The Language of The Language of MathematicsMathematics

The Language of The Language of MathematicsMathematics

Basic GrammarBasic Grammar

A word to the wise

The purpose of this tutorial is to get you to understand what equations and inequalities really are and to get you comfortable with the idea of variables.

However it also introduces a lot of new words and terminology which can be daunting.

Focus on the concepts. You have years to learn the words.

The parts of grammar in elementary math we will look at are:

•Nouns– Adjectives

•Verbs•Sentences

Overview

• In natural languages such as English, Spanish, French, Chinese, etc. nouns are simple and sentences are complicated

• In the language of mathematics it is the other way around. The nouns are complicated and the sentences are simple.

Overview

Nouns

• In mathematics the nouns are called expressions. These are combinations of numbers, variables, operations, and grouping symbols.

• Here are some examples of expressions:

1 7 2 3 2 3 5 2

2 52 22 ( ) 33 4 1( 1) ( 2)

1 (2 3 (4 5 6))( 3) ( 4)

x a y

r su v xr sx xx x

• A noun is the name of a person, place, or thing.

• So London and city are nouns

• Same are 3 and x, since x is shorthand for the word “number”.

• Can you classify the nouns: London, city, 3, and x?

Types of Nouns

• “London” is the name of a specific city and is called a proper noun.

• “city” is not the name of any particular city and is called a common noun.

• “2” is the name of a specific number and is a proper noun.

• “x” is not the name of any particular number and is a common noun.

Common & Proper Nouns

Variables are Common

• So you can see there is no real difference between mathematical variables and common nouns.

• You work with variables (common nouns) in English all the time --- and they don’t bother you at all!!!

• So don’t let x, y a, b, or bother you either!

• We just saw that 2 is a proper noun and so the word “Two” should be capitalized.

• However, when we write “two feet” we use lower case, because “two” is being used to modify the noun “feet”.

• Modifiers of nouns are called adjectives, and are not capitalized.

Numbers as Adjectives

Common Nouns are “Common”

* A classroom could average these to estimate the percent of common nouns in each type of book.

Take home problem*•Open any novel to a full page of text.•Count the number of proper nouns and the number of common nouns.•Compute the ratio:

•Do the same for a textbook.

## #

commoncommon proper

Variables

• y is a variable since it does not name a specific number

• y+2 does not name a specific number either, so y+2 is also a variable.

• y+2 ranges over all the numbers that are 2 more than y.

• Normally any expression containing a variable is also a variable.*

* Really? What about ?yy

Complicated Expressions

• Here are some expressions for the Golden Section of art, architecture, and nature (and the Fibonacci sequence).

1.618033988

1 52

1 1 1 1

11

11

11

1

Don’t worry. In a few short years you will know what these things mean. Meanwhile try computing these on your calculator.

• The simple verbs are the comparisons:= (equals), < (less than), > (greater than)

• We discuss the compound verbs shortly≤ (less than or equal to) and ≥ (greater than or equal to)

• The remaining comparison, ≠ (not equal to), has few uses and will not be covered.

Verbs

• Simple sentences are of the formnoun / verb / noun

where the nouns are expressions.

• We often refer to the two nouns as the “left hand side” (lhs) and the “right hand side (rhs).– For the sentence , the left

hand side is , and the rhs is .

Sentences

3 5 2z z 3z 5 2 z

Sentences may be true, false, or neither. •True or false sentences are called statements.

•Those that are neither true nor false are called equations or inequalities depending upon the verb.

True or False?

Statements, Equations and Inequalities

Compound Sentences

Work it OutLabel each as statement, equation or inequality.

5 7

2 5 7

5 7

3 5 7

2 201+

equation

statement

inequality

statement

statement

equatio

4 1935

3 2 2 n

inequali

5 7 y

1

t

x

x

z z

u u

statement (false)

Work it OutLabel each as statement, equation or inequality.

5 7 equation

2 5 7 statement (true)

5 7 inequality

3 5 7 statement (false)

2 20 291+ statement (true --- lhs )

4 19 1935

3 2 2 equatio

x

x

z z

n

5 7 inequality

1 statement (always false) u u