MATHEMATICAL-LANGUAGE-AND-SYMBOLS-ch-2-Lecture-2024.pdf

MATHEMATICAL LANGUAGE AND SYMBOLS  Is Mathematics a language? Is Mathematics a universal language? “If Mathematics is a universal language, then Mathematics is the language of the universe”  MATHEMATICAL LANGUAGE AND SYMBOLS LANGUAGE – systematic means of communicating ideas or feel...

MATHEMATICAL LANGUAGE AND SYMBOLS  Is Mathematics a language? Is Mathematics a universal language? “If Mathematics is a universal language, then Mathematics is the language of the universe”  MATHEMATICAL LANGUAGE AND SYMBOLS LANGUAGE – systematic means of communicating ideas or feelings by the use of conventionalized signs, sounds, gestures or marks having understood meanings (Merriam-Webster, 2017). Importance of Language Language was invented to communicate ideas to others. The language of mathematics was designed: § numbers § functions § sets § perform operations  CHARACTERISTICS OF MATHEMATICAL LANGUAGE The use of language in mathematics differs from the language of ordinary speech in three important ways (Jamison, 2000) 1. non-temporal – no past, present or future 2. devoid of emotional content 3. precise Know the Convention GREEK LETTERS  A letter that represents an unknown number. e.g. 𝒙 𝒚 𝒏 - system used to communicate mathematical ideas - it has its own grammar, syntax, vocabulary, word order, synonyms, negations, conventions, idioms, abbreviations, sentence structure and paragraph structure  CHARACTERISTICS OF MATHEMATICAL LANGUAGE According to Dr. Carol Burns, Mathematics language is:  CHARACTERISTICS OF MATHEMATICAL LANGUAGE Able to make very fine distinctions. e.g. A square is different to a circle based on definition.  CHARACTERISTICS OF MATHEMATICAL LANGUAGE e.g.  CHARACTERISTICS OF MATHEMATICAL LANGUAGE Able to say things briefly. We can convert mathematical language into expressions or equations.  CHARACTERISTICS OF MATHEMATICAL LANGUAGE e.g. Twice the number eight is sixteen. 𝟐 𝒙 𝟖 = 𝟏𝟔  Mathematical language is CONCISE ADDITION SUBTRACTION MULTIPLICATION DIVISION [+] [-] [ x, ( ), * ] [ ÷, / ] Plus, the sum Minus, the Multiplied by, the Divided by, of, increased difference of, product of , times, of, the quotient by, total, added decreased by, twice, thrice of, per, ratio to, more than, subtracted from, of greater than less, less than  OPERATIONAL TERMS AND SYMBOLS Literal coefficient – the unknown quantity in a term (variable) Numerical coefficient – the constant which determines the number of times a variable is to be multiplied. e.g. 𝟐𝝅𝒓 + 𝟏 𝟐𝝅 – numerical coefficient 𝒓 – literal coefficient (variable) 𝟏 – constant  OPERATIONAL TERMS AND SYMBOLS e.g. 3𝑥, 3𝑥 + 2𝑦, 3𝑥 + 2𝑦 – 5 3𝑥, 𝜋𝑟 ! – one term (monomial) 3𝑥 + 2𝑦, 𝑎! + 𝑏 ! – two terms (binomial) 3𝑥 + 2𝑦 – 5 – three terms (trinomial) 5𝑥 + 6𝑦 – 4𝑧 + 1 – polynomial Examples: Represent the given words or phrases in symbols. 1. The sum of two numbers is 5. Let x be the first number, then 𝑥 − 5 = the second number 2. Two more than thrice a certain number Let x be the certain number, then 3𝑥 + 2 = the required number Examples: Represent the given words or phrases in symbols. 3. Ten less than twice a certain number Let x be the certain number, then 2𝑥 − 10 = the required number 4. The difference of two numbers is five. Let x be the first number(larger), then 𝑥 − 5 = the second number(smaller) Examples: Represent the given words or phrases in symbols. 5. Three consecutive integers Let x be the first integer, then 𝑥 + 1 = the second integer 𝑥 + 2 = the third integer 6. Three consecutive even integers Let x be the first even integer, then 𝑥 + 2 = the second even integer 𝑥 + 4 = the third even integer Examples: Represent the given words or phrases in symbols. 7. Three consecutive odd integers Let x be the first odd integer, then 𝑥 + 2 = the second odd integer 𝑥 + 4 = the third odd integer 8. Ten exceeds a given number Let x be the number 10 − 𝑥 = the excess of a number Examples: Represent the given words or phrases in symbols. 9. The square of the sum of a and b 𝑎 + 𝑏 =the sum of a and b (𝑎 + 𝑏)! = the square of the sum of a and b 10. The sum of the squares of a and b 𝑎! = square of a 𝑏 ! = square of b 𝑎! + 𝑏 ! = the sum of the squares of a and b Examples: Represent the given words or phrases in symbols. 11. Mark is twice as old as Ken, and Ken is three times as old as Ian. Express each of the ages in terms of x. Let x be Ian’s age, then 3𝑥 = Ken’s age 2(3𝑥) = 6𝑥 =Mark’s age 12. The sum of x and y subtracted from the sum of a and b. 𝑎 + 𝑏 =the sum of a and b 𝑥 + 𝑦 =the sum of x and y 𝑎 + 𝑏 − 𝑥 + 𝑦 = the sum of x and y subtracted from the sum of a and b Examples: Represent the given words or phrases in symbols. 13. The perimeter of the isosceles triangle if the base is two centimeters less than the two equal sides. Let x be the length of one side, then 𝑥 − 2 = the length of the base 𝑥 + 𝑥 + 𝑥 − 2 =the perimeter of the isosceles triangle 14. Jet is 4 years younger than his brother Jef. Find the difference of the squares of their ages. Let x be the age of Jef, then 𝑥 − 4 = the age of Jet 𝑥 ! =the square of Jef ’s age (𝑥 − 4)!= the square of Jet’s age 𝑥 ! − (𝑥 − 4)!= the difference of the squares of their ages Examples: Represent the given words or phrases in symbols. 15. The difference between the squares of two consecutive odd integers Let x be the first odd integer, then 𝑥 + 2 = the second odd integer 𝑥 ! = square of the first odd integer (𝑥 + 2)!= the square of the second odd integer 𝑥 ! − (𝑥 + 2)!= difference between the squares of two consecutive odd integers 16. Carl has two times as many 10-peso coins than 5-peso coins. Let x be the number of 5-peso coins, then 2𝑥 = the number of 10-peso coins  CHARACTERISTICS OF MATHEMATICAL LANGUAGE Able to express complex thoughts with relative ease. e.g. 𝟐 + 𝟒 means we need to add 𝟐 and 𝟒 to get 𝟔. Mathematical Language vs. Ordinary Language Mathematical Language Ordinary Language highly compact – conveying a lot full of ambiguities, innuendoes, of information and ideas in a hidden agenda and unspoken very little space cultural assumptions(Jamison, focused – conveying the 2000) important information for the current situation and omitting the rest MATHEMATICAL SENTENCE Is the mathematical analogue of an English sentence. That is, it is a correct arrangement of mathematical symbols that state a complete thought. MATHEMATICAL EXPRESSION Is the mathematical analogue of an English noun. That is, it is a correct arrangement of a mathematical symbols used to represent a mathematical object of interest. Open Mathematical Sentence A sentence which could be true or false depending on the value or values of unknown variables. Closed Mathematical Sentence A sentence that is known to be true or knwon to be false. English Language vs. Mathematical Language Mathematical Noun/Phrase Expression e.g. e.g. Jarwin 𝟐𝒙𝟖 𝟒+𝟖 Nestor’s dog 𝟐𝒙 − 𝟓𝒚 Small eyes English Language vs. Mathematical Language Mathematical Sentence Sentence e.g. e.g. Lyka is beautiful. 𝟐 𝒙 𝟖 = 𝟏𝟔 Heaven has a dog 𝟒 + 𝟖 = 𝟏𝟐 named Happy. 𝟐𝒙 − 𝟓𝒚 = 𝟎 Peter is handsome. OPERATIONAL TERMS AND SYMBOLS MATHEMATICAL SENTENCE – combines two mathematical expressions using a comparison operator which include equal ( = ), not equal ( ≠ ), greater than ( > ), greater than or equal to ( ≥ ), less than ( < ), less than or equal to ( ≤ ). EQUATION – mathematical sentence containing the equal sign ( = ) INEQUALITY – mathematical sentence containing the inequality signs ( ≠, >, ≥ ,