1 12/08/03sw abingdon and witney college binary converting to and from decimal
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12/08/03 SW Abingdon and Witney College 1
Binary
Converting to and from decimal
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Decimal
We normally use the decimal (denary) system, also called base 10
There are 10 different symbols (digits) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 To count higher than nine we re-use the
symbols by putting them in columns The value of a symbol depends on its
position
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Decimal positions
1000 100 10 1
Thousands Hundreds Tens Units
8 2 5 3
Eight thousand two hundred and fifty three
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Binary
Computers use the binary system, also called base 2
There are two different symbols (digits) 0, 1 To count higher than one we re-use the
symbols by putting them in columns The value of a symbol depends on its
position
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Binary positions
8 4 2 1
Eights Fours Twos Units
1 1 0 1
One eight, one four, no twos and one unitThat makes thirteen
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Counting in binary and decimal
0 0 1000 8
1 1 1001 9
10 2 1010 10
11 3 1011 11
100 4 1100 12
101 5 1101 13
110 6 1110 14
111 7 1111 15
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Why do computers use binary?
Computer components that store or handle data are often two-state devices
This is like a switch that can be on or off A memory unit could exist in two voltage
states, high or low A voltage on a cable could be high or low A light could be on or off Two states can be coded by binary 0 and 1
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Why am I learning about binary?
You will learn about IP addresses and how to split up a network into subnets
You need to work out subnet addresses and workstation addresses
For this you need to use binary
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Bytes or octets
We often handle binary digits (bits) in groups of eight
Sometimes these groups are called bytes Sometimes they are called octets We shall often be calling them octets Examples of octets:
00101101
10110010
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Coding data into binary
Decimal numbers can be converted into binary numbers
Characters (letters, punctuation, digits) can be coded using ASCII or EBCDIC
Graphics, sounds and videos have several different and complicated methods for coding them
Program instructions are coded in a machine code that depends on the type of processor
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ACSII
American Standard Code for Information Interchange used for our alphabet
Uses 8 bits (one byte/octet) for each character
7 bits for the basic character and one bit for error checking
Chinese, Arabic and some other languages require 16 bits (2 bytes) for each character – they use Unicode, related to ASCII
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Convert binary to decimal
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary
Convert 11001010 binary to decimalWrite in the binary digits under their values
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Convert binary to decimal
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary 1 1 0 0 1 0 1 0
Convert 11001010 binary to decimalWrite in the binary digits under their valuesNext write in the value for each binary 1 digit
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Convert binary to decimal
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary 1 1 0 0 1 0 1 0
128 64 8 2
Convert 11001010 binary to decimalWrite in the binary digits under their valuesNext write in the value for each binary 1 digitAdd up the values 128 + 64 + 8 + 2 = 202
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One for you to try
Convert 10010101 from binary (base 2) to decimal (base 10)
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary
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Convert decimal to binary
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary
Check
Convert 185 decimal to binaryCan you take 128 from 185? Yes.Put 1 under 128What is left? 185-128 = 57
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Convert decimal to binary
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary 1
Check
Converting 185: we have 57 leftCan you take 64 from 57? No.Put 0 under 64What is left? Still 57
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Convert decimal to binary
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary 1 0
Check
Converting 185: we have 57 leftCan you take 32 from 57? Yes.Put 1 under 32What is left? 57 – 32 = 25
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Convert decimal to binary
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary 1 0 1
Check
Converting 185: we have 25 leftCan you take 16 from 25? Yes.Put 1 under 16What is left? 25 – 16 = 9
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Convert decimal to binary
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary 1 0 1 1
Check
Converting 185: we have 9 leftCan you take 8 from 9? Yes.Put 1 under 8What is left? 9 – 8 = 1
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Convert decimal to binary
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary 1 0 1 1 1
Check
Converting 185: we have 1 leftCan you take 4 from 1? No.Put 0 under 4What is left? Still 1
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Convert decimal to binary
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary 1 0 1 1 1 0
Check
Converting 185: we have 1 leftCan you take 2 from 1? No.Put 0 under 2What is left? Still 1
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Convert decimal to binary
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary 1 0 1 1 1 0 0
Check
Converting 185: we have 1 leftCan you take 1 from 1? Yes.Put 1 under 1What is left? Nothing. Finished.
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Convert decimal to binary
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary 1 0 1 1 1 0 0 1
Check
185 decimal is 10111001 binary
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Convert decimal to binary
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary 1 0 1 1 1 0 0 1
Check 128 32 16 8 1
Check: write in the values of the1 digits and add them up128 + 32 + 16 + 8 + 1 = 185That’s the number we started with. It’s correct.
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One for you to try
Convert 248 from decimal to binary Check your answer
Position 8 7 6 5 4 3 2 1
Value 128 64 32 16 8 4 2 1
Binary
Check
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End