The concept of subtraction is best understood with an example. You decide to drink tea with sweets. There were 10 sweets in the vase. You ate 3 candies. How many candies are left in the vase? If we subtract 3 from 10, there will be 7 sweets left in the vase. Let's write the problem mathematically:
Let's look at the entry in detail:
10 is the number from which we subtract or decrease, which is why it is called reducible.
3 is the number we are subtracting. That's why they call him deductible.
7 is the result of subtraction or is also called difference. The difference shows how much the first number (10) is greater than the second number (3) or how much the second number (3) is less than the first number (10).
If you doubt whether you found the difference correctly, you need to do check. Add the second number to the difference: 7+3=10
When subtracting l, the minuend cannot be less than the subtrahend.
We draw a conclusion from what has been said. Subtraction- this is an action by which the second term is found from the sum and one of the terms.
In literal form, this expression will look like this:
a—b =c
a – minuend,
b – subtrahend,
c – difference.
Properties of subtracting a sum from a number.
13 — (3 + 4)=13 — 7=6
13 — 3 — 4 = 10 — 4=6
The example can be solved in two ways. The first way is to find the sum of the numbers (3+4), and then subtract from total number(13). The second way is to subtract the first term (3) from the total number (13), and then subtract the second term (4) from the resulting difference.
In literal form, the property of subtracting a sum from a number will look like this:
a - (b + c) = a - b - c
The property of subtracting a number from a sum.
(7 + 3) — 2 = 10 — 2 = 8
7 + (3 — 2) = 7 + 1 = 8
(7 — 2) + 3 = 5 + 3 = 8
To subtract a number from a sum, you can subtract this number from one term, and then add the second term to the resulting difference. The condition is that the summand will be greater than the number being subtracted.
In literal form, the property of subtracting a number from a sum will look like this:
(7 + 3) — 2 = 7 + (3 — 2)
(a +b) —c=a + (b - c), provided b > c
(7 + 3) — 2=(7 — 2) + 3
(a + b) - c=(a - c) + b, provided a > c
Subtraction property with zero.
10 — 0 = 10
a - 0 = a
If you subtract zero from a number then it will be the same number.
10 — 10 = 0
a—a = 0
If you subtract the same number from a number then it will be zero.
Related questions:
In example 35 - 22 = 13, name the minuend, subtrahend and difference.
Answer: 35 – minuend, 22 – subtrahend, 13 – difference.
If the numbers are the same, what is their difference?
Answer: zero.
Do the subtraction test 24 - 16 = 8?
Answer: 16 + 8 = 24
Subtraction table natural numbers from 1 to 10.
Examples for problems on the topic “Subtraction of natural numbers.”
Example #1:
Insert the missing number: a) 20 - ... = 20 b) 14 - ... + 5 = 14
Answer: a) 0 b) 5
Example #2:
Is it possible to subtract: a) 0 - 3 b) 56 - 12 c) 3 - 0 d) 576 - 576 e) 8732 - 8734
Answer: a) no b) 56 - 12 = 44 c) 3 - 0 = 3 d) 576 - 576 = 0 e) no
Example #3:
Read the expression: 20 - 8
Answer: “Subtract eight from twenty” or “subtract eight from twenty.” Pronounce words correctly
Can be written using letters.
1. The commutative property of addition is written as follows: a + b = b + a.
In this equality, the letters a and b can take any natural value and the value 0.
3. The property of zero during addition can be written as follows: Here the letter a can have any meaning.
4. The property of subtracting a sum from a number is written using letters as follows:
a - (b + c) = a - b - c. Here b + c< а или b + с = а.
5. The property of subtracting a number from a sum is written using letters like this:
(a + b) - c = a + (b - c), if c< Ь или о = b;
(a + b) - c = (a - c) + b, if c< а или с = а.
6. The properties of zero during subtraction can be written as follows: a - 0 = a; a - a = 0.
Here a can take any natural values and the value 0.
Read the properties of addition and subtraction written using letters.
337. Write the combining property of addition using the letters a, b and c. Replace the letters with their values: a = 9873, b = 6914, c = 10,209 - and check the resulting numerical equality.
338. Write down the property of subtracting a sum from numbers using the letters a, b and c. Replace the letters with their values: a = 243, b = 152, c = 88 - and check the resulting numerical equality.
339. Write down the property of subtracting a number from a sum in two ways. Check the resulting numerical equations by replacing the letters with their values:
a) a = 98, b = 47 and c = 58;
b) a = 93, b = 97 and c = 95.
340. a) In Figure 42, use a compass to find the points M(a + b) and N(a - b).
b) Using Figure 43, explain the meaning of the associative property of addition.
c) Explain with the help of pictures the other properties of addition and subtraction.
341. From the properties of addition it follows:
56 + x + 14 = x + 56 + 14 = x + (56 + 14) = x + 70.
Simplify according to this example expression:
a) 23 + 49 + m; c) x + 54 + 27;
b) 38 + n + 27; d) 176 4- y + 24.
342. Find the meaning of the expression after simplifying it:
a) 28 + m + 72 with m = 87; c) 228 + k + 272 with k = 48;
b) n + 49 + 151 with n = 63; d) 349 + p + 461 with p = 115.
343. From the properties of subtraction it follows:
28 - (15 + s) = 28 - 15 - s = 13 - s,
a - 64 - 26 = a - (64 + 26) = a - 90.
What property of subtraction is used in these examples? Using this property of subtraction, simplify the expression:
a) 35 - (18 + y);
b) m- 128 - 472.
344. From the properties of addition and subtraction it follows:
137 - s - 27 « 137 - (s + 27) = 137 - (27 + s) = 137 - 27 - s = 110 - s.
What properties of addition and subtraction are used in this example?
Using these properties, simplify the expression:
a) 168 - (x + 47);
b) 384 - m - 137.
345. From the properties of subtraction it follows:
(154 + b) - 24 = (154 - 24) + b = 130 + b;
a - 10 + 15 = (a - 10) + 15 = (a + 15) - 10 = a + (15 - 10) = a + 5.
Which property of subtraction is used in this example?
Using this property, simplify the expression:
a) (248 + m) - 24; c) b + 127 - 84; e) (12 - k) + 24;
b) 189 + n - 36; d) a - 30 + 55; e) x - 18 + 25.
346. Find the meaning of the expression after simplifying it:
a) a - 28 - 37 at a = 265; c) 237 + c + 163 with c = 194; 188;
b) 149 + b - 99 with b = 77; d) d - 135 + 165 with d = 239; 198.
347. Points C and D are marked on segment AB, and point C lies between points A and D. Write an expression for length segment:
a) AB if AC = 453 mm, CD = x mm and DB = 65 mm. Find the value of the resulting expression at x = 315; 283.
b) AC, if AB = 214 mm, CD = 84 mm and DB = y mm. Find the value of the resulting expression when y = 28; 95.
348. A turner completed an order for the production of identical parts in three days. On the first day he made 23 parts, on the second day - b parts more than on the first day, and on the third day - four parts less than on the first day. How many parts did the turner produce in these three days? Write an expression to solve the problem and find its value for b = 7 and b = 9.
349. Calculate orally:
350. Find half, quarter and third of each of the numbers: 12; 36; 60; 84; 120.
a) 37 2 and 45 - 17;
b) 156: 12 and 31 7.
362. A pedestrian and a cyclist are moving towards each other on the road. Now the distance between them is 52 km. The speed of a pedestrian is 4 km/h, and the speed of a cyclist is 9 km/h. What will be the distance between them after 1 hour; after 2 hours; in 4 hours? How many hours later will the pedestrian and the cyclist meet?
363. Find the meaning of the expression:
1) 1032: (5472: 19: 12);
2) 15 732: 57: (156: 13).
364. Simplify the expression:
a) 37 + m + 56; c) 49 - 24 - k;
b) n - 45 - 37; d) 35 - t - 18.
365. Simplify the expression and find its meaning:
a) 315 - p + 185 at p = 148; 213;
b) 427 - l - 167 at I = 59; 260.
366. The motorcycle racer covered the first section of the track in 54 s, the second in 46 s, and the third one p s faster than the second. How long did the motorcycle racer take to complete these three sections? Find the value of the resulting expression if n = 9; 17; 22.
367. In a triangle, one side is 36 cm, the other is 4 cm less, and the third is x cm more than the first side. Find the perimeter of the triangle. Write an expression to solve the problem and find its value at x = 4 and x = 8.
368. A tourist traveled 40 km by bus, which is 5 times more than the distance he traveled on foot. What is the total route taken by the tourist?
369. From city to village 24 km. A man comes out of the city and walks at a speed of 6 km/h. Draw on the distance scale (one scale division - 1 km) the position of the pedestrian 1 hour after leaving the city; after 2 hours; in 3 hours, etc. When will he come to the village?
370. True or false inequality:
a) 85 678 > 48 - (369 - 78);
b) 7508 + 8534< 26 038?
371. Find the meaning of the expression:
a) 36,366-17,366: (200 - 162);
b) 2 355 264: 58 + 1 526 112: 56;
c) 85 408 - 408 (155 - 99);
d) 417 908 + 6073 56 + 627 044.
N.Ya. VILENKIN, V. I. ZHOKHOV, A. S. CHESNOKOV, S. I. SHVARTSBURD, Mathematics grade 5, Textbook for general education institutions
Planning mathematics, materials for grade 5 mathematics download, textbooks online
So, in general, subtracting natural numbers does NOT have the commutative property. Let's write this statement using letters. If a and b are unequal natural numbers, then a−b≠b−a. For example, 45−21≠21−45.
The property of subtracting the sum of two numbers from a natural number.
The next property is related to subtracting the sum of two numbers from a natural number. Let's look at an example that will give us an understanding of this property.
Let's imagine that we have 7 coins in our hands. We first decide to keep 2 coins, but thinking that this will not be enough, we decide to keep another coin. Based on the meaning of adding natural numbers, it can be argued that in this case we decided to save the number of coins, which is determined by the sum 2+1. So, we take two coins, add another coin to them and put them in the piggy bank. In this case, the number of coins remaining in our hands is determined by the difference 7−(2+1) .
Now imagine that we have 7 coins, and we put 2 coins into the piggy bank, and after that another coin. Mathematically, this process is described by the following numerical expression: (7−2)−1.
If we count the coins that remain in our hands, then in both the first and second cases we have 4 coins. That is, 7−(2+1)=4 and (7−2)−1=4, therefore, 7−(2+1)=(7−2)−1.
The considered example allows us to formulate the property of subtracting the sum of two numbers from a given natural number. Subtracting a given sum of two natural numbers from a given natural number is the same as subtracting the first term of a given sum from a given natural number, and then subtracting the second term from the resulting difference.
Let us recall that we gave meaning to the subtraction of natural numbers only for the case when the minuend is greater than the subtrahend or equal to it. Therefore, we can subtract a given sum from a given natural number only if this sum is not greater than the natural number being reduced. Note that if this condition is met, each of the terms does not exceed the natural number from which the sum is subtracted.
Using letters, the property of subtracting the sum of two numbers from a given natural number is written as equality a−(b+c)=(a−b)−c, where a, b and c are some natural numbers, and the conditions a>b+c or a=b+c are met.
The considered property, as well as the combinatory property of addition of natural numbers, make it possible to subtract the sum of three or more numbers from a given natural number.
The property of subtracting a natural number from the sum of two numbers.
Let's move on to the next property, which is associated with subtracting a given natural number from a given sum of two natural numbers. Let's look at examples that will help us “see” this property of subtracting a natural number from the sum of two numbers.
Let us have 3 candies in the first pocket, and 5 candies in the second, and let us need to give away 2 candies. We can do it different ways. Let's look at them one by one.
Firstly, we can put all the candies in one pocket, then take out 2 candies from there and give them away. Let us describe these actions mathematically. After we put the candies in one pocket, their number will be determined by the sum 3+5. Now, out of the total number of candies, we will give away 2 candies, while the remaining number of candies will be determined by the following difference (3+5)−2.
Secondly, we can give away 2 candies by taking them out of the first pocket. In this case, the difference 3−2 determines the remaining number of candies in the first pocket, and the total number of candies remaining in our pocket will be determined by the sum (3−2)+5.
Thirdly, we can give away 2 candies from the second pocket. Then the difference 5−2 will correspond to the number of remaining candies in the second pocket, and the total remaining number of candies will be determined by the sum 3+(5−2) .
It is clear that in all cases we will have the same number of candies. Consequently, the equalities (3+5)−2=(3−2)+5=3+(5−2) are valid.
If we had to give away not 2, but 4 candies, then we could do this in two ways. First, give away 4 candies, having previously put them all in one pocket. In this case, the remaining number of candies is determined by an expression of the form (3+5)−4. Secondly, we could give away 4 candies from the second pocket. In this case, the total number of candies gives the following sum 3+(5−4) . It is clear that in both the first and second cases we will have the same number of candies, therefore, the equality (3+5)−4=3+(5−4) is true.
Having analyzed the results obtained from solving the previous examples, we can formulate the property of subtracting a given natural number from a given sum of two numbers. Subtracting a given natural number from a given sum of two numbers is the same as subtracting a given number from one of the terms, and then adding the resulting difference and the other term. It should be noted that the number being subtracted must NOT be greater than the term from which this number is being subtracted.
Let's write down the property of subtracting a natural number from a sum using letters. Let a, b and c be some natural numbers. Then, provided that a is greater than or equal to c, the equality is true (a+b)−c=(a−c)+b, and if the condition is met that b is greater than or equal to c, the equality is true (a+b)−c=a+(b−c). If both a and b are greater than or equal to c, then both of the last equalities are true, and they can be written as follows: (a+b)−c=(a−c)+b= a+(b−c) .
By analogy, we can formulate the property of subtracting a natural number from the sum of three or more numbers. In this case, this natural number can be subtracted from any term (of course, if it is greater than or equal to the number being subtracted), and the remaining terms can be added to the resulting difference.
To visualize the sounded property, you can imagine that we have many pockets and there are candies in them. Suppose we need to give away 1 candy. It is clear that we can give away 1 candy from any pocket. At the same time, it does not matter from which pocket we give it away, since this does not affect the amount of candy that we will have left.
Let's give an example. Let a, b, c and d be some natural numbers. If a>d or a=d, then the difference (a+b+c)−d is equal to the sum (a−d)+b+c. If b>d or b=d, then (a+b+c)−d=a+(b−d)+c. If c>d or c=d, then the equality (a+b+c)−d=a+b+(c−d) is true.
It should be noted that the property of subtracting a natural number from the sum of three or more numbers is not a new property, since it follows from the properties of adding natural numbers and the property of subtracting a number from the sum of two numbers.
Bibliography.
- Mathematics. Any textbooks for 1st, 2nd, 3rd, 4th grades of general education institutions.
- Mathematics. Any textbooks for 5th grade of general education institutions.
Integers
The numbers used for counting are called natural numbers Number zero does not apply to natural numbers.
Single digits numbers: 1,2,3,4,5,6,7,8,9 Double digits: 24.56, etc. Three-digit: 348,569, etc. Multiple-valued: 23,562,456789 etc.
Dividing a number into groups of 3 digits, starting from the right, is called classes: the first three digits are the class of units, the next three digits are the class of thousands, then millions, etc.
By segment call a line drawn from point A to point B. Called AB or BA A B The length of segment AB is called distance between points A and B.
Length units:
1) 10 cm = 1 dm
2) 100 cm = 1 m
3) 1 cm = 10 mm
4) 1 km = 1000 m
Plane is a surface that has no edges, extending limitlessly in all directions. Straight has no beginning or end. Two straight lines having one common point - intersect. Ray– this is a part of a line that has a beginning and no end (OA and OB). The rays into which a point divides a straight line are called additional each other.
Coordinate beam:
0 1 2 3 4 5 6 O E A B X O(0), E(1), A(2), B(3) – coordinates of points. Of two natural numbers, the smaller is the one that is called earlier when counting, and the larger is the one that is called later when counting. One is the smallest natural number. The result of comparing two numbers is written as an inequality: 5< 8, 5670 >368. The number 8 is less than 28 and greater than 5, can be written as a double inequality: 5< 8 < 28
Adding and subtracting natural numbers
Addition
Numbers that add are called addends. The result of addition is called the sum.
Addition properties:
1. Commutative property: The sum of the numbers does not change when the terms are rearranged: a + b = b + a(a and b are any natural numbers and 0) 2. Combination property: To add the sum of two numbers to a number, you can first add the first term, and then add the second term to the resulting sum: a + (b + c) = (a + b) +c = a + b + c(a, b and c are any natural numbers and 0).
3. Addition with zero: Adding zero does not change the number:
a + 0 = 0 + a = a(a is any natural number).
The sum of the lengths of the sides of a polygon is called the perimeter of this polygon.
Subtraction
An action that uses the sum and one of the terms to find another term is called by subtraction.
The number from which it is subtracted is called reducible, the number that is being subtracted is called deductible, the result of the subtraction is called difference. The difference between two numbers shows how much first number more second or how much second number less first.
Subtraction properties:
1. Property of subtracting a sum from a number: In order to subtract a sum from a number, you can first subtract the first term from this number, and then subtract the second term from the resulting difference:
a – (b + c) = (a - b) –With= a – b –With(b + c > a or b + c = a).
2. The property of subtracting a number from a sum: To subtract a number from a sum, you can subtract it from one term and add another term to the resulting difference
(a + b) – c = a + (b - c), if with< b или с = b
(a + b) – c = (a - c) + b, if with< a или с = a.
3. Zero subtraction property: If you subtract zero from a number, it will not change:
a – 0 = a(a – any natural number)
4. The property of subtracting the same number from a number: If you subtract this number from a number, you get zero:
a – a = 0(a is any natural number).
Numeric and alphabetic expressions
Action records are called numeric expressions. The number obtained as a result of performing all these actions is called the value of the expression.
Multiplication and division of natural numbers
Multiplication of natural numbers and its properties
Multiplying the number m by the natural number n means finding the sum of n terms, each of which is equal to m.
The expression m · n and the value of this expression are called the product of the numbers m and n. The numbers m and n are called factors.
Properties of Multiplication:
1. Commutative property of multiplication: The product of two numbers does not change when the factors are rearranged:
a b = b a
2. Combinative property of multiplication: To multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor:
a · (b · c) = (a · b) · c.
3. Property of multiplication by one: The sum of n terms, each of which is equal to 1, is equal to n:
1 n = n
4. Property of multiplication by zero: The sum of n terms, each of which is equal to zero, is equal to zero:
0 n = 0
The multiplication sign can be omitted: 8 x = 8x,
or a b = ab,
or a · (b + c) = a(b + c)
Division
The action by which the product and one of the factors is used to find another factor is called division.
The number being divided is called divisible; the number being divided by is called divider, the result of division is called private.
The quotient shows how many times the dividend is greater than the divisor.
You can't divide by zero!
Division properties:
1. When dividing any number by 1, the same number is obtained:
a: 1 = a.
2. When dividing a number by the same number, the result is one:
a: a = 1.
3. When zero is divided by a number, the result is zero:
0: a = 0.
To find an unknown factor, you need to divide the product by another factor. 5x = 45 x = 45: 5 x = 9
To find the unknown dividend, you need to multiply the quotient by the divisor. x: 15 = 3 x = 3 15 x = 45
To find an unknown divisor, you need to divide the dividend by the quotient. 48: x = 4 x = 48: 4 x = 12
Division with remainder
The remainder is always less than the divisor.
If the remainder is zero, then the dividend is said to be divisible by the divisor without a remainder or, in other words, by an integer. To find the dividend a when dividing with a remainder, you need to multiply the partial quotient c by the divisor b and add the remainder d to the resulting product.
a = c b + d
Simplifying Expressions
Properties of multiplication:
1. Distributive property of multiplication relative to addition: To multiply a sum by a number, you can multiply each term by this number and add the resulting products:
(a + b)c = ac + bc.
2. Distributive property of multiplication relative to subtraction: To multiply the difference by a number, you can multiply the minuend and the subtracted by this number and subtract the second from the first product:
(a - b)c = ac - bc.
3a + 7a = (3 + 7)a = 10a
Procedure
Addition and subtraction of numbers are called operations of the first stage, and multiplication and division of numbers are called actions of the second stage.
Rules for the order of actions:
1. If there are no parentheses in the expression and it contains actions of only one stage, then they are performed in order from left to right.
2. If the expression contains actions of the first and second stages and there are no parentheses in it, then the actions of the second stage are performed first, then the actions of the first stage.
3. If there are parentheses in the expression, then first perform the actions in the parentheses (taking into account rules 1 and 2)
Each expression specifies a program for its calculation. It consists of teams.
Degree of. Square and cube numbers
A product in which all factors are equal to each other is written shorter: a · a · a · a · a · a = a6 Read: a to the sixth power. The number a is called the base of the power, the number 6 is the exponent, and the expression a6 is called the power.
The product of n and n is called the square of n and is denoted by n2 (en squared):
n2 = n n
The product n · n · n is called the cube of the number n and is denoted by n3 (n cubed): n3 = n n n
The first power of a number is equal to the number itself. If a numerical expression includes powers of numbers, then their values are calculated before performing other actions.
Areas and volumes
Writing a rule using letters is called a formula. Path formula:
s = vt, where s is the path, v is the speed, t is the time.
v=s:t
t = s:v
Square. Formula for the area of a rectangle.
To find the area of a rectangle, you need to multiply its length by its width. S = ab, where S is the area, a is the length, b is the width
Two figures are called equal if one of them can be superimposed on the second so that these figures coincide. The areas of equal figures are equal. The perimeters of equal figures are equal.
The area of the entire figure is equal to the sum of the areas of its parts. The area of each triangle is equal to half the area of the entire rectangle
Square is a rectangle with equal sides.
The area of a square is equal to the square of its side:
Area units
Square millimeter – mm2
Square centimeter - cm2
Square decimeter – dm2
Square meter – m2
Square kilometer – km2
Field areas are measured in hectares (ha). A hectare is the area of a square with a side of 100 m.
The area of small plots of land is measured in ares (a).
Ar (one hundred square meters) is the area of a square with a side of 10 m.
1 ha = 10,000 m2
1 dm2 = 100 cm2
1 m2 = 100 dm2 = 10,000 cm2
If the length and width of a rectangle are measured in different units, then they must be expressed in the same units to calculate the area.
Rectangular parallelepiped
The surface of a rectangular parallelepiped consists of 6 rectangles, each of which is called a face.
The opposite faces of a rectangular parallelepiped are equal.
The sides of the faces are called edges of a parallelepiped, and the vertices of the faces are vertices of a parallelepiped.
A rectangular parallelepiped has 12 edges and 8 vertices.
A rectangular parallelepiped has three dimensions: length, width and height
Cube- This cuboid, in which all dimensions are the same. The surface of the cube consists of 6 equal squares.
Volume of a rectangular parallelepiped: To find the volume of a rectangular parallelepiped, you need to multiply its length by its width and by its height.
V=abc, V – volume, a length, b – width, c – height
Cube volume:
Volume units:
Cubic millimeter – mm3
Cubic centimeter - cm3
Cubic decimeter – dm3
Cubic meter – mm3
Cubic kilometer – km3
1 m3 = 1000 dm3 = 1000 l
1 l = 1 dm3 = 1000 cm3
1 cm3 = 1000 mm3 1 km3 = 1,000,000,000 m3
Circle and circle
A closed line located at the same distance from a given point is called a circle.
The part of the plane that lies inside the circle is called a circle.
This point is called the center of both the circle and the circle.
A segment connecting the center of a circle with any point lying on the circle is called radius of the circle.
A segment connecting two points on a circle and passing through its center is called diameter of the circle.
The diameter is equal to two radii.
A number of results inherent in this action can be noted. These results are called properties of addition of natural numbers. In this article we will analyze in detail the properties of adding natural numbers, write them using letters and give explanatory examples.
Page navigation.
Combinative property of addition of natural numbers.
Now let's give an example illustrating the associative property of adding natural numbers.
Let's imagine a situation: 1 apple fell from the first apple tree, and 2 apples and 4 more apples fell from the second apple tree. Now consider this situation: 1 apple and 2 more apples fell from the first apple tree, and 4 apples fell from the second apple tree. It is clear that there will be the same number of apples on the ground in both the first and second cases (which can be verified by recalculation). That is, the result of adding the number 1 with the sum of numbers 2 and 4 is equal to the result of adding the sum of numbers 1 and 2 with the number 4.
The considered example allows us to formulate the combinatory property of adding natural numbers: in order to add a given sum of two numbers to a given number, we can add the first term of the given sum to this number and add the second term of the given sum to the resulting result. This property can be written using letters like this: a+(b+c)=(a+b)+c, where a, b and c are arbitrary natural numbers.
Please note that the equality a+(b+c)=(a+b)+c contains parentheses “(” and “)”. Parentheses are used in expressions to indicate the order in which actions are performed - the actions in parentheses are performed first (more about this is written in the section). In other words, expressions whose values are evaluated first are placed in parentheses.
In conclusion of this paragraph, we note that the combinatory property of addition allows us to uniquely determine the addition of three, four or more natural numbers.
The property of adding zero and a natural number, the property of adding zero and zero.
We know that zero is NOT a natural number. So why did we decide to look at the property of adding zero and a natural number in this article? There are three reasons for this. First: this property is used when adding natural numbers in a column. Second: this property is used when subtracting natural numbers. Third: if we assume that zero means the absence of something, then the meaning of adding zero and a natural number coincides with the meaning of adding two natural numbers.
Let us carry out some reasoning that will help us formulate the property of adding zero and a natural number. Let's imagine that there are no objects in the box (in other words, there are 0 objects in the box), and a objects are placed in it, where a is any natural number. That is, we added 0 and a objects. It is clear that after this action there are a objects in the box. Therefore, the equality 0+a=a is true.
Similarly, if a box contains a items and 0 items are added to it (that is, no items are added), then after this action there will be a items in the box. So a+0=a .
Now we can give the formulation of the property of adding zero and a natural number: the sum of two numbers, one of which is zero, is equal to the second number. Mathematically, this property can be written as the following equality: 0+a=a or a+0=a, where a is an arbitrary natural number.
Separately, let us pay attention to the fact that when adding a natural number and zero, the commutative property of addition remains true, that is, a+0=0+a.
Finally, let us formulate the property of adding zero to zero (it is quite obvious and does not need additional comments): the sum of two numbers, each equal to zero, is equal to zero. That is, 0+0=0 .
Now it's time to figure out how to add natural numbers.
Bibliography.
- Mathematics. Any textbooks for 1st, 2nd, 3rd, 4th grades of general education institutions.
- Mathematics. Any textbooks for 5th grade of general education institutions.