The Language of Mathematics

1. The Language of Mathematics Basic Grammar

2. 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.

3. Overview The parts of grammar in elementary math we will look at are: • Nouns • Adjectives • Verbs • Sentences

4. 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.

5. Nouns • In mathematics the nouns are called expressions. These are combinations of numbers, variables, operations, and grouping symbols. • Here are some examples of expressions:

6. Types of Nouns • 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?

7. Common & Proper 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.

8. 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, x, y, a, or b bother you either!

9. Numbers as Adjectives • 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.

10. Common Nouns are “Common” 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. * A classroom could average these to estimate the percent of common nouns in each type of book.

11. 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

12. Complicated Expressions • Here are some expressions for the Golden Section of art, architecture, and nature (and the Fibonacci sequence). Don’t worry. In a few short years you will know what these things mean. Meanwhile try computing these on your calculator.

13. Verbs • 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.

14. Sentences • Simple sentences are of the form noun / 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 .

15. True or False? 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.

16. Statements, Equations and Inequalities

17. Compound Sentences

18. Work it Out Label each as statement, equation or inequality.

19. Work it Out Label each as statement, equation or inequality.