### Education, The science

# An expression that does not make sense: examples

Expression is the broadest mathematical term. In essence, in this science of them consists of everything, and all operations are also carried out over them. Another question is that, depending on the particular species, completely different methods and techniques are used. So, working with trigonometry, fractions or logarithms is three different actions. An expression that has no meaning can refer to one of two types: numeric or algebraic. But what does this concept mean, what does his example look like and other points will be considered further.

## Numerical Expressions

If the expression consists of numbers, brackets, pluses, minuses, and other signs of arithmetic operations, you can safely call it numeric. Which is quite logical: it is only necessary to look once more at the first named component.

A numerical expression can be anything: the main thing is that it does not have letters. And under "anything" in this case is understood everything: from a simple, standing alone, in itself, figures, to a huge list of them and signs of arithmetic operations that require the 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 this is a completely different kind, which will be described later.

## Conditions for an expression that does not make sense

When the task begins with the word "calculate", you can talk about the transformation. The trick is that this action is not always advisable: it is not that much needed, if an expression having no meaning comes to the fore. The examples are endlessly amazing: sometimes, in order to understand that it has overtaken us, it takes a long time and tediously to open the brackets and count-count-counting ...

The main thing to remember is that there is no sense in expressing an expression whose ultimate result is reduced to a forbidden action in mathematics. If it's completely honest, then the transformation itself becomes meaningless, but in order to find out, we have to fulfill it first. Such a paradox!

The most famous, but from that no less important forbidden mathematical action is division by zero.

Therefore, for example, an expression that does not make sense:

(17 + 11): (5 + 4-10 + 1).

If, with the help of simple calculations, reduce the second bracket to one digit, then it will be zero.

On 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 the forbidden letters to it. Then it becomes a complete algebraic one. It can also be of all sizes and shapes. Algebraic expression is a broader concept, including the previous one. But it was meaningful to start a conversation not with him, but with a numerical one, so that it would be clearer and easier to understand. After all, does the expression have an algebraic meaning? The question is not that it is very complicated, but it has more refinements.

## Why is that?

A letter expression, or an expression with variables, is a synonym. The first term is easy to explain: after all, it contains letters! The second is also not the mystery of the century: instead of letters you can substitute different numbers, so the meaning of the expression will change. It is not difficult to guess that the letters in this case are variables. By analogy, numbers are constants.

And then we return to the main theme: 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 analogous, as for a numerical expression, with only one exception, or, to be more precise, a complement. When you convert and calculate the final result, you have to take into account the variables, so the question is not how "what expression does not make sense?", But "for what value of the variable will this expression make no sense?" And "is there such a value for the variable at which the expression will lose its meaning?"

For example, (18-3): (a + 11-9).

The above expression does not make sense at a equal to -2.

And here about (a + 3) :( 12-4-8) we can safely say that this expression has no meaning for any a.

In the same way, what b you substitute in the expression (b - 11): (12 + 1), it still will make sense.

## Typical tasks on the topic "Expression, meaningless"

Grade 7 studies this topic in mathematics among others, and assignments on it are often found both directly after the relevant lesson, and as a "trick" question on the modules and exams.

That is why it is worthwhile to consider typical tasks and methods for their solution.

*Example 1.*

Does the expression make sense:

(23 + 11): (43-17 + 24-11-39)?

Decision:

It is necessary to make all the calculation in parentheses and bring the expression to the form:

34: 0

Answer:

The end result contains division by zero, hence, the expression does not make sense.

*Example 2.*

Which 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 must 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 valid values (ODZ) is all those numbers, which, when substituted, will have meaning instead of variables.

That is, the task sounds like: find the values at which there will be no division by zero.

Answer:

1) b ((-∞; -17) & (-17; + ∞), or b> -17 & b <-17, or b ≠ -17, which means that the expression makes sense for all b except -17 .

2) b ((-∞; 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 does the following expression make no sense?

(Y-3): (y + 3)

Decision:

The second bracket is zero when playing -3.

Answer: y = -3

*Example 4.*

Which of the expressions do not make sense only at 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 for x = -14, then the second bracket equals -28, and not zero, as it sounds in the definition of a meaningless expression.

*Example 5.*

Invent 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, one thing is, there are different levels of complexity. So, we can say that numerical examples are simple, because they are easier than algebraic ones. The number of variables in the latter adds to the difficulty of solving. But they should not confuse their appearance: the main thing is to remember the general principle of the solution and apply it regardless of whether the example is similar to a typical problem or has some unknown additions.

For example, you might wonder how to solve this task.

Find and write down a pair of numbers that are invalid for the expression:

(X ^{3} - x ^{2} y ^{3} + 13x - 38y) / (12x ^{2} - y).

Variants of answers:

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 terrible and cumbersome, because it actually contains what has long been known: the construction of numbers in a square and a cube, some arithmetic operations, such as division, multiplication, subtraction and addition. For convenience, by the way, you can bring the problem to a fractional form.

The numerator does not please the resulting fraction: (x ^{3} - x ^{2} y ^{3} + 13x - 38y). It is a fact. But there is another reason for happiness: it's not necessary to touch it for solving the task! According to the definition, considered earlier, you can not divide by zero, and what exactly will be shared with it is completely unimportant. Therefore, we leave this expression unchanged and substitute pairs of numbers from these variants in the denominator. The third point fits perfectly, turning a small bracket to zero. But to dwell on this is a bad recommendation, because something else can come up. Indeed: the fifth point 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 particularly complicated. It will not be difficult to understand it. But still work out a couple of examples will never hurt!

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