Code & Mathematical Expression Language in Life Notes

Code & Mathematical Expression Language in Life

• Goals:
  • Explore how math systems resemble language.
  • Facilitate language analysis.
  • Model language.
  • Understand the fundamental symbolic underpinnings of mathematical expressions.

Code

• A regularized system of meaningful symbols found in:
  • Human languages
  • Mathematical expressions
  • Programming codes

Mathematics

• Definitions:
  • "The science of quantity"
  • "All mathematics is symbolic logic"
• A system of expressing numeric values.
• Mathematical expressions use a coded system.

Numerals, Numeric Glyphs, and Numbers

• Numeral:
  • A unit of language that expresses a quantity or amount, or number.
• Numeric Glyph:
  • A unit of writing that represents a numeral or its corresponding quantity/number.
• Numeral characters (numeric glyphs):
  • As pictograms:
    • $1$ = the amount of one (number/concept).
    • $2$ = the amount of two.
    • $10$ = the amount of ten.
    • etc.
  • As logograms:
    • $1$ = one
    • $2$ = two
    • $10$ = ten
    • etc.

Numeral Systems

• Hindu-Arabic numeric glyphs:
  • Some with iconic origins: $1, 2, 3, 4$
• Hindu-Arabic numeric glyphs:
  • Some with origins in older systems: $5, 6, 7, 8, 9$
    • Possibly acrophonic (e.g., origin of 5 resembles letter for first consonant of panca).
    • Or, alphanumeric (e.g., 6 has same origin as 6th glyph in Arabic abjad).
  • Compare with Greek numerals:
    • $\alpha = 1, \beta = 2, \gamma = 3, \delta = 4, …$
    • $\iota = 10, \kappa = 20, \lambda = 30, …$

Symbolic Expressions

• Expression:
  • Any meaningful symbol or combo of symbols.
• Statement/Assertion:
  • An expression that can be true or false.
• Question:
  • Checks for truth/falsehood.
• Term:
  • An expression that is a participant/argument of a statement or a question.
  • Some symbol signifying a quantity or amount (e.g., one, two, 1, 2).
  • Terms have values.
  • c.f. nouns identify reference / concepts / categories
• Symbolism of operators:
  • Words/glyphs representing operations, functions.
  • Examples: $, +, –, \times, \div, =, f(), …$
• Complex Term:
  • A quantity expressed through a combination of symbols.
  • Examples: $12, 2 + 3$
  • Interpreted with a grammar, ordering of characters, operators, and parentheses.

Linguistic & Mathematical Statements

• Statements:

  • Expressions that can be true or false.
  • Example:
    • Asif sees Yael.
    • $2 + 3 = 5$
  • Asif sees Yael:
    • Contains a verb "see" with two arguments (subject & object; one who sees, one who is seen).
    • An expression with terms and operators.
  • $2 + 3 = 5$
    • Operator $+$ with two arguments ($2, 3$).
    • $=$ is a relational operator that asserts identity of value for two arguments ($2+3$ and $5$).
    • Mathematical schooling trains how to resolve more complex terms to simpler terms.

• Terms and statements can differ in their structure across systems of expression:

  • $2 + 3$
  • $\text{sum}(2,3)$
  • $x = y$
  • $\text{isEqual}(x,y)$
  • $x.\text{isEqual}(y)$

Order of Magnitude

• A degree of quantity oriented to the numeric base.
• Quantities larger than the base are expressed with symbolic complexity.
• In the decimal system:
  • Each value between 0-9 has a simple glyph.
  • $10$ is "symbolically complex".
• Each additional column represents another order of magnitude (“ones,” “tens,” “hundreds”).
• Base 8 System:
  • b8 b10
  • $0 0$
  • $1 1$
  • $2 2$
  • $3 3$
  • $4 4$
  • $5 5$
  • $6 6$
  • $7 7$
  • $10 8$
  • $11 9$
  • $12 10$
  • $13 11$
  • $14 12$
  • $15 13$
  • $16 14$
  • $17 15$
  • $20 16$
  • … $100 64$
• Base 6 System:
  • b6 b10
  • $0 0$
  • $1 1$
  • $2 2$
  • $3 3$
  • $4 4$
  • $5 5$
  • $10 6$
  • $11 7$
  • $12 8$
  • $13 9$
  • $14 10$
  • $15 11$
  • $20 12$
  • $21 13$
  • $22 14$
  • $23 15$
  • $24 16$
  • $25 17$
  • … $55 35$
  • $100 36$

Binary Numeric Systems

• Binary numbers: base 2 numeral system.

  • b2 b10
  • $0 0$
  • $1 1$
  • $10 2$
  • $11 3$
  • $100 4$
  • $101 5$
  • $110 6$
  • $111 7$
  • $1000 8$
  • $1001 9$
  • $1010 10$
  • $1011 11$
  • $1100 12$
  • $1101 13$
  • $1110 14$
  • $1111 15$
  • $10000 16$

• Binary:

  • Information encoded using on/off or yes/no.
  • on = 1, off = 0.
  • Each “spot” or column is a bit.
  • A chunk of bits is a byte.
    *Binary numbers, as 4-bit expressions:
  • $0000 0$
  • $0001 1$
  • $0010 2$
  • $0011 3$
  • $0100 4$
  • $0101 5$
  • $0110 6$
  • $0111 7$
  • $1000 8$
  • $1001 9$
  • $1010 10$
  • $1011 11$
  • $1100 12$
  • $1101 13$
  • $1110 14$
  • $1111 15$
  • Using symbols:
  • $\Box\Box\Box\Box 0$
  • $\Box\Box\Box\blacksquare 1$
  • $\Box\Box\blacksquare\Box 2$
  • $\Box\Box\blacksquare\blacksquare 3$
  • $\Box\blacksquare\Box\Box 4$
  • $\Box\blacksquare\Box\blacksquare 5$
  • $\Box\blacksquare\blacksquare\Box 6$
  • $\Box\blacksquare\blacksquare\blacksquare 7$
  • $\blacksquare\Box\Box\Box 8$
  • $\blacksquare\Box\Box\blacksquare 9$
  • $\blacksquare\Box\blacksquare\Box 10$
  • $\blacksquare\Box\blacksquare\blacksquare 11$
  • $\blacksquare\blacksquare\Box\Box 12$
  • $\blacksquare\blacksquare\Box\blacksquare 13$
  • $\blacksquare\blacksquare\blacksquare\Box 14$
  • $\blacksquare\blacksquare\blacksquare\blacksquare 15$

• 8-digit system = 8 bits per byte.

  • Range of 00000000 (0) to 11111111 (255).
  • 256 possible values in an 8 bit space.

• Used in many computing contexts:

  • Internet Protocol (IP) addresses: 255.255.255.255
  • RGB (red-green-blue) values: 255.255.255
  • ASCII: 256 glyphs in basic typeset
  • Max # of rupees for Link in Legend of Zelda

• Other binary systems:

  • Looms, music boxes
  • Morse code
  • Punch cards

Extensions of Mathematical Expression in Programming Languages

• Terms can be non-numeric.
• Operators can be non-arithmetic.
• Statements use more operators aside from $, =, <, >$
• Computers interface between levels of code.

Levels of Code

• Low-level vs. high-level:
  • HL: characters, human-readable commands.
  • LL: bits, bytes, cryptic commands.
  • All data & expression is reducible to binary.

Low-Level Code

• Programmatic expressions that interface with circuitry & machinery.
• Expressions in binary notation.
• Statements, commands, terms.
• Lexical expression of command: arbitrary.
• Programmatic expressions that interface with circuitry & machinery.
  • $1/0$ expression matches physical characteristics of the medium.
  • "Phonological" expression of statement: iconic.

Code and Computers

• Code:
  • A system of symbols and rules for expressing information in a form usable by a computer.
  • Generally used to get a computer to do something that a human wants it to do.
  • Code is commands, whereas human language is not (generally).
  • Constructions with embedded statements and terms.

Expressions in Code

• A system with terms, operators, statements.
• Terms may be variables or constants.
• Operators include functions.
• Wider range of statements.
• Statements may resolve to TRUE or FALSE.
• “Control” statements – ask questions.
• Terms:
  • Constants:
    • $12$
    • $2 + 3$
    • $\text{pi}$
  • Variables (representing numbers, arrays, strings):
    • $i = 2$
    • $j = (2, 3, 4)$
    • $k = \text{"pizza"}$
• Terms as commands:
  • > i 2
  • > 2 + 3 5
  • `> k