Work 7 rectangle formulas laws of arithmetic operations. Theoretical foundations of the laws and properties of arithmetic operations. Learning new material
In the course of historical development, of course, they added and multiplied for a long time, without realizing the laws to which these operations are subject. Only in the 20s and 30s of the previous century, mainly French and English mathematicians figured out the basic properties of these operations. Anyone who wants to get acquainted with the history of this issue in more detail, I can recommend here, as I will do this repeatedly below, the large “Encyclopedia of Mathematical Sciences”.
Returning to our topic, I now mean to actually enumerate those five fundamental laws to which addition is reduced:
1) always represents a number, in other words, the action of addition is always feasible without any exceptions (as opposed to subtraction, which is not always feasible in the area of positive numbers);
2) the amount is always uniquely determined;
3) there is a combinational or associative law: , so the brackets can be omitted altogether;
4) there is a commutative or commutative law:
5) the law of monotonicity holds: if , then .
These properties are understandable without further explanation if we have before our eyes a visual representation of number as quantity. But they must be expressed strictly formally, so that they can be relied upon in the future strictly. logical development theories.
As for multiplication, there are, first of all, five laws similar to those just listed:
1) there is always a number;
2) the product is unambiguous,
3) law of combination:
4) the law of mobility:
5) law of monotonicity: if , then
Finally, the connection between addition and multiplication is established by the sixth law:
6) the law of distribution, or distributivity:
It is easy to understand that all calculations are based solely on these 11 laws. I will limit myself to a simple example, say, multiplying the number 7 by 12;
according to the law of distribution
In this short discussion, you will, of course, recognize the individual steps that we perform when calculating in the decimal system. I’ll leave it to you to figure out the more complex examples yourself. Here we will express only a summary result: our digital calculations consist in re-applying the eleven basic provisions listed above, as well as in applying the results of operations on single-digit numbers (the addition table and the multiplication table) learned by heart.
However, where do the laws of monotony find application? In ordinary, formal calculations, we really don’t rely on them, but they turn out to be necessary in problems of a slightly different kind. Let me remind you here of a method that in decimal counting is called estimating the value of the product and the quotient. This is a technique of the greatest practical importance, which, unfortunately, is not yet sufficiently known at school and among students, although on occasion they talk about it already in the second grade; I will limit myself here to just an example. Let's say we need to multiply 567 by 134, and in these numbers the unit digits are set, say, by physical measurements- only very inaccurate. In this case, it would be completely useless to calculate the product with complete accuracy, since such a calculation still does not guarantee us the exact value of the number we are interested in. But what is really important for us is to know the order of magnitude of the product, that is, to determine within what number of tens or hundreds the number lies. But the law of monotonicity actually gives you this estimate directly, because it follows from it that the required number is contained between 560-130 and 570-140. Further development I again leave these considerations to you yourself.
In any case, you see that in “estimating calculations” you have to constantly use the laws of monotonicity.
As for the actual application of all these things in school teaching, there can be no question of a systematic exposition of all these fundamental laws of addition and multiplication. The teacher can only dwell on the laws of combination, commutation and distribution, and then only when moving on to literal calculations, heuristically deducing them from simple and clear numerical examples.
The approach to the addition of non-negative integers allows us to substantiate the well-known laws of addition: commutative and combinational.
Let us first prove the commutative law, i.e. we prove that for any non-negative integers a and b the equality a + b = b + a holds.
Let a be the number of elements in set A, b be the number of elements in set B and A B=0. Then, by definition of the sum of non-negative integers, a + b is the number of elements of the union of sets A and B: a + b = n (A//B). But the set A B is equal to the set B A according to the commutative property of the union of sets, and, Hence, n(AU B) = n(B U A). By the definition of the sum n(BiA) = b + a, therefore a+b=b+a for any non-negative integers a and b.
Let us now prove the combination law, i.e. we prove that for any non-negative integers a, b, c the equality (a + b) + c = a + (b + c) holds.
Let a = n(A), b = n(B), c = n(C), and АУВ = 0, ВУС = 0 Then, by the definition of the sum of two numbers, we can write (a+ b)+ c = n(A/ /)B) + p(C) = p((AUBUC).
Since the union of sets obeys the combination law, then n((AUB)U C) = n(A U(BUC)). From where, by definition of the sum of two numbers, we have n (A J(BUC)) = n (A) + n (BU C) = a + (b + c). Therefore, (a+ b)+ c -- a+(b + c) for any non-negative integers a, b and c.
What is the purpose of the associative law of addition? He explains how you can find the sum of three terms: to do this, just add the first term with the second and add the third term to the resulting number, or add the first term to the sum of the second and third. Note that the combination law does not imply rearrangement of terms.
Both the commutative and associative laws of addition can be generalized to any number of terms. In this case, the commutative law will mean that the sum does not change with any rearrangement of terms, and the associative law will mean that the sum does not change with any grouping of terms (without changing their order).
From the commutative and associative laws of addition it follows that the sum of several terms will not change if they are rearranged in any way and if any group of them is enclosed in brackets.
Let's calculate, using the laws of addition, the value of the expression 109 + 36+ 191 +64 + 27.
Based on the commutative law, we rearrange terms 36 and 191. Then 109 + 36+191+64 + 27= 109+191+36 + 64 + 27.
Let's use the combination law, grouping the terms, and then find the sums in brackets: 109+ 191 +36 + 64 + 27 ==(109 + 191) + (36 + 64) + 27 = 300 + 100 + 27.
Let's apply the combination law again, enclosing the sum of the numbers 300 and 100 in brackets: 300+ 100 + 27 = (300+ 100) + 27.
Let's do the calculations: (300+ 100)+ 27 = 400+ 27 = 427.
With the commutative property of addition, students primary classes become familiar when studying the numbers of the first ten. It is first used to create a single-digit addition table and then to rationalize various calculations.
Combination law of addition in initial course mathematics is not explicitly studied, but is constantly used. Thus, it is the basis for the technique of adding a number by parts: 3 + 2 = 3 + (1 + 1) = (3+ 1)+ 1 =4+ 1 =5. In addition, in cases where it is necessary to add a number to a sum, a sum to a number, a sum to a sum, the associative law is used in combination with the commutative law. For example, adding the sum 2+1 to the number 4 is proposed in the following ways:
1) 4 + (2+1) = 4 + 3 = 7;
4+2+ 1 = 6+1 =7;
4 + (2+1) = 5 + 2 = 7.
Let's analyze these methods. In case 1, calculations are performed in accordance with in the order indicated actions. In case 2 applied associative property addition. Calculations in the latter case are based on the commutative and associative laws of addition, and intermediate transformations are omitted. They are like that. First, based on the commutative law, we swapped terms 1 and 2: 4+(2-1) = 4 + (1+2). Then we used the combination law: 4 + (1 +2) = (4+ 1) + 2. And finally, we made calculations according to the order of operations (4 +1)+ 2 = 5 + 2 = 7.
Rules for subtracting a number from a sum and a sum from a number
Let us justify the known rules for subtracting a number from a sum and a sum from a number.
The rule for subtracting a number from a sum. To subtract a number from a sum, it is enough to subtract this number from one of the terms of the sum and add another term to the resulting result.
Let's write this rule using the symbols: If a, b, c are non-negative integers, then:
a) for a>c we have that (a+b) -- c = (a -- c)+b;
b) for b>c we have that (a+b) -- c==a + (b -- c);
c) for a>c and b>c, you can use any of these formulas.
Let a >c, then the difference a -c exists. Let us denote it by p: a - c = p. Hence a = p+c. Substitute the sum p+-c instead of a into the expression (a+b) -- c and transform it: (a + 6) --c = (p + c+b) -- c = p+b+-c -- c = p+b
But the letter p denotes the difference a - c, which means we have (a + b) - - c = (a - c) + b, which is what needed to be proved.
The same reasoning is carried out for other cases. Let us now illustrate this rule (case “a”) using Euler circles. Let us take three finite sets A, B and C, such that n(A) = a, n(B) = b, n(C) = c and AUB = 0, CUA. Then (a+b) - c is the number of elements of the set (AUB)C, and the number (a - c) + b is the number of elements of the set (AC)UB. On Euler circles, the set (AUB)C is represented by the shaded area shown in the figure.
It is easy to verify that the set (AC)UB will be represented by exactly the same area. So (AUB)C = (AC)UB for the data
sets A, B and C. Consequently, n((AUB)C) = n((AC)UB)u (a + b) - c - (a - c) + b.
Case “b” can be illustrated similarly.
The rule for subtracting a sum from a number. To subtract the sum of numbers from a number, it is enough to subtract from this number each term one by one, i.e. if a, b, c are non-negative integers, then for a>b+c we have a--(b+c ) = (a - b) - c.
The rationale for this rule and its set-theoretic illustration are carried out in the same way as for the rule for subtracting a number from a sum.
The above rules are discussed in primary school using specific examples, visual images are used for justification. These rules allow you to perform calculations rationally. For example, the rule for subtracting a sum from a number underlies the technique of subtracting a number by parts:
5-2 = 5-(1 + 1) = (5-1)-1=4-1=3.
The meaning of the above rules is well revealed when solving arithmetic problems different ways. For example, the problem “In the morning, 20 small and 8 large fishing boats went to sea. 6 boats returned. How many boats with fishermen still have to return? can be solved in three ways:
/ way. 1. 20 + 8 = 28 2. 28 -- 6 = 22
// way. 1. 20 -- 6=14 2. 14 + 8 = 22
III method. 1. 8 -- 6 = 2 2. 20 + 2 = 22
Multiplication laws
Let us prove the laws of multiplication based on the definition of a product through the Cartesian product of sets.
1. Commutative law: for any non-negative integers a and b, the equality a*b = b*a is true.
Let a = n(A), b = n(B). Then, by the definition of the product, a*b = n(A*B). But the sets A*B and B*A are equally powerful: each pair (a, b) from the set AXB can be associated with a single pair (b, a) from the set BxA, and vice versa. This means n(AXB) = n(BxA), and therefore a-b = n (AXB) = n (BXA) = b-a.
2. Combination law: for any non-negative integers a, b, c, the equality (a* b) *c = a* (b*c) is true.
Let a = n(A), b = n(B), c = n(C). Then, by the definition of the product (a-b)-c = n((AXB)XQ, a-(b -c) = n (AX(BXQ). The sets (AxB)XC and A X (BX Q are different: the first consists of pairs of the form ((a, b), c), and the second - from pairs of the form (a, (b, c)), where aЈA, bЈB, cЈC. But the sets (AXB)XC and AX(BXC) are of equal power, since there is a one-to-one mapping of one set to another. Therefore, n((AXB) *C) = n(A*(B*C)), and, therefore, (a*b) *c = a* (b*c).
3. Distributive law of multiplication relative to addition: for any non-negative integers a, b, c, the equality (a + b) x c = ac+ be is true.
Let a - n (A), b = n (B), c = n (C) and AUB = 0. Then, by the definition of a product, we have (a + b) x c = n ((AUB) * C. Whence, based on equality (*) we obtain n ((A UВ) * C) = n((A * C)U(B* C)), and further, by the definition of the sum and product n ((A * C)U(B* C) ) -- = n(A*C) + n(B*C) = ac + bc.
4. Distributive law of multiplication relative to subtraction: for any non-negative integers a, b and c and a^b the equality (a - b)c = = ac - bc is true.
This law is derived from the equality (AB) *C = (A *C)(B*C) and is proven similarly to the previous one.
The commutative and associative laws of multiplication can be extended to any number of factors. As with addition, these laws are often used together, that is, the product of several factors will not change if they are rearranged in any way and if any group of them is enclosed in parentheses.
Distributive laws establish the connection between multiplication and addition and subtraction. Based on these laws, brackets are opened in expressions like (a + b) c and (a - b) c, as well as the factor is taken out of brackets if the expression is of the form ac - be or
In the initial course of mathematics, the commutative property of multiplication is studied; it is formulated as follows: “The product will not change by rearranging the factors” - and is widely used in compiling the multiplication table for single-digit numbers. The commutative law is not explicitly considered in elementary school, but is used together with the commutative law when multiplying a number by a product. This happens as follows: students are asked to consider various ways find the value of the expression 3* (5*2) and compare the results.
Cases are given:
1) 3* (5*2) = 3*10 = 30;
2) 3* (5*2) = (3*5) *2 = 15*2 = 30;
3) 3* (5*2) = (3*2) *5 = 6*5 = 30.
The first of them is based on the rule of order of actions, the second on the associative law of multiplication, the third on the commutative and associative laws of multiplication.
The distributive law of multiplication relative to addition is discussed in school using specific examples and is called the rules for multiplying a number by a sum and a sum by a number. The consideration of these two rules is dictated by methodological considerations.
Rules for dividing a sum by a number and numbers by a product
Let's get acquainted with some properties of dividing natural numbers. The choice of these rules is determined by the content of the initial mathematics course.
The rule for dividing a sum by a number. If numbers a and b are divisible by number c, then their sum a + b is divisible by c; the quotient obtained by dividing the sum a + b by the number c is equal to the sum of the quotients obtained by dividing a by c and b by c, i.e.
(a + b): c = a: c + b: c.
Proof. Since a is divisible by c, there is such a natural number t = a:c, that a = c-t. Similarly, there is a natural number n - b:c such that b = c-n. Then a+b = c-m + c-/2 = c-(m + n). It follows that a + b is divisible by c and the quotient obtained by dividing a + b by the number c is equal to m + n, i.e. a: c + b: c.
The proven rule can be interpreted from a set-theoretic point of view.
Let a = n(A), b = n(B), and AGV = 0. If each of the sets A and B can be divided into equal subsets, then the union of these sets allows the same partition.
Moreover, if each subset of the partition of set A contains a:c elements, and each subset of set B contains b:c elements, then each subset of set A[)B contains a:c+b:c elements. This means that (a + b): c = a: c + b: c.
The rule for dividing a number by a product. If a natural number a is divisible by natural numbers b and c, then to divide a by the product of numbers b and c, it is enough to divide the number a by b (c) and divide the resulting quotient by c (b): a: (b * c) --(a: b): c = (a: c): b Proof. Let's put (a:b):c = x. Then, by definition of the quotient a:b = c-x, hence similarly a - b-(cx). Based on the associative law of multiplication a = (bc)-x. The resulting equality means that a:(bc) = x. Thus a:(bc) = (a:b):c.
The rule for multiplying a number by the quotient of two numbers. To multiply a number by the quotient of two numbers, it is enough to multiply this number by the dividend and divide the resulting product by the divisor, i.e.
a-(b:c) = (a-b):c.
The application of the formulated rules makes it possible to simplify calculations.
For example, to find the value of the expression (720+ 600): 24, it is enough to divide the terms 720 and 600 by 24 and add the resulting quotients:
(720+ 600): 24 = 720:24 + 600:24 = 30 + 25 = 55. The value of the expression 1440:(12* 15) can be found by first dividing 1440 by 12, and then dividing the resulting quotient by 15:
1440: (12 * 15) = (1440:12): 15 = 120:15 = 8.
These rules are discussed in the initial mathematics course using specific examples. When you first become acquainted with the rule of dividing the sum 6 + 4 by the number 2, illustrative material is used. In the future, this rule is used to rationalize calculations. The rule of dividing a number by a product is widely used when dividing numbers ending in zeros.
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Slide captions:
10.22.15 Cool work
Find the length of the segment AB a b A B b a B A AB= a + b AB= b + a
11 + 16 = 27 (fruits) 16 + 11 = 27 (fruits) Will the total number of fruits change if the terms are rearranged? Masha collected 11 apples and 16 pears. How many fruits were in Masha's basket?
Make up a letter expression to record the verbal statement: “the sum will not change by rearranging the terms” a + b = b + a Commutative law of addition
(5 + 7) + 3 = 15 (toys) Which method of counting is easier? Masha was decorating the Christmas tree. She hung 5 Christmas balls, 7 pine cones and 3 stars. How many toys did Masha hang up? (7 + 3) + 5 =15 (toys)
Make up a letter expression to record the verbal statement: “To add a third term to the sum of two terms, you can add the sum of the second and third terms to the first term” (a + b) + c = a + (b + c) Combination law of addition
Let's count: 27+ 148+13 = (27+13) +148= 188 124 + 371 + 429 + 346 = = (124 + 346) + (371 + 429) = = 470 + 800 = 1270 Let's learn to count quickly!
Are the same laws valid for multiplication as for addition? a b = b a (a b) c = a (b c)
b=15 a =12 c=2 V = (a b) c = a (b c) V = (12 15) 2= =12 (15 2)=360 S = a b= b a S = 12 15 = 15 12 =180
a · b = b · a (a · b) · с = a · (b · с) Commutative law of multiplication Combinative law of multiplication
Let's count: 25 · 756 · 4 = (25 · 4) · 756= 75600 8 · (956 · 125) = = (8 · 125) · 956 = = 1000 · 956 = 956000 Let's learn to count quickly!
LESSON TOPIC: What are we working with in today's lesson? Formulate the topic of the lesson.
212 (1 column), 214 (a, b, c), 231, 230 In the classroom Homework 212 (2 columns), 214(d,e,f), 253
On the topic: methodological developments, presentations and notes
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lesson development Mathematics 5th grade Laws of arithmetic operations
lesson development Mathematics 5th grade Laws of arithmetic operations No. Structure of the annotation Contents of the annotation 1231 Full name Malyasova Lyudmila Gennadievna 2 Position, subject taught Ma...
Topic No. 1.
Real numbers. Numerical expressions. Converting Numeric Expressions
I. Theoretical material
Basic Concepts
· Integers
· Decimal notation of number
· Opposite numbers
· Whole numbers
· Common fraction
Rational numbers
· Infinite decimal
· Period of number, periodic fraction
· Irrational numbers
· Real numbers
Arithmetic operations
Numeric expression
· Expression value
· Conversion of a decimal fraction to an ordinary fraction
Converting a fraction to a decimal
Conversion of a periodic fraction into an ordinary fraction
· Laws of arithmetic operations
· Signs of divisibility
Numbers used when counting objects or to indicate the serial number of an object among similar objects are called natural. Any natural number can be written using ten numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This notation of numbers is called decimal
For example: 24; 3711; 40125.
The set of natural numbers is usually denoted N.
Two numbers that differ from each other only by sign are called opposite numbers.
For example, numbers 7 and – 7.
The natural numbers, their opposites, and the number zero make up the set whole Z.
For example: – 37; 0; 2541.
Number of the form , where m – integer, n – natural number, called ordinary fraction. Note that any natural number can be represented as a fraction with a denominator of 1.
For example: , .
The union of sets of integers and fractions (positive and negative) constitutes a set rational numbers. It is usually denoted Q.
For example: ; – 17,55; .
Let the given decimal fraction be given. Its value will not change if you add any number of zeros to the right.
For example: 3,47 = 3,470 = 3,4700 = 3,47000… .
Such a decimal is called an infinite decimal.
Any common fraction can be represented as an infinite decimal fraction.
A sequentially repeated group of digits after the decimal point in a number is called period, and an infinite decimal fraction having such a period in its notation is called periodic. For brevity, it is customary to write a period once, enclosing it in parentheses.
For example: 0,2142857142857142857… = 0,2(142857).
2,73000… = 2,73(0).
Infinite decimal non-periodic fractions are called irrational numbers.
The union of the sets of rational and irrational numbers constitutes the set valid numbers. It is usually denoted R.
For example: ; 0,(23); 41,3574…
Number 
is irrational.
For all numbers, the actions of three steps are defined:
· Stage I actions: addition and subtraction;
· Stage II actions: multiplication and division;
· Stage III actions: exponentiation and root extraction.
An expression made up of numbers, arithmetic symbols and parentheses is called numeric.
For example: 
; .
The number obtained as a result of performing actions is called the value of the expression.
Numeric expression doesn't make sense, if it contains division by zero.
When finding the value of the expression, the actions of stage III, stage II and at the end of the action of stage I are performed sequentially. In this case, it is necessary to take into account the placement of brackets in the numerical expression.
Converting a numerical expression consists of sequentially performing arithmetic operations on the numbers included in it using the appropriate rules (the rule for adding ordinary fractions with different denominators, multiplying decimals, etc.). Tasks for converting numeric expressions into textbooks are found in the following formulations: “Find the value of a numerical expression”, “Simplify the numerical expression”, “Calculate”, etc.
When finding the values of some numerical expressions, you have to perform operations with fractions different types: ordinary, decimal, periodic. In this case, it may be necessary to convert an ordinary fraction to a decimal or perform the opposite action - replace the periodic fraction with an ordinary one.
To convert decimal to common fraction, it is enough to write the number after the decimal point in the numerator of the fraction, and one with zeros in the denominator, and there should be as many zeros as there are digits to the right of the decimal point.
For example: ; .
To convert fraction to decimal, you need to divide its numerator by its denominator according to the rule for dividing a decimal fraction by a whole number.
For example: 
;
;
.
To convert periodic fraction to common fraction, necessary:
1) from the number before the second period, subtract the number before the first period;
2) write this difference as a numerator;
3) write the number 9 in the denominator as many times as there are numbers in the period;
4) add as many zeros to the denominator as there are digits between the decimal point and the first period.
For example: 
; 
.
Laws of arithmetic operations on real numbers
1. Traveling(commutative) law of addition: rearranging the terms does not change the value of the sum:
2. Traveling(commutative) law of multiplication: rearranging the factors does not change the value of the product:
3. Conjunctive(associative) law of addition: the value of the sum will not change if any group of terms is replaced by their sum:
4. Conjunctive(associative) law of multiplication: the value of the product will not change if any group of factors is replaced by their product:
.
5. Distribution(distributive) law of multiplication relative to addition: to multiply a sum by a number, it is enough to multiply each addend by this number and add the resulting products:
Properties 6 – 10 are called absorption laws 0 and 1.
Signs of divisibility
Properties that allow, in some cases, without dividing, to determine whether one number is divisible by another, are called signs of divisibility.
Test for divisibility by 2. A number is divisible by 2 if and only if the number ends in even number. That is, at 0, 2, 4, 6, 8.
For example: 12834; –2538; 39,42.
Test for divisibility by 3. A number is divisible by 3 if and only if the sum of its digits is divisible by 3.
For example: 2742; –17940.
Test for divisibility by 4. A number containing at least three digits is divisible by 4 if and only if the two-digit number formed by the last two digits of the given number is divisible by 4.
For example: 15436; –372516.
Divisibility test by 5. A number is divisible by 5 if and only if its last digit is either 0 or 5.
For example: 754570; –4125.
Divisibility test by 9. A number is divisible by 9 if and only if the sum of its digits is divisible by 9.
For example: 846; –76455.
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