Algebra/Chapter 2/Different Types of Relationships

Mathematical Expressions

Algebra Chapter 2: An Introduction to Algebra Section 2: Mathematical Sentences

Sets

2.2: Mathematical Sentences

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In this section, we introduce mathematical sentences and the ways that expressions can be related.

The Language of Mathematics

People usually have trouble understanding mathematical ideas. This is not necessarily because math is difficult, but it is usually because mathematics is often presented as a foreign language.

In much mathematical literature, you will come across statements such as: "Let X, Y, and Z be integers". This kind of language is customary, and will very often appear in mathematical literature. When we are told to "let" things be a certain way, you are essentially being asked to imagine or assume that those things are that way.

Previously, we've been talking about expressions. These were quantities of interest that were being represented by symbols and operations. Algebraic expressions are also quantities, just with several unknown parts. Unfortunately, on their own, expressions do not communicate anything, as they are only a single quantity. Expressions in mathematics can be thought of as being analogous to the nouns used in a language, in which both are used to refer to objects of interest.

On its own, the word "cat" does not communicate anything, but telling someone that "a cat is a type of mammal" would communicate a complete thought to them. Likewise, the expression "x" does not communicate anything. Telling someone that "x is equal to 1" would also communicate a complete thought, in this case communicating what the exact value of the unknown value x is. Much like how a correct arrangement of words is referred to as a "sentence", an arrangement of mathematical symbols that expresses a complete thought is referred to as a sentence.

The diagram below provides the general analogy of mathematics as a language.

Relationships

Perhaps the most common kind of mathematical sentence you will see in all of mathematics are ones that describe how quantities are related. Intuitively, we refer to these sentences as relationships. Up to this point, you have encountered relationships all throughout, such as when we said that quantities were "equal", "greater than", "less than", etc. Here, we will cover what these relationships mean in a broader sense.

Numbers have many different names. In our language analogy, we can say that numbers have "synonyms" like nouns do.

Example 2.6: Refer to the expressions below: $6\ \ \ \ \ \ \ \ 1+1+1+1+1+1\ \ \ \ \ \ \ \ 2+4\ \ \ \ \ \ \ \ {\frac {18}{3}}\ \ \ \ \ \ \ \ (7-2)+1$ Even though all of these expressions look different, they all refer to the same number. The idea that all numbers can be expressed in so many ways is an extremely crucial one to all of mathematics, as we will see later in the section.

In mathematics, when an expression A refers to a quantity B, or when expressions A and B refer to the same value, we say that they are equal. We show this relationship with an equal sign (=).

$A=B$

We can extend this definition even futher with more than two quantities or expressions. If we know that A, B, and C all represent the same quantity, we can express this as follows:

$A=B=C$

Example 2.7: Communicate that all of the expressions from Example 2.6 are equal to 6 using the equal sign. The equality of the expressions can be expressed as: $6=1+1+1+1+1+1=2+4={\frac {18}{3}}=(7-2)+1$