The Language of Mathematics

Learning Objectives

• Identify and use mathematical symbols correctly.

• Translate English expressions and sentences into mathematical expressions and sentences, and vice-versa.

Mathematical Symbols and Conventions

• Purpose of Symbols: Mathematics uses symbols instead of words to express quantitative ideas concisely and unambiguously.

• Digits: There are ten fundamental digits:

  • $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$

• Operation Symbols: Symbols used to represent mathematical operations:

  • $+$ (addition)

  • $-$ (subtraction)

  • $\times$ (multiplication)

  • $/$ (division)

  • $\dots$ (and other operational modifiers)

• Stand-in Symbols (Variables and Constants):

  • $a, b, c, \dots$ (symbols used to stand-in for specific or general values)

• Relation and Special Symbols:

  • $=$ (is equal to)

  • < (is less than)

  • $\le$ (is less than or equal to)

  • $\pi$ (pi, constant ratio of a circle's circumference to its diameter)

  • $\dots$

• Grouping Symbols:

  • $( )$ (parentheses)

  • $[ ]$ (brackets)

  • $\{ \}$ (braces)

• Other Mathematical Symbols:

  • $\Sigma$: representing "the sum of"

  • $\exists$: representing "there exists"

  • $\forall$: representing "for every" or "for any"

  • $\infty$: representing "infinity"

Letter Conventions in Mathematical Language

• Start of the Alphabet ($a, b, c, \dots$):

  • Typically denote constants (fixed values).

• Middle Letters from $i$ to $n$ ($i, j, k, l, m, n$):

  • Typically denote positive integers (used primarily for counting).

• End of the Alphabet ($\dots, x, y, z$):

  • Typically denote variables (unknown values).

• Uppercase versus Lowercase Conventions:

  • Lowercase letters: Commonly used for variables (such as $x$ or $y$) or counting values (such as $m$ or $n$).

  • Uppercase letters: Commonly used for sets (such as $X$ or $Y$) and formulas.

English Language versus the Language of Mathematics

• Mathematical Expression versus Mathematical Sentence:

  • Mathematical Expression: A correct arrangement of mathematical symbols that does not express a complete thought. It functions similarly to a noun or noun phrase in English.

  • Mathematical Sentence: A combination of symbols that expresses a complete thought. Because it constitutes a full statement, it makes logical sense to evaluate whether a mathematical sentence is TRUE or FALSE.

Translation Vocabulary: English to Mathematical Symbols

• Addition Operator ($+$):

  • English terms: plus, sum, total, increased by, more, more than, add, added to, added by, added with, in addition to, combined with, put together, augmented.

• Subtraction Operator ($-$):

  • English terms: minus, less, less than, difference, decreased by, diminished by, subtracted from, subtracted by, exceeds by, lowered by, reduced by, loss, fewer, exceeds.

• Multiplication Operator ($\times$, raised dot $\cdot$, or grouping symbols):

  • English terms: times, the product of, multiplied by, multiplied to, multiplied with, twice, doubled, thrice, tripled, squared, cubed.

• Division Operator ($\div$ or $/$):

  • English terms: ratio, quotient, divided by, half of, average, per, over, all over.

• Equality Operator ($=$):

  • English terms: equals, is equal to, is the same as, is similar to, results in, produces, represents, is equivalent to, is, are, was, were, will be.

Standard Rules for Assigning Variables

• Any variable letter can represent an unknown number.

• To maintain uniformity across mathematical translations, standard assignments are specified as:

  • Let $x$ be the 1st unknown number.

  • Let $y$ be the 2nd unknown number.

  • Let $z$ be the 3rd unknown number.

Representations of Algebraic Statements

• Representation of "The sum of two numbers is 12":

  • Using One Unknown Variable:

  • Let $x$ = the first number

  • Let $12 - x$ = the second number

  • Using Two Unknown Variables:

  • Let $x$ = first number

  • Let $y$ = second number

• Representation of Consecutive Integers:

  • Consecutive Integers:

  • Let $x$ = first integer

  • Let $x + 1$ = second integer

  • Let $x + 2$ = third integer

  • (and so forth)

  • Consecutive Odd or Even Integers:

  • Let $x$ = first integer

  • Let $x + 2$ = second integer

  • Let $x + 4$ = third integer

  • (and so forth)

• Representation of Ages:

  • Past Age Expressions (indicate subtraction of past years from current age):

  • Standard key phrases: "years ago", "years back", "was at that time", "during or in the last years".

  • If Carlo's age now is $x$, then Carlo's age 10 years ago / 10 years back is $x - 10$.

  • If Bernard's age now is $x$, then Bernard's age 5 years back is $x - 5$.

  • Future Age Expressions (indicate addition of future years to current age):

  • Standard key phrases: "years from now", "years hence", "years after or in more years".

  • If Carlo's age now is $x$, then Carlo's age 3 years from now is $x + 3$.

  • If Bernard's age now is $x$, then Bernard's age 8 years hence is $x + 8$.

• Representation of Two-Digit and Three-Digit Numbers:

  • Variable definitions:

  • Let $h$ = first digit or hundreds digit

  • Let $t$ = second digit or tens digit

  • Let $u$ = third digit or unit digit

  • Two-Digit Number Formulations:

  • Sum of the digits = $t + u$

  • Value of the number = $10t + u$

  • Value of the number in reverse = $10u + t$

  • Three-Digit Number Formulations:

  • Sum of the digits = $h + t + u$

  • Value of the number = $100h + 10t + u$

  • Value of the number in reverse = $100u + 10t + h$

Translation Exercises and Solutions

• Translation 1:

  • English Phrase: "the sum of x and y"

  • Mathematical Expression: $x + y$

• Translation 2:

  • English Phrase: "the product of x and y"

  • Mathematical Expression: $xy$

• Translation 3:

  • English Phrase: "the difference of x and y"

  • Mathematical Expression: $x - y$

• Translation 4:

  • English Phrase: "the sum of x and the difference of y and z"

  • Mathematical Expression: $x + (y - z)$

• Translation 5:

  • English Phrase: "the sum of x and the sum of y and z"

  • Mathematical Expression: $x + (y + z)$

• Translation 6:

  • English Phrase: "the product of x and the sum of y and z"

  • Mathematical Expression: $x(y + z)$

• Translation 7:

  • English Phrase: "the product of x and the difference of y and z"

  • Mathematical Expression: $x(y - z)$

• Translation 8:

  • English Phrase: "the difference of the product of x and y and z"

  • Mathematical Expression: $xy - z$

• Translation 9:

  • English Phrase: "the product of the sum of x and y and the difference of x and y"

  • Mathematical Expression: $(x + y)(x - y)$

• Translation 10:

  • English Phrase: "the product of x and the sum of y and z"

  • Mathematical Expression: $x(y + z)$

• Translation 11:

  • English Phrase: "x more than y"

  • Mathematical Expression: $y + x$

• Translation 12:

  • English Phrase: "x less than y"

  • Mathematical Expression: $y - x$