Today’s lecture

 Understand the difference between analogue , and and digital information  Convert between decimal numbers and binary digital numbers information Lecture 2 – COMPSCI111/111G SS 2018

Analogue vs digital information Encoding information

 Information in the real world is continuous  Real world information is stored by a computer  Continuous signal using numbers

Weight  Visual information shown 11111111111111111111111 01111111111111111111111 00001111111111111111111 Real Weight 00000011111111111111111 00000000011111111111111 44444000001111111111111  75444000000011111111111 Information stored by a computer is digital 55554401000000111111111 33367544000000011111111  Represented by discrete numbers 22283554444000000111111 99928357544000000011111 99999233657504000001111 99999983666554400000011 Weight 99999928338674400000001 shown Image Pixels 1. Give each pixel colour a number. 2. Let the computer draw the numbers as Real Weight coloured pixels (eg. black = 0). Encoding information Numbers and

 Sound information  Numbers are used to represent all information manipulated by a computer.

Sound  Computers use the system: ― Binary values are either 0 or 1. Waveform Samples

1. Give each sample a number (height of  We use the decimal number system: green box). ― 0 to 9 are decimal values. 2. Let the computer move the loudspeaker membrane according to the samples.

Number Systems Positional Notation

 Base:  Any number can be expressed as: ― Specifies the number of digits used by the system. ― Binary is base 2. ― Decimal is base 10. ∗ ∗ ⋯ ∗

 Positional notation: where is the digit at position , and is the ― Describes how numbers are written. base.

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Most significant digit Least significant digit Decimal Examples Storing Decimal Numbers in a Computer

 657  Series of dials: 6∗10 5∗10 7∗10 ― Each dial goes from 0 to 9.

 Information is stored digitally: 600 50 7 657 ― Finite number of states – 10 per dial. ― No in-between states.

 Decimal number system:  9308 ― 1st dial from right: 10 9∗10 3∗10 0∗10 8∗10 ― 2nd dial from right: 10 6∗10 3∗10 8∗10 638 ― 3rd dial from right: 10 ― etc. 9000 300 0 8 9308

Exercises Switches

The following two questions relate to dials that  A dial is complicated. have 10 different states, as discussed in the ― Each dial has 10 different states (0 – 9). previous slide. ― Physically creating circuits that distinguish all states is complicated.  Given a machine that uses 4 dials, how many ― Would need to distinguish 10 different strengths of different numbers can we represent? electricity (voltages).

 Switches are simple.  If we want to represent 256 different values, ― Each switch is off or on (0 or 1). how many dials do we need? ― Physically creating the circuits is easy. ― Switch off: electrical current cannot flow. ― Switch on: electrical current can flow. Bits and Bytes Using Binary Numbers

 Each binary number is known as a Binary digIT, How many different values/states can we have or . with:

 A bit can be either a 0 or a 1 1 bit: 2bits: 3 bits: 100 0 00 000  Bits are used in groups. 01 001 101 1 10 010 110

 A group of eight bits is referred to as a . 11 011 111

Exercises Converting binary to decimal

 How many different values can we represent  110 with a byte? 1∗2 1∗2 0∗2

4206

 If we want to represent 30 different values, how many bits would we need?  10110 1∗2 0∗2 1∗2 1∗2 0∗2

16042022 Converting from decimal to binary Exercises

 35  106  What is the decimal equivalent of 101111?

2 106 235 253 0 2171 226 1 28 1 Read the remainders 213 0 24 0 Read the remainders from the bottom up. 26 1 from the bottom up. 22 0 23 0 21 0  What is the binary equivalent of 123? 21 1 01 01

 35 is 100011 in binary  106 is 1101010 in binary

Prefixes Decimal prefixes

 A group of 8 bits is a byte 10n Prefix Symbol Decimal  A group of 4 bits is a 1 none 1

103 kilo K 1000  Bytes are the common for memory capacity 106 mega M 1,000,000 109 giga G 1,000,000,000  There are two sets of prefixes: 1012 tera T 1,000,000,000,000  Decimal 15  Binary 10 peta P 1,000,000,000,000,000 1018 exa E 1,000,000,000,000,000,000

1021 zetta Z 1,000,000,000,000,000,000,000 Binary prefixes Prefixes in Computer Science

2n Prefix Symbol Decimal  Both decimal and binary prefixes are used in 20 none 1 Computer Science

10 2 kibi Ki 1024  Decimal prefixes are preferred because they are 220 mebi Mi 1,048,576 easier to calculate, however binary prefixes are

230 gibi Gi 1,073,741,824 more accurate

240 tebi Ti 1,099,511,627,776 Decimal prefix Value (bytes)

250 pebi Pi 1,125,899,906,842,624 8 bits 1 byte same

60 2 exbi Ei 1,152,921,504,606,846,976 1 KiB 1 KB 1024 ≠ 1000 (1 x 210 bytes) (1 x 103 bytes) 270 zebi Zi 1,180,591,620,717,411,303,424 1 MiB 1 MB 1,048,576 ≠ 1,000,000 (1 x 220 bytes) (1 x 106 bytes)

Example – hard disk sizes Exercises

 A 160GB hard disk is equivalent to 149.01GiB  Which has more bytes, 1KB or 1KiB?  160GB = 160 x 109  149.01GiB = (160 x 109) ÷ 230

 How many bytes are in 128MB? Summary

 Computers use the binary number system  We can convert numbers between decimal and binary

 Decimal prefixes and binary prefixes are used for counting large numbers of bytes