The Hexadecimal NumberSystem and Memory

Addressing

ISAT 121

2

Familiar Number Systems

Roman numerals None, one, few, many Positional value systems

Each position in a number has a value

Includes concept of zero Example: decimal system

3

Characteristics of Numbering Systems

1) The digits are consecutive.2) The number of digits is equal to the

size of the base.3) Zero is always the first digit.4) The base number is never a digit.5) When 1 is added to the largest digit,

a sum of zero and a carry of one results.

6) Numeric values determined by the have implicit positional values of the digits.

4

Significant Digits

Binary: 11101101

Most significant digit Least significant digit

Hexadecimal: 1D63A7A

Most significant digit Least significant digit

5

History

Necessity for understanding numbering systems

Historical numbering systems in computing Decimal On/Off Binary (Ada)

Bits & bytes ASCII

6

Common terminology

Term DefinitionBit A numeral in the binary number system: a 0

or a 1.

Byte 8 bits (the smallest addressable memory location).

Kilobyte 1024 bytes, which is 2 to the 10th power, often rounded to 1000 bytes.

Megabyte

Either 1024 kilobytes or 1000 kilobytes, depending on what has come to be standard practice in different situations. For example, when calculating floppy disk capacities, 1 megabyte = 1000 kilobytes; when calculating hard drive capacity, traditionally, 1 megabyte = 1024 kilobytes.

7

Common terminology

Term DefinitionGigabyte 1000 megabytes or 1024 megabytes,

depending on what has come to be standard practice in different situations.

ASCII American Standard Code for Information Interchange coding scheme used for microcomputers, which assigns a 7- or 8-bit code to all characters and symbols. See Appendix B for more information.

Hex Short for hexadecimal. A number system based on 16 values (called base 16), which is explained in this appendix. Uses the 16 numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Hex numbers are often followed by a lowercase h to indicate they are in hex (example: 78h).

8

Example

/ 230 =

In the lab…1. Double click on My Computer2. Right click on C:3. Click on Properties

9

Decimal Number System

Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Base

Ten Each position is power of 10

Value = Σ i=0..n-1 di * 10i, n is count of digits in number

10

Decimal Number Example

Number: 1946 n = 4 and n – 1 = 3 Positions

1 * 103 = 1000 9 * 102 = 900 4 * 101 = 40 6 * 100 = 6

Sum: 1946

11

Binary Number System

Also called the “Base 2 system” The binary number system is used to

model the series of electrical signals computers use to represent information

0 represents the no voltage or an off state

1 represents the presence of voltage or an on state

12

Binary Number System

Digits: 0, 1 Base:

Two Each position is power of 2

Example: 1010 1000[2] 1*27 + 0*26 + 1*25 + 0*24 + 1*23 +

0*22 + 0*21 + 0*20

128[10] + 0 + 32[10] + 0 + 8[10] + 0 + 0 + 0

Sum: 168[10]

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Hexadecimal (Hex) Number System

Digits: 0..9, A, B, C, D, E, F Base:

Sixteen Each position is power of 16

Example: A8[16] A * 161 = 160[10]

8 * 160 = 8[10]

Sum: 168[10]

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Notation for Number Systems

Subscript used to denote base 25[10] decimal 19[16] hexadecimal 0001 1001[2] binary

Web pages and some programming languages Decimal: no additional symbols Hexadecimal: use pound sign “#”

Other notation conventions Hexadecimal: “h” or 0x Octal: “o” Binary: “b”

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Binary to Hexadecimal Conversion

The easiest method for converting binary to hexadecimal is to use a substitution code

Each hex digit converts to 4 binary digits

16

Conversion to Hexadecimal

Binary to hex: Convert each nibble to hex digit (see previous slide)

Decimal to hex: Use repeated division by powers of 16 to find

hex digit at each position 163 = 4096, 162 = 256, 161 = 16, 160 = 1

1946[10] to hex: (1946\256 = 7) + (154\16 = 9) + (10\1) = A Sum: 79A (7*256=1792) + (9*16=144) + (10*1=10) =

1946 Or… Windows Calculator

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Bits, Bytes, and Nibbles

Computers use 0 and 1 states to represent data and instructions

Each stored state or pathway to transport a state is a bit

Eight bits are a byte Basic grouping of bits in computers Can be represented by two hex digits Values: 00..FF or 0..255

Half of byte is nibble (nybble) Can be represented by one hex digit Values: 0..F or 0..15

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Binary Integer Addition

Binary has only two digits: 0, 1 Addition rules:

0 + 0 = 0 1 + 0 = 10 + 1 = 1 1 + 1 = 10

Examples: 3 + 3 = 0011 + 0011 = 0110 5 + 7 = 0101 + 0111 = 1100 9 + 7 = 1001 + 0111 = 10000 –

Overflow

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Binary Integer Subtraction

Two digits: 0, 1 Number of bits dependent Rules for two bits and A≥B

00–00 = 00; 01–00 = 01; 01–01 = 00; 10–00 = 10; 10–01 = 01; 10–10 = 00;11–00 = 11; 11–01 = 10; 11–10 = 01; 11–11 =

00

A<B governed by signed representation used

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Two’s Complement

Rule for storing signed integersRule for storing signed integers Results in “clock” of numbers Rule for negating a number

Take complement of number Value 6[10] = 0110[2] Complement 1001

Add one 1 Yields 2’s complement 1010

“Clock” values dependent on number of bits 4 bits: -8 .. 7 8 bits: -128 .. 127 32 bits: -2,147,483,648 .. 2,147,483,647 64 bits: -9,223,372,036,854,775,808 ..

9,223,372,036,854,775,807

01

2

3

5

6

4

-87

-1-2

-3

-4

-5

-6

-7

0123

56

4

8 7

151413

121110

9

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Overflow and Underflow

Overflow caused by operation generating number greater than register can hold 0111 + 0001 = 1000 7 + 1 ≠ -8

Underflow caused by operation generating number less than register can hold 1000 – 0001 = 0111 -8 – 1 ≠ 7

Either will cause returned value to wrap around “clock”

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Example

Colors on web pages are often expressed in hex “#00FF80”

Graphics editors often represent the color components in decimal values Red: 0 Green: 255 Blue: 128

Graphics editors may represent the color components as percentages Red: 0 Green: 100 Blue: 50

To blend a graphic into the background of a web page, the correct hex value must be calculated

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Conclusion

Questions?