After receiving general information about equalities in mathematics, we move on to narrower topics. The material of this article will give an idea of the properties of numerical equalities.
What is numerical equality
The first time we encounter numerical equalities in elementary school, when we get acquainted with numbers and the concept of "the same". Those. the most primitive numerical equalities are: 2 = 2, 5 = 5, etc. And at that level of study, we called them simply equalities, without specifying "numerical", and laid in them a quantitative or ordinal meaning (which natural numbers carry). For example, the equation 2 = 2 will correspond to an image with two flowers and two bumblebees perched on each. Or, for example, two queues, where Vasya and Vanya are second in order.
As knowledge of arithmetic operations appears, numerical equalities become more complicated: 5 + 7 \u003d 12; 6 - 1 = 5; 2 1 = 2; 21: 7 = 3, etc. Then equalities begin to occur, in the recording of which numerical expressions of various kinds participate. For example, (2 + 2) + 5 = 2 + (5 + 2) ; 4 (4 − (1 + 2)) + 12: 4 − 1 = 4 1 + 3 − 1, etc. Then we get acquainted with other types of numbers, and numerical equalities become more and more interesting and diverse.
Definition 1
Numerical equality is an equality, both parts of which consist of numbers and/or numerical expressions.
Properties of numerical equalities
It is difficult to overestimate the importance of the properties of numerical equalities in mathematics: they are the basis for many things, determine the principle of working with numerical equalities, solution methods, rules for working with formulas, and much more. Obviously, there is a need for a detailed study of the properties of numerical equalities.
The properties of numerical equalities are absolutely consistent with how actions with numbers are defined, as well as with the definition of equal numbers through the difference: number a is equal to the number b only when the difference a-b there is zero. Further in the description of each property, we will trace this connection.
Basic properties of numerical equalities
Let's start studying the properties of numerical equalities with three basic properties that are inherent in all equalities. We list the main properties of numerical equalities:
- reflexivity property: a = a;
- symmetry property: if a = b, then b = a;
- transitivity property: if a = b and b=c, then a = c, where a , b and c are arbitrary numbers.
The property of reflexivity denotes the fact that a number is equal to itself: for example, 6 = 6, - 3 = - 3, 4 3 7 = 4 3 7, etc.
Proof 1
It is easy to demonstrate the validity of equality a − a = 0 for any number a: difference a - a can be written as a sum a + (− a), and the addition property of numbers gives us the opportunity to assert that any number a corresponds to the only opposite number − a, and their sum is zero.
Definition 3
According to the symmetry property of numerical equalities: if the number a is equal to the number b,
that number b is equal to the number a. For example, 4 3 = 64
, then 64 = 4 3
.
Proof 2
You can justify this property through the difference of numbers. condition a = b corresponds to equality a − b = 0. Let's prove that b − a = 0.
Let's write the difference b - a as - (a - b), relying on the rule for opening brackets preceded by a minus sign. The new entry for the expression is - 0 , and the opposite of zero is zero. Thus, b − a = 0, hence: b = a.
Definition 4
The property of transitivity of numerical equalities states that two numbers are equal to each other if they are simultaneously equal to a third number. For example, if 81 = 9 and 9 = 3 2 , then 81 = 3 2 .
The property of transitivity also corresponds to the definition of equal numbers through the difference and properties of operations with numbers. Equalities a = b and b=c correspond to the equalities a − b = 0 and b − c = 0.
Proof 3
Let us prove the equality a − c = 0, from which the equality of numbers will follow a and c. Since adding a number to zero does not change the number itself, then a - c write in the form a + 0 − c. Instead of zero, we substitute the sum of opposite numbers −b and b, then the final expression becomes: a + (− b + b) − c. Let's group the terms: (a − b) + (b − c). The differences in brackets are equal to zero, then the sum (a − b) + (b − c) there is zero. This proves that when a − b = 0 and b − c = 0, the equality a − c = 0, where a = c.
Other important properties of numerical equalities
The main properties of numerical equalities discussed above are the basis for a number of additional properties that are quite valuable in the context of practice. Let's list them:
Definition 5
By adding to (or subtracting from) both parts of the numerical equality, which is true, the same number, we obtain the correct numerical equality. Let's write it literally: if a = b, where a and b are some numbers, then a + c = b + c for any c.
Proof 4
As a justification, we write the difference (a + c) − (b + c).
This expression can easily be converted to the form (a − b) + (c − c).
From a = b by condition it follows that a − b = 0 and c − c = 0, then (a - b) + (c - c) = 0 + 0 = 0. This proves that (a + c) − (b + c) = 0, hence, a + c = b + c;
Definition 6
If both parts of the correct numerical equality are multiplied with any number or divided by a number that is not equal to zero, then we get the correct numerical equality.
Let's write it down literally: when a = b, then a c = b c for any number c. If c ≠ 0 then and a:c = b:c.
Proof 5
Equality is true: a c − b c = (a − b) c = 0 c = 0, and it implies the equality of the products a c and b c. And division by a non-zero number c can be written as a multiplication by the reciprocal of 1 c ;
Definition 7
At a and b, different from zero and equal to each other, their reciprocals are also equal.
Let's write: when a ≠ 0 , b ≠ 0 and a = b, then 1 a = 1 b. The extreme equality is not difficult to prove: for this purpose, we divide both sides of the equality a = b by a number equal to the product a b and not equal to zero.
We also point out a couple of properties that allow the addition and multiplication of the corresponding parts of the correct numerical equalities:
Definition 8
With term-by-term addition of the correct numerical equalities, the correct equality is obtained. This property is written as follows: if a = b and c = d, then a + c = b + d for any numbers a , b , c and d.
Proof 6
It is possible to substantiate this useful property based on the previously mentioned properties. We know that any number can be added to both sides of a true equality.
Towards equality a = b add the number c, and to equality c = d- number b, the result will be the correct numerical equalities: a + c = b + c and c + b = d + b. We write the last one in the form: b + c = b + d. From equalities a + c = b + c and b + c = b + d according to the property of transitivity, the equality follows a + c = b + d. Which is what needed to be proven.
It is necessary to clarify that term by term it is possible to add not only two true numerical equalities, but also three or more;
Definition 7
Finally, we describe such a property: term-by-term multiplication of two correct numerical equalities gives the correct equality. Let's write in letters: if a = b and c = d, then a c = b d.
Proof 7
The proof of this property is similar to the proof of the previous one. Multiply both sides of the equation by any number, multiply a = b on the c, a c = d on the b, we obtain the correct numerical equalities a c = b c and c b = d b. We write the last as b c = b d. The property of transitivity makes it possible from the equality a c = b c and b c = b d derive equality a c = b d which we needed to prove.
And again, we clarify that this property is applicable for two, three or more numerical equalities.
Thus, one can write: if a = b, then a n = b n for any numbers a and b, and any natural number n.
Let's finish this article by collecting all the considered properties for clarity:
If a = b , then b = a .
If a = b and b = c , then a = c .
If a = b , then a + c = b + c .
If a = b, then a c = b c.
If a = b and c ≠ 0, then a: c = b: c.
If a = b , a = b , a ≠ 0 and b ≠ 0 , then 1 a = 1 b .
If a = b and c = d, then a c = b d.
If a = b , then a n = b n .
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
Having a general idea of equalities in mathematics, we can proceed to a more detailed study of this issue. In this article, we will, firstly, explain what numerical equalities are, and, secondly, we will study.
Page navigation.
What is numerical equality?
Acquaintance with numerical equalities begins at the very initial stage of studying mathematics at school. This usually happens in grade 1 right after the first numbers from 1 to 9 become known and after the phrase "the same" takes on meaning. Then the first numerical equalities appear, for example, 1=1, 3=3, etc., which at this stage are usually called simply equalities without a clarifying definition of "numerical".
Equalities of the specified type at this stage are given a quantitative or ordinal meaning, which is embedded in . For example, the numerical equation 3=3 corresponded to the picture, which shows two branches of a tree, each of which has 3 birds sitting on it. Or when our comrades Petya and Kolya are third in line in two lines.
After studying arithmetic operations, more diverse records of numerical equalities appear, for example, 3+1=4, 7−2=5, 3 2=6, 8:4=2, etc. Further, numerical equalities of an even more interesting form begin to occur, containing various ones in their parts, for example, (2 + 1) + 3 = 2 + (1 + 3) , 4 (4−(1+2))+12:4−1=4 1+3−1 and the like. Then there is an acquaintance with other types of numbers, and numerical equalities become more and more diverse.
So, it’s enough to beat around the bush, it’s time to give a definition of numerical equality:
Definition.
Numerical equality is an equality, in both parts of which there are numbers and / or numerical expressions.
Properties of numerical equalities
The principles of working with numerical equalities are determined by their properties. And a lot is tied to the properties of numerical equalities in mathematics: from the properties of solving equations and some methods for solving systems of equations to the rules for working with formulas that connect various quantities. This explains the need for a detailed study of the properties of numerical equalities.
The properties of numerical equalities are in full agreement with how operations with numbers are defined, and are also in agreement with definition of equal numbers through the difference: the number a is equal to the number b if and only if the difference a−b is equal to zero. Below, when describing each property, we will trace this connection.
Basic properties of numerical equalities
A review of the properties of numerical equalities should begin with three basic properties that are characteristic of all equalities without exception. So, basic properties of numerical equalities This:
- reflexivity property: a=a ;
- symmetry property: if a=b , then b=a ;
- and the transitivity property: if a=b and b=c , then a=c ,
where a , b and c are arbitrary numbers.
The reflexivity property of numerical equalities refers to the fact that a number is equal to itself. For example, 5=5 , −2=−2 , etc.
It is easy to show that for any number a the equality a−a=0 is true. Indeed, the difference a−a can be rewritten as the sum a+(−a) , and from the properties of number addition we know that for any number a there is a unique −a , and the sum of opposite numbers is equal to zero.
The symmetry property of numerical equalities states that if the number a is equal to the number b, then the number b is equal to the number a. For example, if 2 3 =8 (see ), then 8=2 3 .
We justify this property through the difference of numbers. The condition a=b corresponds to the equality a−b=0 . Let us show that b−a=0 . The rule for opening brackets preceded by a minus sign allows us to rewrite the difference b−a as −(a−b) , which in turn is equal to −0 , and the number opposite to zero is zero. Therefore, b−a=0 , which implies that b=a .
The property of transitivity of numerical equalities states that two numbers are equal when they are both equal to a third number. For example, it follows from the equalities (see ) and 4=2 2 that .
This property is also consistent with the definition of equal numbers through the difference and the properties of operations with numbers. Indeed, the equalities a=b and b=c correspond to the equalities a−b=0 and b−c=0 . Let's show that a−c=0 , whence it will follow that the numbers a and c are equal. Since adding zero does not change the number, a−c can be rewritten as a+0−c . Zero is replaced by the sum of opposite numbers −b and b , while the last expression takes the form a+(−b+b)−c . Now we can group the terms as follows: (a−b)+(b−c) . And the differences in brackets are zeros, hence the sum (a−b)+(b−c) is equal to zero. This proves that, under the condition a−b=0 and b−c=0, the equality a−c=0 holds, whence a=c .
Other important properties
From the basic properties of numerical equalities, analyzed in the previous paragraph, a number of properties that have tangible practical value follow. Let's break them down.
Let's start with this property: if you add (or subtract) the same number to both parts of a true numerical equality, then you get a true numerical equality. Using letters, it can be written like this: if a=b , where a and b are some numbers, then a+c=b+c for any number c .
To justify, we compose the difference (a+c)−(b+c) . It can be converted to the form (a−b)+(c−c) . Since a=b by convention, then a−b=0 , and c−c=0 , so (a−b)+(c−c)=0+0=0 . This proves that (a+c)−(b+c)=0 , hence a+c=b+c .
We go further: if both parts of a true numerical equality are multiplied by any number or divided by a non-zero number, then we get the correct numerical equality. That is, if a=b , then a c=b c for any number c , and if c is a non-zero number, then a:c=b:c .
Indeed, a·c−b·c=(a−b)·c=0·c=0 , which implies that the products of a·c and b·c are equal. And division by a non-zero number c can be thought of as multiplying by 1/c.
From the analyzed property of numerical equalities, one useful consequence follows: if a and b are different from zero and equal numbers, then their reciprocals are also equal. That is, if a≠0 , b≠0 and a=b , then 1/a=1/b . The last equality is easy to prove: for this, it is enough to divide both parts of the original equality a=b by a non-zero number equal to the product a b .
And let's dwell on two more properties that allow us to add and multiply the corresponding parts of the correct numerical equalities.
If you add the correct numerical equalities term by term, then you get the correct equality. That is, if a=b and c=d , then a+c=b+d for any numbers a , b , c and d .
Let us justify this property of numerical equalities, starting from the properties already known to us. It is known that we can add any number to both parts of a true equality. In the equality a=b we add the number c, and in the equality c+d we add the number b, as a result we get the correct numerical equalities a+c=b+c and c+b=d+b, the last of which we rewrite as b+c= b+d. From the equalities a+c=b+c and b+c=b+d, by the property of transitivity, the equality a+c=b+d follows, which was to be proved.
Note that it is possible to add term by term not only two correct numerical equalities, but also three, and four, and any finite number of them.
We complete the review of the properties of numerical equalities with the following property: if we multiply two correct numerical equalities term by term, we get the correct equality. Let's formulate it formally: if a=b and c=d , then a c=b d .
The proof of this property is similar to the proof of the previous one. We can multiply both sides of the equality by any number, multiply a=b by c, and c=d by b, we get the correct numerical equalities a c=b c and c b=d b , the last of which we rewrite as b c=b d . Then, by the property of transitivity, the equalities a·c=b·c and b·c=b·d imply the required equality a·c=b·d .
Note that the voiced property is true for term-by-term multiplication of three or more correct numerical equalities. It follows from this statement that if a=b , then a n =b n for any numbers a and b , and any natural number n .
At the end of this article, we write all the analyzed properties of numerical equalities in a table:
Bibliography.
- Moro M.I.. Mathematics. Proc. for 1 cl. early school At 2 p. Part 1. (First half year) / M. I. Moro, S. I. Volkova, S. V. Stepanova. - 6th ed. - M.: Enlightenment, 2006. - 112 p.: ill. + App. (2 separate l. ill.). - ISBN 5-09-014951-8.
- Algebra: textbook for 7 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 17th ed. - M. : Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
And now let's analyze this task in detail.
Consider the next cell in the pyramid.
We know that 11 is the sum of 7 and another unknown number. Obviously, the second number is 4, so we can fill in the cell on the right in the first row.
There is one empty cell left in the pyramid. It should contain a number, adding to which 7 should get 12. Thus. in the empty cell on the left in the first row should be the number 5.
Consider the cells in the second row. There should be two numbers in the sum of which should be equal to 24. At the same time, note that in order to get the desired two numbers in the second column, you need to add 3 and 5 to some unknown number, which is located in the middle cell of the first row, that is, the difference these two numbers should equal 2. The numbers 11 and 13 are suitable for these conditions, because 11 + 13 \u003d 24, and on the other hand 13 - 11 \u003d 2. Thus, we can fill in the cells of the 2nd row.
And it remains to find the last number in the first row. This number can be obtained if it is added to 3 and then we get 11. Thus. this number is 8.
