Numbers and Counting

What you may have forgotten about our number system

Base 10

012345678910 or 1+0 - One 10 and no

units

11 or One ten and one unit12 or One ten and two units…20 or Two tens and no units…30 or Three tens and no

units…99100 or One hundred, no

tens and no units

Base 2This time, we have only two digits to work with: 0 and 1

0

1

10 or One Two and no Units

11 or One Two and one Unit (2+1=3)

100 or One Four (2^2), no Two and no Units

101 or One Four, no Two, and One Unit (4+1=5)

110 or One Four, One Two and no Units (4+2=6)

111 or One Four, One Two and One Unit (4+2+1=7)

1000 or One Eight (2^3), no Four, no Two and no Unit

…

10,000 or One Sixteen (2^4), no Eight, no Four, no Two and no Unit

Binary Coded Decimal (BCD)We group Binary Digits according to Decimal system

1

10

11

100

101

110

111

1000

1001

1 0000

1 0001 (11)1 00101 00111 0100….10 0000…..10 0001 (21)….1001 1001 (99)1 0000 0000 (100)1 0000 0001 (101)

1

2

3

4

5

6

7

8

9

10

Hexadecimal – Base 16

Again, we group four Binary Digits0 00001 00012 00103 00114 01005 01016 01107 01118 10009 1001We are out of digits, so we use letters10 A11 B12 C13 D14 E15 F

Now,

10 or One Sixteen and no Units

11 or One Sixteen and one Unit (17 in Base 10)

….

1A or One Sixteen and A (10) units

1B

1C

1D

1E

1F

20 or Two Sixteens and no Units (32 in Base 10)

More Hex98999A9B9C9D…A9AAAB…AEAFB0B1…

F9FAFBFCFDFEFF100 or One 256 (16^2), no 16, no Unit

When we are talking Hex Numbers,We either add 0x before the numberOr we add an “h” at the end of the Number. So, F7B in Hex would beWritten as 0xF7B or F7Bh.

Conversion

Binary to Hexadecimal:

1. Group the Binary into sections of four digits each.

2. Convert the four Binary digits to one digit of Hexadecimal.

Binary to Hex

10010101

1001 0101

9 5

• Group into fours

• Convert to Hex

Binary to Hex, again

110110100001

1101 1010 0001

D A 1

Group into fours

Convert to Hex

Hex value

010001010110011110001001101010111100110111101111

456789ABCDEF

Hex to Binary

EF8

E F 8

1110 1111 1000

Make some room

Convert to Binary

010001010110011110001001101010111100110111101111

456789ABCDEF

Binary to Decimal

Start with Binary Number

1001 1100

Add up the 1’s:

128

16

8

4

128 64 32 16 | 8 4 2 1

X X X X | X X X X

Binary to Decimal

Start with Binary Number

1001 1100

Add up the 1’s:

128

16

8

4

156 in Decimal

128 64 32 16 | 8 4 2 1

X X X X | X X X X

Decimal to Binary

A lot of subtracting:

252

128 64 32 16 | 8 4 2 1

X X X X | X X X X

252-128124- 64 60- 32 28 -16 12 -8 4 -4 0

What 1’s do we have?

Decimal to Binary

A lot of subtracting:

252

128 64 32 16 | 8 4 2 1

X X X X | X X X X

252-128124- 64 60- 32 28 -16 12 -8 4 -4 0

What 1’s do we have?

1111 1100

Decimal to Binary

A lot of subtracting:

89

128 64 32 16 | 8 4 2 1

X X X X | X X X X

89-64 25 -16 9- 8 1 -1 0

What 1’s do we have?

Decimal to Binary

A lot of subtracting:

89

128 64 32 16 | 8 4 2 1

X X X X | X X X X

89-64 25 -16 9- 8 1 -1 0

What 1’s do we have? 0101 1001

Grouping Numbers

• We start with a bit – one binary digit

• A bit is shortened to “b”

• Group eight bits together to form a Byte

• A Byte is shortened to “B”

• Eight bits (one Byte) is the ASCII character set and the foundation of desktop computing

• IBM uses a 7-3 code on its big computers

Some ASCII

• A capital “A” is (decimal) 65, in Binary it is 0 1 0 0 0 0 0 1

• A capital “B” is 66, in Binary it is0 1 0 0 0 0 1 0

* A lower case “a” is 97, in Binary it is

0 1 1 0 0 0 0 1

* So the alphabet becomes numbers

More Grouping

• Kilo (K) is 1000 in Decimal, 1024 in Binary (2^10)

• Mega (M) is one million in Decimal, or 1,048,576 in Binary (2^20)

• Giga (G) is one billion in Decimal and slightly larger in Binary (2^30)

How it is Used

• Modem speed (limit) is 56K bps – 56,000 bits per second maximum speed

For some reason, the FCC limits speed to 54K

• RAM capacity is measured in “G” as 2GBOr 2 billion Bytes of memory

* Hard disk capacity in G’s: 200GB or

200 Billion Bytes

Bytes

• Notice that both RAM and Hard Disk capacity are measured in Bytes. That gets confusing, so let’s spend a moment and work through what each means.

RAM for a moment

• RAM = Random Access Memory orRead And Write Memory

* Think of RAM as “square somethings” used to measure the size of a desk top. The more RAM, the bigger the desk top and the more piles of things you can make.

Hard Disk Drives

• Think of Bytes here as some sort of volume measure.

• Hard Disk size is like talking about a file cabinet size – how many Bytes (characters) can it hold (how many drawers).

• Storing more information on your Hard Disk does NOT make it run slower!

Brief History of Computing

• 1946ENIAC introduced

• Big, took up lots of floor space for vacuum tubes.

• Folk legend has it that “bug” was first seen here.

• Used to compute where a shell fired from a cannon would land

First Desktop

• 1977 Apple introduces the Apple II • Now we had computers on our desks!

• By 1980, we had two word processors and VisiCalc (spreadsheet).

• Caused a lot of pain in the “IT” department and here is why …..

How Things Worked

• You had the computer

• You had US

• You had THEM

Computer

US

THEM

THEM talked to US, who talked to the Computer (using Assembler Language)And US gave THEM the computer output.

Just a Fad

• IBM was certain that personal computers would fade out after a couple years.

• Just to keep customers happy, they put together a Personal Computer (PC) and stole the name from Apple.

• IBM Personal Computer introduced in 1981 with two floppy disk drives.

Side Note

• IBM did not want to build an Operating System, so they hired a “kid” who said he had one – his name is Bill Gates.

• Bill bought QDOS (Quick and Dirty O S) for $75,000 and then licensed it (NOT sold it) to IBM – hence his first millions. Gary Kidall still shakes his head!

Inside

• The core of a computer, or its personality, is in the BIOS – Basic Input\Output System.

• So, IBM got lawyers to put up a patent “fence” around the BIOS.

BIOS

Some More Numbers

• There were about $900 in parts in a PC

• It sold for $3500.

• Let’s see … 3500 – 900 = 2600

• That number is called PROFIT

• Made a lot of people want to build PCs!

In Texas - 1983

• Two very smart guys got together and figured out a BIOS that would get around all the IBM patents – and Compaq computer was born.

More in 1983

• Apple Computer introduces the Lisa – a graphical/mouse computer.

And more 1983

• IBM introduces the XT model, still running at 4.77 MHz

but with a hard disk drive (10 MB).

* Still $3500. Monitor extra.

1984

• Apple introduces the Macintosh

• IBM introduces the AT model computer with 80286 processor at 6MHz.

1987

• IBM tries to retake the market by introducing the PS/2 line of computers. Lots of new patents and expensive licensing agreement forms.

• Technically superior, but a dud in the marketplace. Required “personality” disks for each expansion card.

1993

• The Pentium processor introduced. Potential for 32-bit computing in a 16-bit world (DOS).

1995

• Microsoft launches Windows 95

• Clock speeds reach 266 MHz.

2000

• Clock speeds reach 1000 MHz, or 1 Gig