This is a very interesting question. The fact is that expression definition math is the basis of all mathematics. All mathematics consists of expressions and their transformations. It is a broad term that includes everything you learn in math – fractions, formulas, equations, tasks, and more.
For example, you need to solve a very complex mathematical task. Even if you’re very good in math, will you be able to find an answer for the task right away? The answer is no.
You will need to find the solution for the task doing multiple mathematical transformations. Consistently, step by step, you will be simplifying this task according to certain rules, i.e. converting the mathematical expressions. Depending on how well you’re going to do the modifications, depends how strong you are in mathematics. If you can’t deal with such modifications, it will be very difficult for you in the future.
Expression definition math is a very broad concept. Almost everything you’re dealing with in mathematics is a set of mathematical expressions. All the tasks, formulas, fractions, equations, and so on – all of these consist of mathematical expressions.
3 + 2 is a mathematical expression. c2 - d2 is also a mathematical expression. And a huge fraction and even one number – all of them are mathematical expressions. For example, the equation: 5x + 2 = 12 consists of two mathematical expressions, connected by an equal sign. One expression is on the left, the other is on the right.
Each type of mathematical expression has its own set of rules and methods that should be used in the solution. There is one set for the work with fractions. There is another set for the work with trigonometric expressions, and another set for the work with logarithms, and so on. Sometimes these rules are the same, sometimes they are totally different.
Below, we will discuss two main types of expression definition math: numerical expressions and algebraic expressions.
Numerical expression definition math is a very simple concept. The name itself suggests that this is the expression with numbers. This is the mathematical expression made up of numbers, parentheses, and other mathematical signs. For example, 7-3 is a numeric expression; (8 + 3.2) x 5.4 is also the numerical expression. The expressions with fractions are also numerical.
Average numbers, fractions, any tasks for the calculation without X and other characters are numerical expressions.
The main sign of a numeric expression definition math is the lack of letters, only numbers and mathematical signs.
As a rule, you need to calculate numeric expressions. In order to do this, you sometimes need to open the brackets, to change signs, simplify, rearrange terms – i.e., convert the expressions.
Sometimes it happens that nothing should be done with a numerical expression. This transaction is implemented when the expression does not make sense.
Understandably, if you some abracadabra, like 3+:) (+) - + there is no need to do anything with such expression, as it is unclear what to do with this expression.
But sometimes at first sight you can see quite decorous expressions, for example, (2 + 3) / (16 - 2 x 8). However, this expression doesn’t make sense for the simple reason that the second parenthesis (if you count) equals to zero. And it is impossible to divide by zero. This is a restricted operation in mathematics. So, there is nothing you can do with this expression. There can be only one possible answer for the task like that «The expression does not make sense».
If there are letters in a numerical expression, then such expression definition math is called an algebraic expression. For example: 5a2; 3x-2y; 3 (z-2); 3,4m / n; x2 + 4x-4; (A + b)2... and so on.
Such expressions are also called literal expressions or expressions with variables.
The notion of an algebraic expression is wider than a numeric expression, as it includes all the numerical expressions, i.e. numeric expression is also an algebraic expression, but without letters.
For example, in the expression 5 + y, y is a variable, unlike the number 5 that is a constant.
The algebraic expression definition math means that in order to solve such expression one needs to use the laws and rules of algebra.
With the numeric expressions everything is clear – you can’t divide by zero. What about letters? For example, there is an expression with variables: 2 / (a - 5). Does it make sense? Well, who knows? Because a can be any number.
However, there is one value of a, at which this expression does not make sense. This value is number 5. If you replace variable a with the number 5, in brackets you will get zero. And you can’t divide by zero. So, it turns out that this expression does not make sense if a = 5. This expression makes sense for all values of a, except a = 5.
The entire set of numbers that can be substituted into the given expression is called the domain of acceptable values of this expression.
Now let’s figure out what the transformation of expressions is. The answer is simple – it is any action with expression definition math. For example, let’s take a numerical expression 3 + 5. How can you convert it? It's very simple – by calculating: 3 + 5 = 8. This calculation will be the transformation of the expression.
You can also write the same expression in a different way: 3 + 5 = 5 + 3. I this case, you didn’t calculate anything; you simply wrote an expression in another form. However, it is also the transformation of the expression. It is possible to make many such transformations.
Any action with the expression, any record of it in another form is called a conversion of the expression. However, it is important to keep in mind one very important rule when doing transformations with expression definition math. Let’s discuss the rule.
Suppose, you have transformed the expression haphazardly, like 3 + 5 = 2 + 1. Is it a transformation? Of course, it is. You have written the expression in another form, haven’t you?
The fact that the random transformations are not interested for the math at all. All mathematics is based on transformations, when not only the appearance of the expressions changes, but also the essence of the expression. 3 + 5 can be written in any desired form, but it should be equal to 8.
Conversions that do not change the essence of the expression are called identical.
It is identical transformations that allow us, step by step, to transform complex examples in a simple expression, while retaining the essence of the example. If in the chain of transformations we make a mistake and don’t do the identical transformation, then we’ll be dealing with another example and other answers that are not irrelevant to the task.
In algebraic expressions, the identical transformations are solved in formulas and rules. For example, there is the following formula in algebra: a (b + c) = ab + ac. This means that instead of expression a (b + c) we can boldly write the expression ab + ac, and vice versa. This is the identical transformation. Mathematics provides us with the choices of these two expressions.
One of the most important and necessary transformations is the main property of fractions. If the numerator and denominator are multiplied (divided) by the same number, or the expression that is not equal to zero, the fraction will not change.
There are many formulas that define the identical transformations. One of the basic transformations is the factoring. It is used in all of mathematics, from the elementary to the highest.