binary system and logical gates the foundation of all computing devices are built on hexadecimal...

Binary System and Logical Gates

The foundation of all computing devices are built on

Hexadecimal numbers are more convenient for working on hardware.

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Objectives

After studying, students will be able to: Translate between base-ten and base-two

numbers, and represent negative numbers using both sign-magnitude and two’s complement representations

Explain how floating-point numbers, character, sounds, and images are represented inside the computer

Build truth tables for Boolean expressions and determine when they are true or false

Describe the relationship between Boolean logic and computer hardware/circuits

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The Binary Numbering System

How can an electronic (or magnetic) machine represent information?

Key requirements: clear, unambiguous, reliable

External representation is human-oriented base-10 numbers keyboard characters

Internal representation is computer-oriented base-2 numbers base-2 codes for characters

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Invitation to Computer Science, 6th Edition4

Two Important Factors

Two types of integers: Unsigned: not negative Signed: any numbers: positive, zero,

negative The number of bits used to hold the

integer 4 bits, 8 bits (1 byte), 2 bytes, 4 bytes,

any other sizes It is crucially important for signed

integers

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Base 2 and base 10

The binary numbering system is a base-2 positional numbering system

Base ten: Uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Each place corresponds to a power of 10 1,943 = 1 * 103 + 9 * 102 + 4 * 101 + 3 * 100

Base two: Uses 2 digits: 0, 1 Each place corresponds to a power of 2 1101 = 1 * 23 + 1 * 22 + 0 * 21 + 1 * 20 = 13

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Conversion

Converting from binary to decimal Add up powers of two where a 1 appears in

the binary number Converting from decimal to binary

Repeatedly divide by two and record the remainder

Example, convert 9: 9/2 = 4 remainder 1, binary number = 1 4/2 = 2 remainder 0, binary number = 01 2/2 = 1 remainder 0, binary number = 001 1/2 = 0 remainder 1, binary number = 1001

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Binary to decimal (1)

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Binary to decimal (2)Binary number 10011011

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Binary to decimal (3)Binary number 10011011

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Binary number 10011011


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Binary number 10011011

Decimal to Binary (1)Decimal number 156

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15


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Arithmetic operations

Computers use fixed-length binary numbers for integers, e.g., with 4 bits could represent 0 to 15

Arithmetic overflow: when computer tries to make a number that is too large, e.g. 14 + 2 with 4 bits

Binary addition: 0+0=0, 0+1=1, 1+0=1, 1+1=0 with carry of 1

Example: 0101 + 0011 = 1000Invitation to Computer Science, 6th Edition

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Signed integers

Signed integers include negative numbers

Method I: Sign/magnitude notation uses

1 bit for sign, the rest for value.+5 = 0101, -5 = 11010 = 0000 and 1000!

Two expressions for 0. No Good!

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Signed integers

Method II: Two’s complement The MSB bit is a sign bit:

0 --- positive number 1 --- negative number

Steps: flip every bit Add 1 +5 = 0101, -5 = 1010 + 1 = 1011 0 = 0000, -0 = 1111 + 1 = 0000

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Two’s Complement

Determine number of bits used to hold the number

The most significant bit (the left-most bit) is the sign bit. 0: positive; 1: negative

Bits flip and add 1

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Two’s Complement: B to D

Interpret 11011011 as a two's complement binary number, and give its decimal equivalent. Notice we use 8 bits. First, note that the number is negative, since it

starts with a 1. Change the sign to get the magnitude of the

number. 1 1 0 1 1 0 1 1 ¬ 0 0 1 0 0 1 0 0 + 1---------------- 0 0 1 0 0 1 0 1

Convert the magnitude to decimal: 001001012 = 2516 = 2×16 + 5 = 3710.

Since the original number was negative, the final result is -37. 22

Two’s Complement: D to B

Convert the magnitude, -72 to 8-bit binary

72 --- 01001000 Conversion:

0 1 0 0 1 0 0 0 ¬ 1 0 1 1 0 1 1 1 + 1

---------------1 0 1 1 1 0 0 0

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Binary Numbers: Using 3 bits

000 ------------ 0

001 ------------ 1

010 ------------ 2

011 ------------ 3

100 ------------ -4

101 ------------ -3

110 ------------ -2

111 ------------ -1

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Binary Numbers: -1

3 bits: 111 ------------ -1 4 bits: 1111 ----------- -1 8 bits: 11111111 ------- -1 16 bits: 1111111111111111 -- -1

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Overflow

If a location is too small for a number, overflow occurs.

for unsigned integers, a byte can hold a number between 0 and 255. If using one byte to store 256, we get 0.

For signed integers, a byte can hold a number between -128 and 127. If using one byte to store -130 or 128, overflows occurs.

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Hexadecimal Numbers

Based on 16 The 16 digits are 0, 1, 2, 3, 4, 5, 6, 7,

8, 9, A, B, C, D, E, F Examples:

Hexadecimal Binary Decimal 1 1

1 F 1111

15 11 00010001

17 ff 11111111

255

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Hexadecimal Numbers

Hexadecimal Binary Decimal 1 0001 1 2 0010 2 3 0011 3 4 0100 4 5 0101 5 6 0110 6 7 0111 7 8 1000 8 9 1001 9 A 1010 10 B 1011 11 C 1100 12 D 1101 13 E 1110 14 F 1111 15

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Hexadecimal Numbers

Decimal to hexadecimal: repeatedly divide the decimal number by 16

Hexadecimal to decimal:Right most number + 16 * second number from the right + 256*third number from the right + …

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Hexadecimal Numbers

Large binary number to decimal number through hexadecimal number: Group the binary number into groups of

4 digits starting from right. Convert each group into a hexadecimal

digit Convert the hexadecimal number to

decimal.

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Hexadecimal Numbers

Large decimal number to binary number through hexadecimal number: Convert the decimal number to

hexadecimal number Convert each digit of the hexadecimal

number into corresponding 4 digits

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Floating numbers (optional)

Floating point numbers use binary scientific notation Scientific notation, base 10: 1.35 ×

10-5

Base 2: 3.2510 = 11.012 = 1.101 × 21

Characters and text: map characters onto binary numbers in a standard way ASCII (8-bit numbers for each character) Unicode (16-bit numbers for each

character)32


Conversion from binary numbers to decimal: adding exponentialsUsing 4 bits to represent a binary number

Binary number Decimal0000 --- 00001 --- 10010 --- 20011 --- 30100 --- 4…0111 --- 7 unsigned signed1000 --- 8 -8 (8 – 16)1001 --- 9 -7 (9 – 16)…1111 --- 15 -1 (15 – 16)

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Conversion from decimal numbers to binary: recursively division by 2: 10, 20, 50, -40

Operations of binary operations Addition

Carry Overflow

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Sounds and Images

Sounds and images require converting naturally analog representations to digital representations

Sound waves characterized by: amplitude: height of the wave at a

moment in time period: length of time until wave

pattern repeats frequency: number of periods per time

unit35

Digitization

Digitize: to convert to a digital form Sampling: record sound wave values at

fixed, discrete intervals To reproduce sound, approximate using

samples Quality determine by:

Sampling rate: number of samples per second More samples = more accurate wave form

Bit depth: number of bits per sample More bits = more accurate amplitude

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Images

Image sampling: record color or intensity at fixed, discrete intervals in two dimensions

Pixels: individual recorded samples RGB encoding scheme:

Colors are combinations of red, green, and blue

One byte each for red, green, and blue Raster graphics store picture as

two-d grid of pixel values39

Digitizatin

Data size: how much to store: 1000 integer values 10-page text paper 60-second sound file 480 by 640 image

Data compression: storing data in a reduced-size form to save space/time Lossless: data can be perfectly restored Lossy: data cannot be perfectly restored

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Computers use binary because “bistable” systems are reliable current on/off magnetic field left/right

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Transistors

Transistors Solid-state

switches Change on/off

when given power on control line

Extremely small (billions per chip)

Enable computers that work with gigabytes of data

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Boolean Logic and Gates

Boolean logic: rules for manipulating true/false Boolean expressions can be converted to circuits Hardware design/logic design pertains to the

design and construction of new circuits Note that 1/0 of binary representations maps to

true/false of Boolean logic Boolean expressions: x ≤ 35, a = 12 Boolean operators: (0 ≤ x) AND (x ≤ 35), (a =

12) OR (a = 13), NOT (a = 12)(0 ≤ x) (x ≤ 35), (a = 12) + (a = 13), ~(a = 12)

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Boolean Logic and Gates (continued)

Truth tables lay out true/false values for Boolean expressions, for each possible true/false input

Example: (a b) + (a ~b)

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a b ~b (a b) (a ~b)(a b) + (a ~b)

true true false true false true

true false true false true true

false true false false false false

false false true false false false

Boolean Logic and Gates (continued)

Gate: an electronic device that operates on inputs to produce outputs

Each gate corresponds to a Boolean operator

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Boolean Logic and Gates (continued) Gates are built from

transistors NOT gate: 1 transistor AND gate: 3

transistors OR gate: 3 transistors NAND and NOR: 2

transistors

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Building Computer Circuits

Circuit: has input wires, contains gates connected by wires, and has output wires

Outputs depend only on current inputs: no state

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To convert a circuit to a Boolean expression: Start with output and work backwards

Find next gate back, convert to Boolean operator Repeat for each input, filling in left and/or right side

To convert a Boolean expression to a circuit: Similar approach

To build a circuit from desired outcomes: Use standard circuit construction algorithm:

e.g., sum-of-products algorithm

Invitation to Computer Science, 6th Edition

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Building Computer Circuits (continued)

Example from text Build truth table:


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a b c Output1 Output2

0 0 0 0 1

0 0 1 0 0

0 1 0 1 1

0 1 1 0 1

1 0 0 0 0

1 0 1 0 0

1 1 0 1 1

1 1 1 0 0


Example from text Find true rows for Output1


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a b c Output1 Output2

0 0 0 0 1

0 0 1 0 0

0 1 0 1 1

0 1 1 0 1

1 0 0 0 0

1 0 1 0 0

1 1 0 1 1

1 1 1 0 0


Example from text For each true row, AND input sto

make 1 a = 0, b = 1, c = 0: (~a b ~c) a = 1, b = 1, c = 0: (a b ~c)

Combine row subexpressions with OR (~a b ~c) + (a b ~c)

Build circuit from expression (and repeat for other output)


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Compare-for-equality (CE) circuit Input is two unsigned binary numbers Output is 1 if inputs are identical, and

0 otherwise Start with one-bit version (1-CE) and

build general version from that


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1-CE circuit: compare two input bits for equality

Truth table:

Boolean expression: (a b) + (~a ~b)


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a b Output

0 0 1

0 1 0

1 0 0

1 1 1


N-bit CE circuit Input: a0a2…an-1 and b0b2…bn-1, where ai

and bi are individual bits

Pair up corresponding bits: a0 with b0, a1 with b1, etc.

Run a 1-CE circuit on each pair AND the results


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Full adder circuit Input is two unsigned N-bit numbers Output is one unsigned N-bit number, the

result of adding inputs together Example:

Start with one-bit adder (1-ADD)


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1

0 0 1 0 1

+ 0 1 0 0 1

0 1 1 1 0


Sum digit, si, has Boolean expression:(~ai ~bi ci) + (~ai bi ~ci) +

(ai ~bi ~ci) + (ai bi ci)

Carry digit, ci+1, has Boolean expression:(~ai bi ci) + (ai ~bi ci) +

(ai bi ~ci) + (ai bi ci)


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N-bit adder circuit Input: a0a2…an-1 and b0b2…bn-1, where ai and bi

are individual bits a0 and b0 are least significant digits: ones place

Pair up corresponding bits: a0 with b0, a1 with b1, etc.

Run 1-ADD on a0 and b0, with fixed carry in c0 = 0

Feed carry out c1 to next 1-ADD and repeat


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Control Circuits

Control circuits make decisions, determine order of operations, select data values

Multiplexor selects one from among many inputs 2N input lines N selector lines 1 output line

Each input line corresponds to a unique pattern on selector lines

That input value is passed to output


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Control Circuits (continued)

Decoder sends a signal out only one output, chosen by its input N input lines 2N output lines

Each output line corresponds to a unique pattern on input lines

Only the chosen output line produces 1, all others output 0


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Control Circuits (continued)

Decoder circuit uses To select a single arithmetic instruction,

given a code for that instruction Code activates one output line, that line

activates corresponding arithmetic circuit Multiplexor circuit uses

To choose one data value from among a set, based on selector pattern

Many data values flow into the multiplexor, only the selected one comes out


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Summary

Computers use binary representations because they maximize reliability for electronic systems

Many kinds of data may be represented at least in an approximate digital form using binary values

Boolean logic describes how to build and manipulate expressions that are true/false

We can build logic gates that act like Boolean operators using transistors

Circuits may be built from logic gates: circuits correspond to Boolean expressions


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Summary

Sum-of-products is a circuit design algorithm: takes a specification and ends with a circuit

We can build circuits for basic algorithmic tasks: Comparisons (compare-for-equality

circuit) Arithmetic (adder circuit) Control (multiplexor and decoder

circuits)


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Pseudo language

Pseudo languageStep Operation

Step 1 operation 1Step 2 operation 2……Step N operation N

Pseudo language

Input Get value(s) of var1(, var2,…)

Output Print value(s) of var1(, var2,…)

Pseudo language

Processing Computation

Set the value of “variable” to “arithmetic expression”

Conditional If “condition” is true then

fist set of operations else

second set of opeations

Pseudo language

Processing Computation

Set the value of “variable” to “arithmetic expression”

Conditional if “condition” is true then

fist set of operations else

second set of opeations Using indentations

Pseudo language

Processing/computation (cont.) Iterative (Loops)

While (condition) do step I through step jSteps i: operation

…Step j operation

[ End of the loop ] --- optional

Pseudo language & Examples

The calculation of gas mileage1. Get the value for gallons used, starting

mileage, ending mileage2. Set the value of distance driven to (ending

mileage – starting mileage)3. Set the value of gas mileage to (distance

driven / gallons used)4. Print the value of gas mileage5. [ Stop ] --- optional


Computing the sum of first N integers1. Get the value of N2. Set sum to 0, count to 03. While count is <= N do step 4 to 54. Set the sum to (sum + count)5. Set the value of count to (count + 1)6. Print the value of sum


Get the value of largest of n numbers1. Get the value of n and values of numbers T1

T2…Tn

2. Set the value of largest to the value of T1

3. Set the value of i to 24. While i is <= n do step 4 to 55. If the value of Tn is larger than the value of

largest then Set the value largest to the value of Tn

6. Set the value of i to (i + 1)7. Print the value of largest


Sort n numbers: discussing in class Sequential sort Quick sort

Search: discussing in class Sequential search Binary search

Efficiency of an Algorithm

The Efficiency of an Algorithm is calculated by the computations (arithmetic operations, comparisons…) needed to complete the rask

Usually the variable is the magnitude of the task. For example, the size of the array

O(1), O(log(n)), O(n), O(n2), …,O(en)…

binary system and logical gates the foundation of all computing devices are built on hexadecimal...

Documents

binary system

number of bits

binary numbering system

bits binary addition

positive number

numbers base

fixedlength binary numbers

negative number steps