as before"). We call these "powers of 10".

The number is constructed by indicating "how many times we need each power of 10" (that's what base 10 means):

\[ 1729 = 1*10^3 + 7*10^2 + 2*10^1 + 9*10^0 = 1000 + 700 + 20 + 9 \]

Binary

The binary system is a positional number system with two symbols: 0 and 1;

the number is constructed by indicating "how many times we need each power of 2" (that's why base 2) using only the valid symbols: \[ \begin{alignat}{2} 0b11011000001 &=& \: 1*2^{10} \: & + & \: 1*2^9 \: &+& \: 0*2^8 &+& \: 1*2^7 \: &+& \: 1*2^6 \: &+& \: 0*2^5 \: &+& \: 0*2^4 \: &+& \: 0*2^3 \: &+& \: 0*2^2 &+& \: 0*2^1 \: &+& \: 1*2^0 \: & &\qquad\\ \qquad &=& \: 1024 \: & + & \: 512 \: &+& \: 0 &+& \: 128 \: &+& \: 64 \: &+& \: 0 \: &+& \: 0 \: &+& \: 0 \: &+& \: 0 &+& \: 0 \: &+& \: 1 \: &=& \: 1729 \end{alignat} \] The first 8 natural numbers (starting at 0) are:

decimal	binary
0	000
1	001
2	010
3	011
4	100
5	101
6	110
7	111

Notation: To indicate that we're writing a binary number, we usually prepend "0b" to the number, so the binary number:
101010 should be written 0b101010 to distinguish it from "one hundred and one thousand and ten".

You might also see this as 101010b.

Activity 7 [3 minutes]:

Hexadecimal

The last useful positional system we'll use in this class is hexadecimal.
The hexadecimal (or hex) system is a positional number system with 16 symbols: 0-9 plus a, b, c, d, e, and f;

the number is constructed by indicating "how many times we need each power of 16" (that's why base 16) using only the valid symbols. \[ 6C1 = 6*16^2 + C*16^1 + 1*16^0 = 6*16^2 + (12)*16^1 + 1*16^0 = 1536 + 192 + 1 = 1729\]

Notation: To indicate that we're writing a hex number, we usually prepend "0x" to the number, so the hex number:
1234 should be written 0x1234 to distinguish it from "one thousand, two hundred and thirty four".

Activity 8 [Group; 3 minutes]:

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Binary Arithmetic

Any operation we can do with decimals, we can do with binary.

Addition

\[ \begin{aligned} 0b1011&\\ + 0b1001& \\ \hline 0b10100& \end{aligned} \]
Note that we can keep track of the carry in order to get the proper addition:
\[ \begin{aligned} \texttt{carry:}\mathit{1011} \texttt{ }&\\ \texttt{A: } 1011&\\ + \texttt{ B: } 1001& \\ \hline \texttt{Out: }10100& \end{aligned} \]

Activity 9 [Group: 1 minute (if we have time)]:

Subtraction

Subtraction is tricky because the way we represent negative numbers.

You can solve it in two ways:

• "borrowing" from the digit on the left (instead of "carrying"): \[ \begin{aligned} 0b1 1 0 1 1 1 0&\\ - 0b1 0 1 1 1& \\ \hline 0b1 0 1 0 1 1 1& \end{aligned} \]
• Using a "complement" approach: instead of subtracting, add the "additive complement".
  The result is calculated like this: \[ A - B = A + (B \text{'s two's complement }) + 1 \] In practice, this looks like this: \[ \begin{align*} 0b0111& \rightarrow & 0b0111& & &\qquad\\ -0b0001& \rightarrow & + \mathit{0b1111}& & &\qquad\\ \hline 0b0110& \qquad & 0b\mathbf{1}0110& \rightarrow & &\mathbf{0b0110} \end{align*} \]
  With this approach, we disregard any carry 1s generated by the addition of 1.

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Logical Operators

Logical operations refer to operations on variables that can contain truth values: true and false, usually denoted 1 and 0, respectively.

Think of these as operations on the "truth-value of a statement".

NOT

NOT is an operation that can be applied to a single bit; since the operation applies to a "single" operand,
it is called a unary operation.

When NOT is applied to a single bit (lets call it p), we can get the following table:

p	NOT p
1	0
0	1

Note that we're labeling on the outside for clarity, but the label refers to the interior of each circle

Note that, if \(p\) denotes the operand, \(\text{NOT } p\) can be written as \(\sim p\), \(\neg p\), \(\bar{p}\) or even \(!p\).

OR

OR (AKA: the disjunction operator) is an operation that is applied to pairs of operands,
it is called a binary operation.

it is true if any of its operands is true.

p	q	p OR q
0	0	0
0	1	1
1	0	1
1	1	1

Note: other symbols you might see are: \( p\lor q \text{, } p+ q \text{, } p\mid q \).

AND

AND (AKA: the conjunction operator) is a binary operator; it is true if and only if both of its operands are true.

We can show the effect of this operator using a table:

p	q	p AND q
0	0	0
0	1	0
1	0	0
1	1	1

Note that we're labeling on the outside for clarity, but the label refers to the interior of each circle

Note: other symbols you might see are: \( p \wedge q \text{, } p\cdot q \text{, } p\&q \).

Examples

Activity 10 [Group: 1 minute (if we have time)]:

It turns out that AND, OR, and NOT are necessary and sufficient to implement any boolean function.

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Before next class (Thursday 09/09)

[Optional]

• Review the class notes
• Come to online OHs; bring questions, jokes, or simply come and listen.

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1-Minute Debrief

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