LESSON 2: Mathematical Language and Symbols

Mathematical Language and Symbols

CHARACTERISTICS OF MATHEMATICAL LANGUAGE 

        The language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express. It is; 

• precise (able to make very fine distinctions) 
• concise (able to say things briefly) 
• powerful (able to express complex thoughts with relative ease)

EXPRESSIONS VS SENTENCES 

English Language

• Nouns are used to name things we want to talk about (like people, places and things). 

        -NOUN: Sogod, Southern Leyte, province

• Sentences are used to state complete thoughts

        -SENTENCE: Sogod is a municipality of Southern Leyte.

Mathematical Language

• A “noun” is called an expression in mathematics. Thus, an expression is a name given to mathematical object of interest.

        -NOUN: sin x, 𝑥−3 3 − 2�

• A mathematical sentence, just like the English sentence, must state a complete thought.

        -SENTENCE: 2𝑥 − 𝑦 ≤ 5, 2𝑥 2 − 6𝑥 + 12 = 0

Mathematical Expression

        Numbers have lots of different names. For example, the expressions 

 5 2+3 10÷2 (6 – 2) + 1 1+1+1+1+1 

 All look different, but all just different names for the same number. 

Mathematical sentences have verbs. For example, in the sentence 3+7 = 12, the verb is ‘=’.

Expression 

        In order to communicate effectively, people must agree on the meanings of certain words and phrases. When there is ambiguity, confusion can result. The primary way that ambiguity is avoided is by the use of definitions. By defining words and phrases, it is assured that everyone agrees on their meaning.

An example of a definition in mathematics: DEFINITION: Expression

        An expression is the mathematical analogue of an English noun; it is a correct arrangement of mathematical symbols used to represent a mathematical object of interest. An expression does not state a complete thought; in particular, it does not make sense to ask if an expression is true or false. 

        Three of the most common types of expressions are numbers, sets and functions. Expressions have lots of different names; it depends on what we are doing with the expression.

Example: The number 1 goes by all the following names: 

SIMPLIFY: (some expression) 

To simplify an expression means to get a different name for the expression, that in some way simpler.

The notion of ‘simpler’ can have different meanings: 

• FEWER SYMBOLS: ‘3+1+5’ and ‘9’ are both names for the same number, but ‘9’ uses fewer symbols. 
• FEWER OPERATIONS: ‘5 ∙ 3’ is simpler than ‘3+3+3+3+3’ 
• PREFERRED STYLE / FORMAT: 2 4 and 1 2 are both names for the same number, but people usually prefer the name1 2 ; it is said to be in ‘reduced form’ or ‘simplest form’.

DEFINITION: Mathematical Sentence. A mathematical sentence is the analogue of an English sentence; it is a correct arrangement of mathematical symbols that states a complete thought. It makes sense to ask about the TRUTH of a sentence. Sentences have verbs 

• The sentence ‘1 + 2 = 3’ is read as ‘one plus 2 equals to three’. 
•  A complete thought is being stated, which in this case is true. 

Connectives 

• If ‘=’ is the verb, then what is ‘+’? 
• The symbol ‘+’ is a connective; a connective is used to ‘connect’ objects of a given type to get a ‘compound’ object of the same type.