Transcript The Language of Mathematics

The Language of
Mathematics
Basic 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.
Overview
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.
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 x  a 2  y
25 r s
2
2  (u  v )  3 2
3 4 1 r  s
x
( x  1)  ( x  2)
1  (2  3  (4  5  6))
( x  3)  ( x  4)
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?
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.
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!
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.
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.
# common
• Compute the ratio: # common  # proper
• Do the same for a textbook.
* A classroom could average these to estimate the
percent of common nouns in each type of book.
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
y
?
y
Complicated Expressions
• Here are some expressions for the
Golden Section of art, architecture, and
nature (and the Fibonacci sequence).
  1.618033988

1 5
2
 1 1 1 1
1
 1
1
1
1
1
1
Don’t worry. In a few
short years you will know
what these things mean.
Meanwhile try computing
these on your calculator.
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.
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 z  3  5  2  z, the left
hand side is z  3, and the rhs is 5  2  z .
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.
Statements, Equations and
Inequalities
Compound Sentences
Work it Out
Label each as statement, equation or
inequality.
x5 7
equation
25 7
x5 7
statement
inequality
3 5  7
statement
1+
2

20
19
statement
4
3
5
3 z  2  z  2
a 5 7
equation
inequality
u  u 1
statement (false)
Work it Out
Label each as statement, equation or
inequality.
x5 7
equation
25 7
x5 7
statement (true)
inequality
3 5  7
statement (false)
1+
2

20
19
statement (true --- lhs 
4
5
3 z  2  z  2
a 5 7
equation
inequality
u  u 1
statement (always false)
3
29
)
19