Mathematical Language and Symbols

The Language of Mathematics

Mathematics is a language with its own notation that we are very well acquainted with; but it is often neglected to teach the origin of this notation.

A long time ago, mathematicians used full sentences to express their formulas.

$\text{ex. The sum of two real numbers is also a real number.}$

We can clearly see the problem with this notation– it’s a hassle to write down ––and that is the reason our modern mathematical notation was born.

The Characteristics of the Mathematical Language

The language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express; it is inherently intuitive to the group that uses them most.

Mathematics’ language is often described as the following:

• Precise
  • It is able to make very fine distinctions.
• Concise
  • It is able to say things briefly.
• Powerful
  • It is able to express complex thoughts with relative ease.

If we were to use our previous example and translate it into mathematical notation, we would get the following expression:

$\forall a,b\in \mathbb{R},a+b\in \mathbb{R}$

The following are the equivalent meanings of the following parts of the expression above.

• $\forall a,b\in \mathbb{R}$
  • This portion translates to “For all real numbers a and b…”
    • The first symbol ( $\forall$ ) stands for “for all” or “for any”.
    • The second and third symbols ( $a$ and $b$ ) are variables that refer to hypothetical numbers involved in the expression.
    • The fourth symbol ( $\in$ ) stands for “is in” or “is a member of”.
    • The fifth symbol ( $\mathbb{R}$ ) refers to the set of real numbers. ( You may refer here.).
• $a+b\in \mathbb{R}$
  • This portion translates to “…the sum of $a$ and $b$ is a real number”.
    • The first three symbols ( $a+b$ ) refers to the sum of $a$ and $b$.
    • The last two symbols ( $\in \mathbb{R}$ ) functions the same way it did in the previous section. It can be read as “is in the set of real numbers” or “is a member of the set of real numbers”.

Expressions vs Sentences

When formulating a sentence one must contain a complete though, the same is true in mathematics; a mathematical sentence must state a complete thought.

An expression, on the other hand, refers to a mathematical object of interest.

Truth of Sentence

Mathematical sentences may either be True or False, but never Both.

Unary and Binary Operations

We are mostly familiar with binary operators in mathematics, which are operators that require two operands ( addition for example needs one value on each side ). But there are such things as unary operators which only needs one value (usually on the operator’s right).

Unary Operations

These are operations that only need one value.

A well known example of unary operators is the negative sign.

$-5$

Another example of unary operators are trigonometry functions.

$\sin \theta ,\cos \theta ,\tan \theta$

Binary Operations

These are operations that only need one value.

An operations is considered binary if it takes two real numbers as arguments to produce another real number.

Properties of Binary Operations (Addition and Multiplication)

Closure of Binary Operations

• The product and the sum of any two real numbers is also a real number.

Expression for Addition: $\forall a,b\in \mathbb{R},a+b\in \mathbb{R}$

Expression for Multiplication: $\forall a,b\in \mathbb{R},a\cdot b\in \mathbb{R}$

Commutativity of Binary Operations

Expression for Addition: $\forall a,b\in \mathbb{R},a+b=b+a$

Expression for Multiplication: $\forall a,b\in \mathbb{R},a\cdot b=b\cdot a$

Associativity of Binary Operations

Expression for Addition: $\forall a,b,c\in \mathbb{R},(a+b)+c=a+(b+c)$

Expression for Multiplication: $\forall a,b,c\in \mathbb{R},(a\cdot b)\cdot c=a\cdot (b\cdot c)$

Distributivity of Binary Operations

$\forall a,b\in \mathbb{R},c(a+b)=ca+cb$

Identity Elements of Binary Operations

Expression for Addition ( $e$ is equivalent to $0$ ): $\forall a\in \mathbb{R},a+e=e+a=a$

Expression for Addition ( $e$ is equivalent to $1$ ): $\forall a\in \mathbb{R},a\cdot e=e\cdot a=a$

Inverse of Binary Operations

$\forall a\in \mathbb{R},a+(-a)=-a+a$