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Binary & Hex. The number systems of Computer Science. What you will learn. Decimal Binary Hexadecimal Arithmetic Converting between number bases Counting 2’s Compliment. Number Systems. Prehistory Unary, or marks: / /////// = 7 /////// + ////// = /////////////

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binary hex

Binary & Hex

The number systems of

Computer Science

what you will learn
What you will learn
  • Decimal
  • Binary
  • Hexadecimal
  • Arithmetic
  • Converting between number bases
  • Counting
  • 2’s Compliment
number systems
Number Systems

Prehistory

Unary, or marks: /

/////// = 7

/////// + ////// = /////////////

Babylonian (Mesopotamia) around 2000 BC

Used 60 numerical symbols – sexadecimal. Used to measure time

5; 25, 30 = 5hr 25min 30 secs. Was positional based by had no concept of zero. 5,4560 = 5 x 60 + 45 x 1 = 34510

Roman Numerals: (non positional, no concept of zero)

VII + V = VVII = XII

MCMXCIX = 1999

MCMXCVIIII = 1999

MCMLXXXXVIIII = 1999

MDCCCCLXXXXVIIII = 1999

quote from pierre simon laplace
Quote from Pierre Simon Laplace
  • “The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius. “
arabic indian numerals decimal
Arabic/Indian Numerals (Decimal)
  • Better, Arabic/Indian based Numerals:

7 + 5 = 12 = 1 x 10 + 2

  • 345 is really
    • 3 x 100 + 4 x 10 + 5 x 1
    • 3 x 102 + 4 x 101 + 5 x 100
    • 3 is the most significant symbol (carries the most weight)
    • 5 is the least significant symbol (carries the least weight)
  • Digits (or symbols) allowed: 0-9
  • Base (or radix): 10
slide6

Try multiplication in (non-positional) Roman numerals(!):

    • XXXIII (33 in decimal)
    • XII (12 in decimal)
    • ---------
    • XXXIII
    • XXXIII
    • CCCXXX
    • -----------
    • CCCXXXXXXXXXIIIIII
    • CCCLXXXXVI
    • CCCXCVI = 396
  • The Babylonians wouldn’t have had this problem.
slide7

There are many ways to “represent” a number

  • Representation does not affect computation result
    • LIX + XXXIII = LXXXXII (Roman)
    • 59 + 33 = 92 (Decimal)
  • Representation affects difficulty of computing results
  • No concept of “Zero”
  • Computers need a representation that works with fastelectronic circuits
what is the base
What is the base ?
  • The decimal numbering system is also known as base 10. The values of the positions are calculated by taking 10 to some power.
  • Why is the base 10 for decimal numbers ?
    • Because we use 10 digits. The digits 0 through 9.
main memory
Main Memory
  • Original computers used holes/no hole to represent two state, 1/0.
  • Modern computers use transistor on/off translates to values 1/0
  • Requires use of Binary number system
slide10

Binary: positional numbers work great with

2-state devices

  • Digits (symbols) allowed: 0, 1
    • Binary Digits, or bits
  • Base (radix): 2
  • 10012 is really
    • 1 x 23 + 0 x 22 + 0 X 21 + 1 X 20
    • 910
  • 110002 is really
    • ?
    • ?
  • Computers usually multiply Arabic numerals by convertingto binary, multiplying and converting back for display
counting in binary
Counting in Binary
  • Binary
  • 0
  • 1
  • 10
  • 11
  • 100
  • 101
  • 110
  • 111
  • Decimal equivalent
  • 0
  • 1
  • 2
  • 3
  • 4
  • 5
  • 6
  • 7
slide12

Decimal to Binary

  • Divide decimal value by 2 until the value is 0
    • E.g.
    • 47 /2 = 23 Remainder 1
    • 23/2 = 11 Remainder 1
    • 11/2 = 5 Remainder 1
    • 5/2 = 2 Remainder 1
    • 2/2 = 1 Remainder 0
    • 1/2 = 0 Remainder 1
    • Read from the bottom up
    • Ans: 101111
binary to decimal
Binary to Decimal
  • 1000001 1 X 20 = 1
  • 26 25 24 23 22 21 20 0 X 21 = 0
  • 0 X 22 = 0
  • 20 = 1 24 = 16 0 X 23 = 0
  • 21 = 225 = 320 X 24 = 0
  • 22 = 426 = 640 X 25 = 0
  • 23 = 81 X 26 = 6465
binary to decimal alternative
Binary to Decimal (alternative)

Converting the binary 100101 to 37

Multiply running total by two, add in next digit. Repeat for all digits.

some buzz words
Some buzz words
  • Single Binary Digit =Bit
  • 4 bits = nibble
  • 8 bits = byte
  • 16 bits = word
  • 32 bits = longword
  • 2 nibbles = 1 byte
  • 2 bytes = 1 word

The word size depends on the processor used, above is shown the word size for the 68000

addition of binary numbers
Addition of Binary Numbers
  • Examples:
  • 1 0 0 1 0 0 0 1 1 1 0 0
  • + 0 1 1 0 + 1 0 0 1 + 0 1 0 1
  • 1 1 1 1 1 0 1 0 1 0 0 0 1

Carry a one

addition large numbers
Addition (large numbers)
  • Example showing larger numbers:
  • 1 0 1 0 0 0 1 1 1 0 1 1 0 0 0 1
  • + 0 1 1 1 0 1 0 0 0 0 0 1 1 0 0 1
  • 1 0 0 0 1 0 1 1 1 1 1 0 0 1 0 1 0
working with large numbers
Working with large numbers
  • 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1
  • Humans can’t work well with binary numbers. We will make errors.
  • Shorthand for binary that’s easier for us to work with - Hexadecimal
hexadecimal
Hexadecimal
  • Binary 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1
  • Hex 5 0 9 7
  • Written: 509716
what is hexadecimal really
What is Hexadecimal really ?
  • Binary 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1
  • Hex 5 0 9 7
  • A number expressed in base 16. It’s easy to convert binary to hex and hex to binary because 16 is 24.
counting in hex
Counting in Hex
  • Binary Hex Binary Hex
  • 0 0 0 0 0 1 0 0 0 8
  • 0 0 0 1 1 1 0 0 1 9
  • 0 0 1 0 2 1 0 1 0 A
  • 0 0 1 1 3 1 0 1 1 B
  • 0 1 0 0 4 1 1 0 0 C
  • 0 1 0 1 5 1 1 0 1 D
  • 0 1 1 0 6 1 1 1 0 E
  • 0 1 1 1 7 1 1 1 1 F
decimal to hex
Decimal to Hex
  • Repeat dividing by 16 and noting the remainder until the quotient = 0

E.g. 54401d to Hex

54401/16 = 3400 Remainder 1

3400/16 = 212 Remainder 8

212/16 = 16 Remainder 4

13/16 = 0 Remainder 13

Remainders are expressed Hex numbers

Answer: $D481

hex to decimal
Hex to Decimal
  • Write out the 1st 16 numbers
  • Same as for binary to decimal

E.g. F9E116

1x160 =1

Ex161=14x16 =224

9x162 =9x256 =2304

Fx163=15x4096= 61440

= 63969

another binary to hex conversion
Another Binary to Hex Conversion
  • Binary 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1
  • Hex 7 C 3 F
  • 7C3F16
binary coding schemes
Binary Coding Schemes
  • To represent characters we use Binary Coding Schemes
  • Two of the most popular are
    • ASCII (American Standard Code for Information Interchange
    • EBCDIC (Extended Binary Coded Decimal Interchange Code)
  • Here are some ASCII codes
    • CHARACTER ASCII
    • A 01000001 – 41 in hex
    • B 01000010 - 42 in hex
    • C 01000011 - 43 in hex
    • D 01000100 - 44 in hex
    • BELL 00000111 – 7 in hex
    • ESC 00011101 – 1B in hex
binary coded decimal bcd
Binary Coded Decimal (BCD)
  • Each digit is converted individually to it’s binary equivalent
  • 2634 = 0010 0110 0011 0100
  • 3825 = 0011 1000 0010 0101
  • Disadvantages are that arithmetic is more complicated and storage is inefficent.
  • Used only in simple device where memory is not a problem e.g. Calculators, clocks, etc.
negative integers
Negative Integers
  • Most humans precede number with “-” (e.g., -2000)
    • Accountants, however, use parentheses: (2000)
  • Sign-magnitude
  • Example: -1000 in hex?
    • 100010 = 3 x 162 + e x 161 + 8 x 160
    • -3E816
2 s compliment
2’s Compliment
  • Used for negative binary numbers
  • Procedure:

Invert all the bits: 0 becomes 1 and 1 becomes 0

Add 1

  • E.g.

To get –1

110 = 000000012

Invert all bits 11111110

Add 1 11111111

Ans: -110 = 111111112

2 s compliment1
2’s Compliment
  • Using the TC notation, the Most significant indicates the sign. 1 means a negative number
  • With 8 bits, can represent signed numbers in the range:

+127 -1 0 -128

01111111 11111111 00000000 10000000

  • With N bits, can represent signed numbers in the range:

(2N-1)-1 0 -(2N-1)

  • TC is widely used because it is easy to manipulate. The sign bit is treated the same as the others
2s compliment clock
2s Compliment Clock

Stepping clockwise increases the binary number

Stepping anti-clockwise decreases the binary number

Notice how when we add one to a negative number it’s values move closer to zero

addition subtraction
Addition & Subtraction
  • Addition of TC’s numbers same as normal binary addition
  • Subtraction of TC’s numbers can use either

A-B or A + TC(B)

E.g.

01112 – 00112 = ???

Change second number to TC

0111

1101+

0100

Answer: 01112 – 00112 = 01002

decimal fractions to binary
Decimal fractions to binary
  • More info on p158 Clements.

Question: Convert 0.6250 (decimal) to a binary fraction

0.3125 x 2 = 0.6250 yielding 0

0.625 x 2 = 1.2500 yielding 1

0.25 = 0.5000 yielding 0

0.5 x 2 = 1.0000 yielding 1

Ans:0.0101

binary to decimal fractions
Binary to Decimal fractions
  • More info on p158 Clements.
summary
Summary
  • Decimal
  • Binary
  • Hexadecimal
  • Arithmetic
  • Converting between number bases
  • Counting
  • Coding schemes
  • 2’s Compliment