Math notations, formulas, and statements

People all over the world speak different languages, but there is one language that everyone understands: the language of mathematics. Mathematical expressions do not need to be translated in order to be comprehensible to someone on the other side of the world. Unlike natural language, mathematics is used to convey abstract, logical ideas. It is a system that consists of notations, formulas, and statements. Let's take a closer look at each one of these components.

Mathematical notation

There is such a term in mathematics as mathematical notation. Math notations are used to represent various mathematical operations, relations, and elements, and also to combine them into formulas and expressions. In other words, they allow us to easily write all mathematical statements. Think of math notations as an alphabet used by mathematicians to communicate with each other. These notations mainly consist of letters, symbols, numbers, and signs. There are different types of notations used for specific purposes. Let's take a look at all of them!

The first type of notation is arithmetic notation. They're used to represent all mathematical operations using only several symbols like $+, -,\cdot, \div, \pm$ and some combinations of them. For example, when you were a first-grader, you studied addition. When the teacher told you to write two plus one and count the result, you just wrote $2+1=3$ instead of words! You were very young, but at that moment in time, you began using arithmetic notations.

The second type of notations are equality signs and comparison notations. The most common equality signs are $=, \approx$. You may know only the first equality sign, so let's find out what the second sign means! The approximately equal sign $\approx$ is often used when you round any numbers. For example, Earth's gravitational acceleration $g$ is approximately equal to $9.8\ m/s^2$. You can rewrite this sentence using the $\approx$ sign: $g \approx9.8\ m/s^2$. As for comparison notations, the most common signs are $>, \ge , <, \le$. The $>, <$ are called greater than and less than signs. These signs remind you of a crocodile, don't they? This crocodile is always hungry and eats the biggest possible number, so its mouth is always open toward the biggest number.

The $\ge, \le$ are called greater than or equal to and less than or equal to signs. For example, in many countries, people have to be at least $35$ years old to be elected as a president, so their age must be $\ge35$.

Algebra symbols are also a type of math notation. One of the most famous algebra symbols is $n!$, which is read as $n$ factorial. Its meaning is simple: it stands for the product of all natural numbers from $1$ to $n$, that is, $n! =1\cdot 2 \cdots n$. We all agree that it is far more convenient to write $n!$ instead of a long product, don't we? This is exactly the main purpose why math notations were brought into the game. Here are some more difficult types of algebra symbols.

Here you can see the Greek alphabet. Greek letters are also a part of math notations!

We've discussed the most essential types of math notations. There also are some more complex types, which you'll see as you progress in math, but we won't focus on them now.

Mathematical formulas

Now that you know what math notations are, you are ready to get to know formulas. A mathematical formula is an expression or specific rule written using mathematical symbols. They help to write all mathematical expressions faster and more formally. Mathematical formulas include numbers known as constants, letters representing unknown quantities known as variables, and mathematical symbols known as signs. When we know the value of one quantity, we can find the value of another using a formula. In algebra, geometry, and other subjects, formulas are used to simplify the process of getting an answer and save time.

The main formulas are arithmetic, algebra, and geometry.

Let's begin with arithmetic formulas. These formulas are made of constants and signs. Coming back to the previous paragraph, $2+1=3$ is an example of an arithmetic formula. You see, even first-graders use formulas!

Algebra formulas are performed to determine unknown quantities that are expressed in letters. They are formed by a combination of:

• variables: $x$ is the variable in the expression $x+2$

• constants: $7$ and $5$ are constants in the expression $4x-7=5$

• coefficients: $4$ is the coefficient of $x$ in the expression $4x$

• coefficients of variables: $a$ is the coefficient of the variable $x^2$ in the quadratic expression $ax^2+bx+c$

When students are introduced to algebra, they begin with short multiplication formulas like the following:

• $a^2-b^2=(a-b)(a+b)$

• $(a-b)^2=a^2-2ab+b^2$

Many students wonder why they have to get acquainted with these boring formulas and even lose all motivation to study math. Don't be like them! These formulas are really important, and they will help you to simplify many complex math expressions which you'll learn while progressing in math.

Geometry is a branch of mathematics that deals with the shape, size, mutual arrangement of shapes, and properties of shapes. Thus, formulas in geometry are used to find the dimensions, perimeter, area, surface area, and volume of these shapes. For example,

• The area of the circle = $\pi r^2$, where $r$ is the radius of the circle and $\pi \approx3.14$

• The volume of the cube = $a^3$, where $a$ is the side of the cube.

Mathematical statements

If until now there was no English in math notations or formulas, statements, on the other hand, use natural language. A mathematical statement is a sentence that gives us certain information and is either true or false. It can contain words and symbols. Mathematical proof is a convincing argument for the truth of a statement. Mathematical statements usually consist of two parts: a hypothesis or assumption and a conclusion.

All statements are usually made using the following expressions:

• "Either/Or" — $A$ or $B$ must hold (either A or B).

  

• "And" — both $A$ and $B$ must hold (A and B).

  

• "If...then..." — conditional statement. For instance, "If the weather is nice (hypothesis), then I will go for a walk (conclusion)".

So, let's build a few mathematical statements using these expressions. We'll start with a simple one. You know, that all integers are even or odd and there are no other options. Thus, we can write down the following mathematical expression: If a number is an integer, it's either even or odd. Let's move to something more complex and remember your school days once more. For sure, you've learned some divisibility rules in primary or middle school. One of the easiest to remember was the divisibility condition for $5$. Here it is: A number is divisible by $5$ if its last digit is $5$ or $0$. We've just written another mathematical statement!

Conclusion

In this topic, we learned some basics of mathematics. Let's take a look at the key points:

• Mathematical notation — letters, symbols, numbers, and signs which are used to represent mathematical operations and elements.

• Mathematical formula — an expression written using mathematical symbols, which helps to write all mathematical expressions faster and more formally.

• Mathematical statement — a sentence that gives us certain information and is either true or false.

Now that you know their meaning as well as some examples, you are ready to dive deeper into the fascinating world of mathematics.