Expression is the broadest mathematical term. In essence, everything consists of them in this science, and all operations are also carried out on them. Another question is that, depending on the particular species, completely different methods and techniques are used. So, working with trigonometry, fractions or logarithms are three different actions. An expression that does not make sense can refer to one of two types: numerical or algebraic. But what this concept means, what its example looks like, and other points will be considered later.
Numeric expressions
If an expression consists of numbers, brackets, pluses and minuses and other signs of arithmetic, it can safely be called numerical. Which is quite logical: you just have to take a look at the first component named once.
Any number can be a numerical expression: the main thing is that there are no letters in it. And “anything” in this case means everything: from a simple, lonely, in itself, digit, to a huge list of them and signs of arithmetic operations that require subsequent calculation of the final result. A fraction is also a numerical expression if it does not contain any a, b, c, d, etc., because then it is a completely different kind, which will be discussed later.
Conditions for an expression that does not make sense
When the task begins with the word "calculate", we can talk about the transformation. The thing is that this action is not always advisable: it is not that much needed, if an expression that does not make sense comes to the fore. The examples are infinitely surprising: sometimes, in order to understand that it has overtaken us, it takes a long and tedious time to open the brackets and count-count-count ...
The main thing to remember: it does not make sense that expression whose final result is reduced to an action forbidden in mathematics. If it’s completely honest, then the transformation itself becomes meaningless, but in order to find out, you have to perform it first. Such a paradox!
The most famous, but no less important forbidden mathematical action is the division by zero.
Therefore, for example, an expression that does not make sense:
(17 + 11) :( 5 + 4-10 + 1).
If, using simple calculations, we reduce the second bracket to one digit, then it will be zero.
According to the same principle, "honorary title" is given to this expression:
(5-18) :( 19-4-20 + 5).
Algebraic Expressions
This is the same numeric expression if you add forbidden letters to it. Then it becomes a complete algebraic. It can also be of all sizes and shapes. Algebraic expression is a broader concept that includes the previous one. But it made sense to start a conversation not with him, but with a numerical one, so that it would be easier to understand and understand. After all, does an algebraic expression make sense - the question is not that it is very complex, but having more clarifications.
Why is that?
A literal expression, or an expression with variables, is synonymous. The first term is easy to explain: after all, it, in the end, contains letters! The second is also not a mystery of the century: instead of letters, you can substitute different numbers, as a result of which the meaning of the expression will change. It is easy to guess that the letters in this case are variables. By analogy, numbers are constants.
And here we return to the main topic: what is an expression that does not make sense?
Examples of algebraic expressions that do not make sense
The condition for the meaninglessness of an algebraic expression is similar, as for a numerical one, with only one exception, or more precisely, an addition. When converting and calculating the final result, variables have to be taken into account, so the question is not “what expression does not make sense?”, But “at what value of the variable does this expression make no sense?” and "is there such a value for the variable that the expression loses its meaning?"
For example, (18-3) :( a + 11-9).
The above expression does not make sense with a equal to -2.
And as for (a + 3) :( 12-4-8), we can safely say that this expression does not make sense for any a.
In the same way, whatever b you substitute in the expression (b - 11) :( 12 + 1), it will still make sense.
Typical tasks on the topic "Expression that does not make sense"
Grade 7 studies this topic in mathematics, among others, and tasks on it are often encountered both immediately after the corresponding lesson, and as a “trick” question on modules and exams.
That is why it is worth considering typical tasks and methods for solving them.
Example 1
Does the expression make sense:
(23 + 11) :( 43-17 + 24-11-39)?
Decision:
It is necessary to make all the calculations in brackets and bring the expression to the form
34: 0
Answer:
The final result contains division by zero, therefore, the expression does not make sense.
Example 2
What expressions do not make sense?
1) (9 + 3) / (4 + 5 + 3-12);
2) 44 / (12-19 + 7);
3) (6 + 45) / (12 + 55-73).
Decision:
You should calculate the final value for each of the expressions.
Answer: 1; 2.
Example 3
Find the range of valid values for the following expressions:
1) (11-4) / (b + 17);
2) 12 / (14-b + 11).
Decision:
The range of permissible values (ODZ) is all those numbers, the substitution of which instead of variables, the expression will make sense.
That is, the task sounds like: find the values at which there will be no division by zero.
Answer:
1) b є (-∞; -17) & (-17; + ∞), or b> -17 & b <-17, or b ≠ -17, which means that the expression makes sense for all b except -17 .
2) b є (-∞; 25) & (25; + ∞), or b> 25 & b <25, or b ≠ 25, which means that the expression makes sense for all b except 25.
Example 4
At what values will the following expression not make sense?
(y-3) :( y + 3)
Decision:
The second bracket is zero when the game is -3.
Answer: y = -3
Example 4
Which of the expressions does not make sense only with x = -14?
1) 14: (x - 14);
2) (3 + 8x) :( 14 + x);
3) (x / (14 + x)) :( 7/8)).
Answer:
2 and 3, since in the first case, if we substitute x = -14, then the second bracket is equal to -28, and not to zero, as it sounds in the definition of an expression that does not make sense.
Example 5
Create and write down an expression that does not make sense.
Answer:
18 / (2-46 + 17-33 + 45 + 15).
Algebraic expressions with two variables
Despite the fact that all expressions that do not make sense have one essence, there are different levels of their complexity. So, we can say that numerical ones are simple examples, because they are easier than algebraic ones. The number of variables in the latter adds difficulties to the solution. But they should not be confused by their appearance: the main thing is to remember the general principle of the solution and apply it regardless of whether the example looks like a typical problem or has some unknown additions.
For example, the question may arise how to solve such a task.
Find and write a pair of numbers that are not valid for the expression:
(x 3 - x 2 y 3 + 13x - 38y) / (12x 2 - y).
Answer Options:
1) 3 and 107;
2) 1 and -12;
3) 2 and 48;
4) -2 and 24;
5) -3 and 108.
But in reality it only looks scary and cumbersome, because in reality it contains what has long been known: squaring numbers into a square and a cube, some arithmetic operations, such as division, multiplication, subtraction and addition. For convenience, by the way, the problem can be reduced to a fractional form.
The numerator of the resulting fraction is not happy: (x 3 - x 2 y 3 + 13x - 38y). It is a fact. But there is another reason for happiness: you don’t even need to touch it for solving the task! According to the definition considered earlier, it is impossible to divide by zero, and what exactly will be divided into it is completely unimportant. Therefore, we leave this expression unchanged and substitute pairs of numbers from these options in the denominator. Already the third paragraph fits perfectly, turning a small bracket into zero. But to dwell on this is a bad recommendation, because something else may come up. Indeed: the fifth paragraph also fits well and fits the condition.
We write down the answer: 3 and 5.
Finally
As you can see, this topic is very interesting and not very complicated. To understand it is not difficult. But still, working out a couple of examples will never hurt!