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Chapter 1CSC 180

Dr. Adam Anthony

How Computers Work: Bits and Bytes

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OverviewHow do computers work?Understanding bitsElectronic LogicMemory and StorageData Representation

Text NumbersImagesSound

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The World’s Simplest Computer

• This computer “knows” when B1 is pressed or not (input)• It Lights up, or stays dark, to let us know what it

computed (output)• Amount of information: 1 Bit• Discussion:

– What types of problems can we solve with this computer?– Is this actually a computer?

B1

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About Bits• Bits are great because they are easy to

represent mechanically– A black line on paper (or lack of one)– Water flowing through a tube (or not)– Electricity flowing over a wire (or not)– Electricity flowing over a wire at a high/low

voltage level• Now we just need some coding theory and

some physics to get a “real” computer

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Coding Theory• Examples of Bit-wise or Binary code: – Alarms or buzzers– “Light one if by land, two if by sea”

• Paul Revere– Morse Code:

• A = .- B = -… C = -.-.– Boolean Logic:

• AND: 1 AND 1 = 1; 1 AND 0 = 0; 0 AND 0 = 0• OR: 1 OR 1 = 1; 1 OR 0 = 1; 0 OR 0 = 0• NOT: NOT 1 = 0; NOT 0 = 1

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Physics• Basics of Electricity:– Once generated, electricity is either transmitted or

absorbed– Electricity that is partially absorbed is said to be meeting

resistance• Generates heat

– Voltage is a ratio between the power (watts) and current flow (amps) of an electrical source• Measurable• Can be altered with very little resistance• Since Voltage = Watts/Amps, reducing the voltage also reduces

the power

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Logic Gates• Mechanized Boolean logic, usually electric• A simple, inefficient, and out-dated AND gate:

• Four basic types of logic gates in computers– AND (symbol ) means both inputs are true– OR () means one OR the other is true (could be both)– NOT () provides the opposite of the single input– XOR () means EXCLUSIVELY (X) one input is true OR the

other is true

Diodes allow electricity to pass in one direction, and only if there is capacity on the line for it to pass

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Truth Tables and Diagram Primitives

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Logic Diagrams• One if by land, two if by sea:

SEA!!!

LAND!!!

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Truth Tables Revisited• What would a truth table look like for the

expression:(A B) (B C)

A B C A B B B C (A B) (B C)

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Boolean Expressions• What would a truth table look like for the

expression:(A B) (B C)

A B C A B B B C (A B) (B C)

0 0 00 0 1

0 1 00 1 1

1 0 0

1 0 1

1 1 0

1 1 1

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Boolean Expressions• What would a truth table look like for the

expression:(A B) (B C)

A B C A B B B C (A B) (B C)

0 0 0 0 1 1 00 0 1 0 1 1 0

0 1 0 1 0 0 00 1 1 1 0 1 1

1 0 0 1 1 1 1

1 0 1 1 1 1 1

1 1 0 1 0 0 0

1 1 1 1 0 1 1

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Creating Chips• What would a computer chip look like for

the expression:(A B) (B C)

• Each operator will get a gate (4 total here)• Respect order of operations– (), , , , – Often, () will denote exactly one order– Operations that are “earlier” will be “closer” to

the input; “later” operations closer to output

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Creating Chips• What would a computer chip look like for

the expression:X = (A B) (B C)

A

B

C

X

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Chips Summary• Anything we can reason with logic can be

implemented as a circuit, such as: – Controllers for devices (keyboard, mouse, screen,

etc.)– Addition/Subtraction – Multiplication/Division– Along with a few other operations, we have a whole

microprocessor made just with logic gates!• But wait, there’s more! – What about memory?

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The Point of Memory• Most processors have the power to add

pairs of numbers• How do we compute 9 + 8 + 4?• Answer: – Compute 9 + 8 and store the result in memory– Add 4 to the result stored in memory

• Memory is also implemented using gates!• There are many types/levels of memory

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Circuits that Remember• Behold, the Flip-Flop!

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Setting the Output to 1

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Setting the Output to 1 (cont.)

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Setting the Output to 1 (cont.)

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What Else Can We Do?

What happens if we put a zero on both inputs? …a one on the upper input and a zero on the lower input? …a zero on the upper input and a one on the lower input? …a one on both inputs?

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Storing Data with Flip Flops• One flip-flop stores a bit, which can be

abstracted to mean 0/1, ‘yes/no’ or ‘true/false’• Two flip-flops store two bits – Using a predetermined code, we can now represent

4 things instead of 2• 00 10 01 11

• We could make up a code for numbers: – 00 = 0, 10 = 1, 01 = 2, 11 = 3

• But really, we don’t need a code at all!

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Binary Integers• Number systems are completely arbitrary• Our favorite is Base 10:

– First place = 1’s (100), Second place = 10’s (101)– Third place = 100’s (102), Fourth place = 1000’s (103)– 17 base 10 = 1 * 101 + 7 * 100

– Need 10 digits (0 – 9) to count correctly in each place• Why do we like base 10 so much?• How about base 2?:

– First place = 1’s (20)– Second place = 2’s (21)– Third place = ??? Fourth place = ???– How many digits do we need?

• Base 2 is convenient for computers

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Binary Integer Practice• How do we write 0 in

base-2?– 00

• 1?– 01

• 2?– 10

• 3?– 11

• Wait…isn’t that familiar?

• How many different numbers can we store with 3 flip flops?– 000 – 111 on board

• What about 8?– What’s the biggest

number we can count to?

• What about 32?• 64??

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Base 10 to Base 2 conversion

1. Divide the number by 2 and write down the remainder

2. As long as the quotient obtained is not zero, continue to divide by 2 and write down the remainder to the LEFT of the last number

• 78 • 78/2 = 39 R 0 0• 39/2 = 19 R 1 10• 19/2 = 9 R 1 110• 9/2 = 4 R 1 1110• 4/2 = 2 R 0 01110• 2/2 = 1 R 0 001110• 1/2 = 0 R 1

1001110

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Memory & Abstraction• Flip Flops can be used to implement memory

– These are sometimes called SRAM (Static Random Access Memory)– …meaning that once the circuit is “set” to 1 or 0, it will stay that way

until a new signal is used to re-set it• DRAM (Dynamic Random Access Memory):

– Use a capacitor to store the charge (has to be refreshed periodically)– Faster

• BUT…

• Abstraction tells us that (for most purposes) it really doesn’t matter how we implement memory -- we just know that we can store (and retrieve) “a bit” at a time

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Scaling Up• One Flip-Flop: 1 bit (3 gates on chip)• 8 bits = 1 Byte (24 gates on chip)• 1024 Bytes = 1 Kilobyte (24,000 gates on chip)• 1024 Kilobytes = 1 Megabyte (24,000,000 gates

on chip)• 1024 Megabytes = 1 Gigabyte (24,000,000,000

gates on chip!)• Cheapest laptop on BestBuy.com in 2010: 2 GB

RAM (48,000,000,000 gates!)

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Accessing Memory• Memory is stored in rows of Bytes (8 bits)• Each row has an address represented by a unique

binary number– i.e. memory row number 23 is addressed as 10111 (1*24

+ 0*23 + 1* 22 + 1* 21 + 1* 20)• Amount of memory in a machine is limited by the

biggest number the processor can represent:– 32 bit processors (still most common): 4,294,967,295 ~ 4

GB MAX– 64 bit processors (growing in popularity):

18,446,744,073,709,551,615 ~ 18.45 million TB

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Hexadecimal• It would be very inconvenient to read and

write a 64-bit address in binary:0010100111010110111110001001011000010001110011011110000011100000

• Instead, we group each set of 4 bits together into a hexadecimal (base 16) digit:– The digits are 0, 1, 2, …, 9, A (10), B (11), …, E (14), F

(15) 0010 1001 1101 0110 1111 1000 1001 0110 0001 0001 1100 1101 1110 0000 1110 0000

2 9 D 6 F 8 9 6 1 1 C D E 0 E 0

• …which we write, by convention, with a “0x” preceding the number to indicate it’s heXadecimal:– 0x29D6F89611CDE0E0

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Other Memory Concepts(read the book!!)

• Mass storage: hard disks, CDs, USB/flash drives…– Stores information without a constant supply of electricity– Larger than RAM– Slower than RAM– Often removable– Physically often more fragile than RAM

• CDs, hard drives, etc. actually spin and have tracks divided into sectors, read by a read/write head– Seek time: Time to move head to the proper track– Latency: Time to wait for the disk to rotate into place– Access time: Seek + latency– Transfer rate: How many bits/second can be read/written once you’ve

found the right spot• Flash memory: high capacity, no moving parts, but less reliable for

long-term storage

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Representing Data• We’ve seen how to (basically) process

different forms of data• We’ve seen how to store it• But how do we represent something that is

not a number?

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Representing Text• Use a simple code: 0 = A, 1 = B, 2 = C … • ASCII encoding translates this idea to binary, using a byte (actually 7

bits for 128 total values):

• Unicode encoding uses 2 bytes and allows for more characters, like é, ö and Щ

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Representing Images• A computer screen is technically a dense grid of tiny

multi-colored light bulbs• An image on the screen is a setting of each light bulb to a

specific color• A data representation of such an image is called a bit-map

– Just a row-by-row of color values: red red red redblue blue blue bluered red red redgreen green green green

• If this seems like it takes up a lot of memory, you are right– But memory-saving techniques must still be converted to a

bitmap for display

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Representing Colors• Three Primary Colors: Red, Green, Blue• RGB: “Mixing” system, similar to mixing paints

– Choose the ‘amount’ of each color to mix in– Use 1 byte for each primary (16 bits)

• Minimum amount? Maximum?– Pure Red = (255,0,0) = 11111111 00000000 00000000– “Cleveland” Brown = (56,43,28)

= 00111000 00101011 00100110

• Remember hexadecimal?– Pure Red = 0xFF0000– “Cleveland” Brown = 0x382B26– Common in HTML (as we’ll learn this year!)

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Representing SoundThe sound wave represented by the sequence 0, 1.5, 2.0, 1.5, 2.0, 3.0, 4.0, 3.0, 0

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In-Class Exercise1. Learn how to add two binary integers by

hand2. Learn how to add two one-bit integers

using logic gates3. Combine the one-bit adder into an 8-bit

adder

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Representing Integers Revisited

• So we learned a really slick way to count on a computer from 0 – 2k where k is the number of bits we can use

• And we learned that we can use logic gates to add them

• But that’s only positive numbers!• We can do better

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Two’s Complement• Two’s complement is a clever representation that allows

binary addition to be performed in an elegant way

Left-most bit is called the ‘sign bit’

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Two’s Complement cont.

This works in both directions!

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Adding in Two’s Complement

• Turns out, we can do this with one easy modification to our original algorithm: – If the left-most addition results in a carry, throw

that bit away to get the correct result• Work with a friend to:

1. Convert the numbers to 4-bit two’s complement2. Add them3. Verify the result4. Solve: (1 + 5), (6 + -2), (3 + -5)

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Floating Point Numbers• Non-integers are a problem…• Remember that any rational number can be represented as a fraction

– …but we probably don’t want to do this, since• (a) we’d need to use two words for each number (i.e., the numerator and

the denominator)• (b) fractions are hard to manipulate (add, subtract, etc.)

• Irrational numbers can’t be written down at all, of course• Notice that any representation we choose will by definition have

limited precision, since we can only represent 232 different values in a 32-bit word– 1/3 isn’t exactly 1/3 (let’s try it on a calculator!)

• In general, we also lose precision (introduce errors) when we operate on floating point numbers

• You don’t need to know the details of how “floating point” numbers are represented, but you should know they are represented differently