Chapter 3 Representing Numbers and Text in Binary

Chapter 3Representing Numbers and

Text in Binary

Information Technology in Theory

By Pelin Aksoy and Laura DeNardis

2

Objectives

• Understand the binary numbering system• Mathematically convert numbers between decimal

and binary• Understand binary coded decimal (BCD)

representation


3

Objectives (continued)

• Learn about alternative numbering systems such as octal and hexadecimal and explain their significance in information technology

• Provide real-world examples of binary and hexadecimal representation in information technology

• Convert alphanumeric text into binary


4

The Binary Numbering System

• Any information can be represented in binary• Binary is a code with two symbols, 0 and 1• To understand binary, it’s helpful to think about how

the decimal system works


5

Think About the Decimal System


6

The Base 10 SystemDecimal digits are combined to create larger numbers4268.25 = (4 x 103) + (2 x 102) + (6 x 101) + (8 x 100) + (2 x 10–1) + (5 x 10–2)10 raised to the power of...10–2 =1/(10x10)=0.0110–1 =1/10=0.1100 =1101 =10102 =10x10=100103 =10x10x10=1000104 =10x10x10x10=10,000and so on...Therefore, decimal is also called the base 10 system


7

Base-2 System

• Binary is referred to as the base-2 system• How do we count above 1?

– As in the decimal system, each placeholder has its own weighting

– The weightings are based on powers of 2 instead of powers of 10

– The last digit has a value of 1 and occupies the “8s place”


8

Base-2 System (continued)

• The number 1010 is made up of four digits– The rightmost 0 occupies the “1s place” – The next digit has a value of 1 and occupies the

“2s place” – The next digit has a value of 0 and occupies the

“4s place”– The last digit has a value of 1 and occupies the “8s

place”


9

Binary to Decimal Integer Conversion1011.11 = (1 x 23) + (0 x 22) + (1 x 21) + (1 x 20) + (1 x 2–

1) + (1 x 2–2)2 raised to the power of...2–4 =1/(2x2x2x2)=0.06252–3 =1/(2x2x2)=0.1252–2 =1/(2x2)=0.252–1 =1/2=0.520 =121 =222 =2x2=423 =2x2x2=8and so on....


10

Binary to Decimal Integer Conversion


11

Converting an 8-bit Number into Decimal


12

Real-World Example of Binary to Decimal Conversion

• Every computing device connecting to the Internet uses a unique identifier, known as an IP (Internet Protocol) address

• Addresses are either 32 or 128 bits long• An example of an address is:

– 01000111001111001001100010100000 • Dotted decimal format condenses the addresses

– For example 71.60.152.160


13

Converting to Dotted Decimal Format

• Step 1: – Break the binary address into four groups of

8 bits• Step 2:

– Convert each group of 8 bits into decimal• Step 3:

– Separate each of the four resulting numbers with dots


14

Example

• Convert the following binary IP address into dotted decimal format:01000111001111001001100010100000

• Step 1: Separate the IP address into four octets – 01000111 (Octet 1) – 00111100 (Octet 2) – 10011000 (Octet 3) – 10100000 (Octet 4)


15

Example (continued)

• Step 2: Convert each binary octet into its equivalent decimal number– 01000111 = 0+64+0+0+0+4+2+1 = 71– 00111100 = 0+0+32+16+8+4+0+0 = 60– 10011000 = 128+0+0+16+8+0+0+0 =152– 10100000 = 128+0+32+0+0+0+0+0 = 160

• Step 3: Write the decimal values separated by dots– 71.60.152.160


16

Decimal Integer to Binary Conversion:Some Intuitive Examples

Decimal to binary conversion Explanation of conversion

The decimal number 1(10) = 1(2) The rightmost bit is the 1s place, so a 1 in that placeholder equals 1.

The decimal number 2(10) = 10(2) The second bit from the right is the 2s place, so a 1 in that spot equals 2.

The decimal number 3(10) = 11(2) Combining the preceding examples, a 1 in the 2s place and a 1 in the 1s place adds up to 3.

The decimal number 4(10) = 100(2) The third bit from the right is the 4s place. Placing a 1 in this position equals 4.


17

Method 1 for Converting Decimal to Binary

• Divide number by two, noting the presence or absence of a remainder, and using this information to derive the binary representation of the decimal number

• Construct a table with two columns to do this– First column contains the quotient– Second column contains a decision bit

to indicate the presence or absence of a remainder as a result of the division


18

ExampleConvert the number 30 to binary


19

Another ExampleConvert the number 9 to binary


20

Method 2 for Converting Decimal to Binary

• First determine how many bits are present in the binary equivalent of the decimal number

• x bits can represent 2x decimal numbers ranging from 0 to 2x-1 – A single bit can represent two values: 0 and 1– Two bits can represent four values: 00, 01, 10, and

11, or the decimal numbers 0,1,2,3– Three bits can represent eight values: 000, 001,

010, 011, 100, 101, 110, 111, or the decimal numbers 0 through 7


21

Method 2 for Converting Decimal to Binary (continued)

• As an example, representing the number 30 would require how many bits?– Four bits can represent 24 or 16 values (decimal

values from 0 to 15), not enough to represent the decimal value 30

– Five bits can represent 25 or 32 values (decimal values from 0 to 31), a sufficient amount to represent the decimal value 30

– Converting the decimal number 30 results in a five-bit binary number


22

Converting “9” to Binary____ ____ ____ ____

1s place

2s place

4s plac

e

8s place

Step 1: Compare 9 to 8. Because 9 is larger than 8, place a 1 in the 8s place. Subtract 9-8 = 1.

____ ____ ____ ____ 1s

place2s

place4s

place8s

place

11

Step 2: Compare the remainder 1 to 4. Because 1 is smaller than 4, place a 0 in the 4s place and continue with the remainder 1.

Step 3: Compare the remainder 1 to 2. Because 1 is smaller than 2, place a 0 in the 2s place and continue with the remainder 1.

____ ____ ____ ____ 1s

place

2s plac

e

4s place

8s place

11 00

Step 4: Compare the remainder 1 to 1. Because 1 is equal to 1, place a 1 in the 1s place to complete the conversion.

____ ____ ____ ____ 1s

place2s

place4s

place8s

place

11 00 00

____ ____ ____ ____ 1s

place2s

place4s

place8s

place

11 00 00 11


23

Binary Coded Decimal

• Binary coded decimal (BCD) is a different approach

• Encodes each digit in the decimal number individually rather than converting the entire number

• To convert 30 to BCD, the 3 would first be converted into binary, and then 0 would be converted into binary

• Then the two binary strings would be concatenated (i.e. brought together) to represent the number 30


24

Binary Coded Decimal (continued)


25

BCD Example

Convert the decimal number 7244(10) into binary coded decimal

7 = 01112 = 00104 = 01004 = 0100

7244(10) = 0111001001000100(2) (BCD)


26

BCD Example (continued)

Convert the sequence 0001100001111001 into decimal

First break the sequence into groups of four starting from right to left: 0001 1000 0111 1001

Then convert each group into decimal:0001(2) = 1(10)

1000(2) = 8(10)

0111(2) = 7(10)

1001(2) = 9(10)

The answer is 1879(10)


27

Binary Representation of Positive Nonintegers

• But how can binary represent noninteger numbers?

• Converting a noninteger number into binary is similar to converting an integer into binary

• For example, consider the number 42.4375 • Finding the binary equivalent of this

noninteger number requires the following steps:


28

Binary Representation of Positive Nonintegers (continued)

• The integer part of 42.4375 is first converted into its binary equivalent using the procedures described previously– The binary equivalent of 42 is calculated to be

101010• The fractional part of the decimal number (0.4375)

is converted to binary by multiplying 0.4375 by 2 and comparing it to 1


29


• If the result of the multiplication is greater than or equal to 1, then a bit with a value of 1 is noted, the fractional part of the multiplication result is extracted and multiplied by 2, and the result is compared to 1

• If the result of the multiplication is less than 1, then a bit with a value of 0 is noted and the number is multiplied by 2 again

• Repeat until the result of the multiplication is 1


30



31

Converting the Binary Number 111.110 into Decimal


32

Representing Negative Integers in Binary

• How can negative numbers be represented in binary?• Why not just add a preceding bit indicating the sign?

– 5 might be represented as 0101 – -5 might be represented as 1101

• Limitations of binary addition preclude this approach


33

The Following Rules Apply to Binary Addition

• 0 + 0 = 0• 0 + 1 = 1• 1 + 0 = 1• 1 + 1 = 0 with a carry of 1• 1 + 1 + carry of 1 = 1 with a carry of 1• The addition of 0101 (+5) and 1101 (-5) based on

the above rules results in:

0101

+1101

10010Information Technology in Theory

34

“2’s Complement Notation”

• Step 1: Determine the 8-bit binary representation of 5– 00000101

• Step 2: Take the “complement” of the binary pattern determined in Step 1– Change all the 0s to 1s and change all the 1s to 0s– 11111010

• Step 3: Arithmetically add 1 to the binary pattern determined in Step 2

11111010 + 1

11111011• The 8-bit 2’s complement of -5 is therefore 11111011

Express -5 in 8-bit 2’s complement form


35

“2’s Complement Notation”

• Determine the 16-bit binary representation of 7– 0000000000000111

• Take the “complement” of the binary pattern determined in Step 1– 1111111111111000

• Add 1 to the binary pattern determined in Step 2– 1111111111111001

Express -7 in 16-bit 2’s complement form


36

Alternative Numbering Systems

• Long binary streams need to be made more manageable for humans

• Shorthand notations– Octal (Base-8)– Hexadecimal (Base-16)


37

Octal

• The octal numbering system, also called base 8, uses eight numbers

• Counting in octal is identical to counting in decimal or binary, but octal uses eight numbers (0 through 7) instead of 10 or 2

• Because we run out of numbers at 7, the next number after 7 is 10 in octal

• Counting in octal:– 0, 1, 2, 3, 4, 5, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16,

17, 20, 21, 22, 23, 24, 25, 26, 27, 30 Information Technology in Theory

38

Convert the Octal Number 167 to Decimal

• The seven is in the 80, or “1s place”; the six is in the 81, or “8s place”; and the one is in the 82, or “64s place”

• The octal number 167 can be converted to decimal as follows:(1 × 64) + (6 × 8) + (7 × 1) = 64 + 48 + 7 = 119(10)

• The octal number 167(8) = the decimal number 119(10)


39

Octal as a Shorthand for Binary


40

Binary to Octal Conversion

• Convert the binary stream 111101000000 into octal• Break the number into groups of three starting from

the right: 000, 000, 101, and 111• Convert each group into its octal equivalent and

place the resulting octal numbers in sequential order– 000 = 0– 000 = 0– 101 = 5– 111 = 7

111101000000 = 7500(8) Information Technology in Theory

41

Hexadecimal

• Uses 16 characters• There are only 10 unique numbers, 0-9, so

hexadecimal has to introduce alphabetic letters• Hexadecimal characters are:

– 0, 1, 2, 3, 4, 5, 6, 7, 8 9, A, B, C, D, E, and F – The letter “A” symbolizes the tenth number, “B”

symbolizes the eleventh number, and so forth


42

The Hexadecimal Numbering System

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 1E, 1F, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2A, 2B, 2C, 2D, 2E, 2F, 30, 31, and so on


43

Hexadecimal as Shorthand


44

Hexadecimal as Shorthand (continued)

Convert 1111101010110001(2) into hexadecimal shorthand:

0001 = 1

1011 = B

1010 = A

1111 = F

1111101010110001(2) = FAB1(16)


45

Comparison of Numbering Systems


46

Real-World Example of Hexadecimal as Shorthand Notation

• 48-bit NIC address:

101000011111000001011011001010101100010000000001.

• Break the address into groups of four and convert each group into its equivalent hexadecimal character:

1010 = A 0010 = 20001 = 1 1010 = A1111 = F 1100 = C0000 = 0 0100 = 40101 = 5 0000 = 01011 = B 0001 = 1


47

Real-World Example of Hexadecimal as Shorthand Notation (continued)

• The hexadecimal shorthand representation of the 48-bit address is A1F05B2AC401


48

Representing Text and Other Characters in Binary

• Binary code can represent text and alphanumeric characters

• Two standards: – ASCII – Unicode


49

ASCII• Extended ASCII assigns an 8-bit code for each

alphanumeric character • Recall that an 8-bit code can represent 28, or 256,

unique items


50

Segment of ASCII Chart


51

ASCII Example

H = 01001000e = 01100101l = 01101100l = 01101100o = 01101111! = 100100001

Therefore, “Hello!” = 0100100001100101011011000110110001101111. The hexadecimal shorthand for this binary sequence is 48 65 6C 6C 6F 21


52

Unicode

• Unicode is an important standard that uses 16 bits• Allows for a representation of 216 (more than

65,000) unique characters • Provides sufficient characters to encode many

different major languages (such as English, Arabic, and Chinese)

• Unicode charts may be found by visiting http://unicode.org


53

EBCDIC

• Extended Binary Coded Decimal Interchange Code

• Standard associated with IBM computers

• Assigns 8 bits per character

• An extension of binary coded decimal


54

Summary

• Bits can encode any type of information, including the decimal numbers we use in everyday life and alphanumeric text

• A real-world example of binary to decimal conversion is the unique Internet address– Encoded in dotted decimal format – Makes a long binary string easier for people to

read• You can also convert between binary and positive

integers and represent negative integer and positive noninteger numbers in binary


55

Summary (continued)

• Alternative numbering systems are octal, which uses eight numbers, and hexadecimal, which uses 16 numbers– Not used by digital devices, but by people as a

shorthand convention – Network identification in local area networks is

usually represented in hexadecimal • Binary code also represents alphanumeric

characters – ASCII– Unicode– EBCDIC


Documents

Chapter 3 Representing Numbers and Text in Binary