EXPRESSIONS VERSUS SENTENCES

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For a complete discussion, read the text (and solutions are here).

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People sometimes have trouble understanding mathematical ideas:

not necessarily because the ideas are difficult,

but because they are being presented in a foreign language—the language of mathematics.

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)

Every language has its vocabulary (the words)

and its rules for combining these words into complete
thoughts (the sentences).

Mathematics is no exception.

As a first step in studying the mathematical language,

we will make a very broad classification between the ‘nouns’ of mathematics
(used to name mathematical objects of interest)

and the ‘sentences’ of mathematics (which state complete mathematical thoughts).

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;

it does not make sense to ask if an expression is *true* or *false*.

The most common expression types are
*numbers*, *sets*, and *functions*.

Numbers have lots of different names: for example, the expressions

$5$ | $2 + 3$ | [beautiful math coming... please be patient] $\frac{10}{2}$ | $(6 - 2) + 1$ | $1 + 1 + 1 + 1 + 1$ |

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

This simple idea—that numbers have lots of different names—is extremely
important in mathematics!

DEFINITION
sentence

A mathematical *sentence* is the analogue of an English sentence;
it is a correct arrangement of mathematical symbols that states a complete thought.

Sentences have verbs.

In the mathematical sentence ‘$\,3 + 4 = 7\,$’ ,
the verb is ‘$\,=\,$’.

A sentence can be (always) true, (always) false, or sometimes true/sometimes false.

For example, the sentence ‘$1 + 2 = 3$’ is true.

The sentence ‘$1 + 2 = 4$’ is false.

The sentence ‘$x = 2$’ is sometimes true/sometimes false:
it is true when $\,x\,$ is $\,2\,$,
and false otherwise.

The sentence ‘$x + 3 = 3 + x$’ is (always) true,
no matter what number is chosen for $\,x\,$.

EXAMPLES:

$2$ | is an expression |

$1+1$ | is an expression |

$x+1$ | is an expression |

$1+1=2$ | is a (true) sentence |

$1+1=3$ | is a (false) sentence |

$x+1 = 3$ | is a (sometimes true/sometimes false) sentence |

So, $\,x\,$ is to mathematics as *cat* is to English:

hence the title of the book,

*One Mathematical Cat, Please!*

Master the ideas from this section

by practicing*both* exercises at the bottom of this page.

When you're done practicing, move on to:

Basic Addition Practice

by practicing

When you're done practicing, move on to:

Basic Addition Practice

CONCEPT QUESTIONS EXERCISE:

On this exercise, you will not key in your answer.

However, you can check to see if your answer is correct.

However, you can check to see if your answer is correct.