03 Digitizing Information: Symbols and Binary Systems

• Digitizing is representing information with digits, which are the ten Arabic numerals 0 through 9.
• However, digitizing in computing can use almost any symbol, not just the ten digits.

Alternative Representations

• Digitizing is not limited to just digits; symbols can also be used.
• Digits have short names which is a practical advantage, for example, "one" or "two".
• IT uses short names for symbols, for example, exclamation point is called "bang" and asterisk is called "star".
• There needs to be an agreed ordering for the basic symbols.
• Collating sequence places information in order by using non-digit symbols.
• Today, digitizing represents information by symbols.

Fundamental Information Representation

• A computer represents information by the presence or absence of some physical phenomenon giving a binary system.
• The two states in a binary system are named 1 and 0.
• Larger symbols are constructed using these two basic binary states.
• The physical phenomenon used can be charge, current, magnetization, or other detectable qualities.

Binary Representation Properties

• Only two states are available in binary representation.
• Discrete means "distinct" or "separable" in the binary system.
• It is not possible to transform one value into another by tiny gradations so, there are no "shades of gray".

Binary Systems

• Binary encoding has two patterns: present and absent
• There is no specific law binding what the on and off states can represent, meaning the assignment is arbitrary.
• on can represent "present" and off can represent "absent".

Bits

• The unit in binary representation occupies a specific place in space and time. The presence or absence of phenomenon can be set and detected.
• The binary unit is known as a bit.
• Bit is a contraction for "binary digit".
• Bit sequences can be interpreted as binary numbers.
• Groups of bits form symbols.
• Computer memory is arranged as a long sequence of bits.
• The physical phenomenon can be encoded, the information can be set and detected to be present or absent.

Binary Encodings alternative

• There is no limit to encoding two physical states.
• Stones represent the two states on all squares; white stones for absent and black stones for present
• Multiple stones of two colors per square with more white being 1 and more black being 0.

Combining Bit Patterns

• Two-bit patterns lead to limited resources for digitizing information.
• The number of bits determines the symbols; n bits in sequence yields 2^n symbols.
• To encode the intended information, two pattern must be combined into sequences providing enough symbols.

Binary System Explained

• The base-2 number system is used to represent numbers.
• Only use 0 and 1 when counting in binary.
• Binary counting example = 0, 1, 10, 11, 100, 101, 110, 111, 1000.

Hexadecimal Explained

• Base-16 numbers are known as hex digits, short for hexadecimal digits.
• Hexadecimal digits utilize decimal numbers + the first six Latin letters: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
• Hexadecimal digits provide a better way to represent bit sequences.

Changing from Hex to Binary

• A 32-bit computer instruction can be written like 1000 1110 1101 1000 1010 0011 1010 0000
• The above is tedious and prone to error so convert four-bit groups to hex forming 8E D8 A3 A0.

Hexadecimal Values

• Each hex digit codes a four-bit group
• 0000 = 0
• 0001 = 1
• 0010 = 2
• 0011 = 3
• 0100 = 4
• 0101 = 5
• 0110 = 6
• 0111 = 7
• 1000 = 8
• 1001 = 9
• 1010 = A
• 1011 = B
• 1100 = C
• 1101 = D
• 1110 = E
• 1111 = F

Digitizing Numbers in Binary

• Binary digits' earliest use was for encoding numbers and keyboard characters.
• Representing sound, images, video, and other information types is important.

Place Value in Decimal Number

• To find the quantity expressed by a decimal number, multiply the digit in a place by the place value and add the results.
• 1,010 (base 10) = (0 × 1) + (1 × 10) + (0 × 100) + (1 × 1000).

Place Value in Binary Numbers

• Binary operates the same way as decimal.
• The base is 2 instead of 10.
• Binary place values are 1, 2, 4, 8, 16.

Binary Numbers

• Binary: 1010 = (1 × 8) + (0 × 4) + (1 × 2) + (0 × 1)

Digitizing Text

• The number of bits determines the number of symbols available for representing values, so n bits in sequence yield 2^n symbols.
• The more characters required the more symbols are needed.

Text Considerations

• Roman letters, Arabic numerals, and about a dozen punctuation characters are the minimum needed to digitize English text.
• Consider Basic arithmetic symbols, (+, −, *, /, =?), characters not required for English (ö, é, ñ, ø?), Punctuation symbols? « », ¿, π, ∀)?, and business symbols: ¢, £, ¥, ©, and ®?.

Assigning Symbols in English

• Need to represent:
  • 26 uppercase letters
  • 26 lowercase letters
  • 10 numerals
  • 20 punctuation characters
  • 10 useful arithmetic characters
  • 3 other characters (new line, tab, and backspace)
  • For a total of 95 symbols, which is enough for English

Assigning Symbols Additional Info

• Since 95 distinct symbols need representing, 7 bits are required.
• 6 bits provides 2^6 = 64 symbols, insufficient to represent all the English alphabet and additional symbols
• 7 bits provides 2^7 = 128 symbols
• 128 symbols is ample for the 95 different characters needed for English characters.
• Some additional characters are also represented

ASCII

• ASCII = American Standard Code for Information Interchange.
• A widely used 7-bit (2^7) code.
• Advantages of a standard ensures computer parts built by different manufacturers can be connected and programs can create data and store it where other programs can access it later.

Extended ASCII

• 7-bit ASCII is insufficient due to its inability to represent text from other languages.
• IBM decided to use the next larger set of symbols: 8-bit symbols (2^8).
• Eight bits produce 256 symbols.
• The 7-bit ASCII is the 8-bit ASCII representation with the leftmost bit set to 0.
• The 8-bit ASCII handles many languages with Latin alphabet derivatives.
• IBM give 8-bit sequences a special name, byte
• A byte is a standard unit for computer memory.

Unicode

• The 256 extended ASCII codes cover most Western languages.
• Unicode represents many more characters using up to 32 bits to code characters.
• UTF-8 records Unicode by writing long characters as groups of bytes.

Advantages of Long Encodings

• Computing efficiently to minimize the memory amount with the shortest symbol sequence is the aim.
• NATO Broadcast Alphabet and Bar Codes are examples of the opposite.

NATO Broadcast Alphabet Specifics

• The code for letters used in radio communication is purposefully inefficient.
• The code is distinctive when spoken amid noise.
• The alphabet encodes letters as words where words are the symbols.
• The longer encoding improves recognition chances.
• Digits maintain their usual names, but nine becomes niner.

Barcodes

• Universal Product Codes (UPC) utilize more than the number of bits to encode information.
• The UPC-A encoding utilizes 7 bits to encode the digits 0–9.
• UPC encodes the manufacturer (left side) and the product (right side).
• Different bit combinations are used for each side and one side is the complement of the other.
• The bit patterns were chosen to be as different from each other as possible.

Parity

• Computer memory is subject to errors.
• An extra bit detects errors.
• A ninth bit per byte can detect errors using parity.
• Parity refers to whether a number is even or odd.
• Even parity means that if the sum of 1s even, set the ninth bit to 0; otherwise set it to 1.
• All 9-bit groups have evenparity.
• Any single bit error in a group causes its parity to become odd so, hardware detects an error has occurred.
• It cannot detect specifically which bit is wrong.

Why Byte?

• IBM was building a supercomputer, called Stretch.
• There was a need for a word to describe quantity of memory between a bit and a word.
• A computer memory word is typically the amount required to represent computer instructions; currently a word is 32 bits.
• The i to a y ensure there is no confusion with a bit.

Metadata

• Converting the content into binary is half of the problem of representing information.
• Metadata describes information's properties.
• Content characteristics also require encoding.
  • Content structure.
  • Related content.
  • Collection Location.
  • Assigned Units of measure.
  • Instructions of how it is displayed
  • Time it was created or captured
• Metadata = information describing information.
• It does not require its own binary encoding.
• The most common way to give metadata is with tags.

Properties of Data

• Metadata and what is describing it are separate.
• ASCII representations of letters and How the letters look (for example Comic Sans Font) are discussed.

Data Tags

• Instead of filling a file with the Times New Roman font, the file populates with letters and tags that describe how it should display.
• This avoids locking-in the form of display, that is changed by changing the metadata.
• Metadata is presented at several levels.

Tags for Metadata

• The Oxford English Dictionary (OED) is the definitive source reference for every English etymology, meaning, and usage.
• The OED printed version has 20 volumes, weighs 150 pounds, and uses 4 feet of shelf space.

Using Tags for Metadata Specifics

• The OED conversion to digital form began in 1984.
• To find the definition of the verb, "set” find it in a search, the OED searches many similar words for set such as closet, horsetail, and settle.
• Software can help sort out the words.

Structure Tags Information

• Special tags can handle structure:
  • <hw> the OED's tag for a headword (word being defined) -<pr> handles pronunciation -<ph> does the phonetic notations
  • <ps> parts of speech -<hm> homonym numbers -<e> surrounds the entire entry
  • <hg> surrounds the head group and all definition starting information.
• Software algorithms can use tags.
• Tags do not print.
• They are included only to specify the structure so the computer discerns the dictionary part to use.
• Structure tags format, especially boldface used for headwords
• Knowing the structure generates the formatting information.

Representing Numbers

• Numerical representation uses two fixed bits for technology, which is binary and decimals
• Decimal numbers use base 10 positional notations with 10 digits ranging between 0-9.
• Decimal example: 234.56 = 2x10^2 + 3x10^1 + 4x10^0 + 5x10^-1 + 6x10^-2.
• Positional Notation uses any base.
• Useful bases for positions
  • Hexadecimal: 16
  • Octal: 8
  • Binary: 2

Numerical Types

• Major types:
• Integers.
• Non-integers.
  • Floating points
    • Float (32 bits).
    • Double (64 bits).
    • Doubles allow quantitative precision by sharing common rules.

Internal representations

• Integer uses less space and are processed at higher speed.
• Non-integers require more bytes and complex processing.
• A sum of two integers can be computed in a single CPU clock cycle.
• A sum of two double precision requires 4 CPU clock cycles.
• Numbers:

Integers

• 32 bits
  • From –(2^31-1) to 2^31-1
  • Internal binary representation
• Negatives are represented in two's complement

Floating Point Double Numbers

• 64 bits
  • Limit changes in language implementations differently
    • Represented as
      • Mantissa * BaseExp
  • Internal binary representation; Mantissa = Significant.

Two's complement

• Considers negative integers.
• The following are representations of -1.
• Represents 1 00000000000000000000000000000001 -31 zeros followed by 1 invert all bits -31 one followed by all zeros
  • 11111111111111111111111111111110
  • add +1 11111111111111111111111111111111
  • Representation –(2^31-1) 10000000000000000000000000000001100000000000000000000000000000001

Two’s Compliment Examples

• 8-bit example

Overflow

• 8 bit examples consider sums that goes over an int (integer) amount.
• This can discriminated between negative and non-negative quantities

Calculations With Floats

• The exponent adjusts the point, that is, less than one of first digit to the right from the decimal point.

Normalized Floating-Point Numbers

• Any rational number with a finite number of digits uses 2 numbers:
• Mantissa
• Exponent
• The calculation can be simplified by normalizing the number

IEEE Standards

• Uses bit standards
• Sign
• Exponent
• Mantissa

Parameters for numbers

• Numbers utilize various forms including “double” (neg,pos), integer, and digit forms

R Calculation

• Doubles use 3 types
• Smallest -largest These examples calculate for the double in absolute value, for negatives.

Calculating doubles and floats

• Numbers must not only be fixed and accounted to order of magnitude.
• Numbers can be represented in parts e.g. with limitations to parts like fixed integers.
  • As real numbers increase, computations will substitute specific numbers. Truncation with Doubles
• Numbers near reals will all fall towards doubles i.e. “rounding”

Approximating Errors

Rounding will cause some errors of the actual number, especially at the numbers beyond reals in computation will be substituted. -This error is a calculation that is dependent on the exponential. This can be used to estimate, what the impact of the double on real values are.

• (Relative error) independent in exponent from calculations in computations. Example of operations base 10.
• Numbers are combined to determine correct answer. (truncation happens as needed with double’s) and with greater significance we find improved accuracy. Arithmetical Computation using Double’s
• Commutative has the same validity for doubles or a floating point where it has an equal value or float.
• Calculations depend on order of operations
• Associative property: can be violated depending on how values can be interpreted (+ vs *)
• Distributive property: can be violated depending on how we view it across both (+ vs *)
• Doubles depend heavily on structure so that their data has equal representation