MODULE 3: The Language of Mathematics

Symbols

• Mathematics uses symbols instead of words

Ten Digit Symbols

• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Operational Symbols

• Addition (+)

• Subtraction (-)

• Multiplication (×, ⋅, *)

• Division (÷, / , :)

• Square root

Stand-in for values symbols

• Represents a value

• a, b, c

Relation/Special Symbols

• =, <, >, ≤, π

Grouping symbols

• ( ), [ ], {}

Other symbols

• Σ the sum of (used to add long series of numbers)

• ∃ there exists (states that specific element exists)

• ∀ for every (for any, for all, state a rule for every single item)

• ∞ infinity (represents endless)

Letter Conventions

• Start of the alphabet (a, b, c,…) - Constants/Fixed Values

• From i to n (i, j, k…n) - positive integers for counting

• End of the alphabet (…x, y, z) - variables/unknown value

UPPERCASE vs. lowercase

• Lowercase for variables (x,y)

• UPPERCASE FOR sets and formulas

The English language and language of mathematics

• Nouns = Numbers - 15, 1/2, 4 raised to 2

• Connectives = Operational/grouping symbols - =, -, ()

• Verbs = relation symbols - =, <, >

• Pronouns = Variables (letters representing quantities) - x, y, x

Mathematical Expression

• an expression with a correct arrangement of mathematical symbols but does not express a complete thought

Mathematical Sentence

• Is one that expresses a complete thought. In a mathematical sentence, it makes sense to ask if the sentence is TRUE or FALSE.

Symbols of Operations to English Expression for ADDITION

• Addition (+) - plus, sum, total, increased by, more, more than, add, added to, added by, added with, in addition to, combined with, put together, and augmented

Symbols of Operations to English Expression for SUBTRACTION

• Subtraction (-) - minus, less, less than, difference, decreased by, diminished by, subtracted from, subtracted by, exceeds by, lowered by, reduced by, loss, fewer, exceeds

Symbols of Operations to English Expression for MULTIPLICATION

• Multiplication (Raised dot or *) - times, the product of, multiplied by, multiplied to, multiplied with, twice, doubled, thrice, tripled, squared, cubed

Symbols of Operations to English Expression for DIVISION

• Division (÷) - ratio, quotient, divided by, half of, average, per, over, all over

Symbols of Operations to English Expression for EQUAL

• Equal (=) - equals, is equal to, is the same as, is similar to, results in, produces, represents, is equivalent to, is, are, was, were, will be

Other Useful translations

• EXAMPLE - Representation of the statement "The sum of two numbers is 12".”

• TWO UNKNOWNS = x+y= 12

• ONE UNKNOWN = 12-x =the second number (y)

Representation of Consecutive (succeeding) Integers

1. x= first integer

2. x+1 = second integer

3. x+2 = third integer and so on

Consecutive odd/even integers

• ODD (ex. x=1)

1. x = first integer

2. x + 2 = 3 second odd

3. x + 4 = 5, third odd and so on

• EVEN (x=1)

1. x+1 =2, first even

2. x + 3 = 4, second even

3. x + 5 = 6, third even and so on

Representation of Ages

• OLD AGE - Subtraction of the past number of years from the present age, can be expressed as: years ago, years back, was at that time, during or in the last _ years

• FUTURE AGE - Addition of the future number of years from the present age, can be expressed as: years from now, years hence, years after or in _ more years

EXAMPLE:

Carlo’s age now = x

PAST AGE: Carlo’s age ten years ago = x-10

FUTURE AGE: Carlo’s age 3 years from now = x+3

Representation of a two-digit and a three-digit number

Let,

h = first digit or hundreds digit,

t = second digit or tens digit, and

u = third digit or unit digit

FOR TWO (2) DIGIT NUMBER (EX. 57: T=5, U=7)

• t + u = sum of the digits, 5+7= 12

• 10t + u = The number is self, 10(5) + 7 = 57

• 10u + t = The number in reverse, 10(7) + 5 = 75

FOR THREE (3) DIGIT NUMBER (EX. 538: H=5, T=3, U=8)

• h + t + u = sum of the digits, 5+3+8 = 16

• 100h + 10t + u = the number, 100(5) + 10(3) + 8 = 538

• 100u + 10t + h = the number in reverse, 100(8) + 10(3) + 5 = 835