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 perform the actions in the parentheses first (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 is a rectangular parallelepiped with all dimensions 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.
We have defined addition, multiplication, subtraction and division of integers. These actions (operations) have a number of characteristic results, which are called properties. In this article we will look at the basic properties of adding and multiplying integers, from which all other properties of these actions follow, as well as the properties of subtracting and dividing integers.
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The addition of integers has several other very important properties.
One of them is related to the existence of zero. This property of addition of integers states that adding zero to any integer does not change that number. Let's write this property of addition using letters: a+0=a and 0+a=a (this equality is true due to the commutative property of addition), a is any integer. You may hear that the integer zero is called the neutral element in addition. Let's give a couple of examples. The sum of the integer −78 and zero is −78; If you add the positive integer 999 to zero, the result is 999.
Now we will give a formulation of another property of addition of integers, which is associated with the existence of an opposite number for any integer. The sum of any integer with its opposite number is zero. Let's give the literal form of writing this property: a+(−a)=0, where a and −a are opposite integers. For example, the sum 901+(−901) is zero; similarly, the sum of opposite integers −97 and 97 is zero.
Basic properties of multiplying integers
Multiplication of integers has all the properties of multiplication of natural numbers. Let us list the main of these properties.
Just as zero is a neutral integer with respect to addition, one is a neutral integer with respect to integer multiplication. That is, multiplying any integer by one does not change the number being multiplied. So 1·a=a, where a is any integer. The last equality can be rewritten as a·1=a, this allows us to make the commutative property of multiplication. Let's give two examples. The product of the integer 556 by 1 is 556; the product of one and the negative integer −78 is equal to −78.
The next property of multiplying integers is related to multiplication by zero. The result of multiplying any integer a by zero is zero, that is, a·0=0 . The equality 0·a=0 is also true due to the commutative property of multiplying integers. In the special case when a=0, the product of zero and zero is equal to zero.
For the multiplication of integers, the inverse property to the previous one is also true. It claims that the product of two integers is equal to zero if at least one of the factors is equal to zero. In literal form, this property can be written as follows: a·b=0, if either a=0, or b=0, or both a and b are equal to zero at the same time.
Distributive property of multiplication of integers relative to addition
Joint addition and multiplication of integers allows us to consider the distributive property of multiplication relative to addition, which connects the two indicated actions. Using addition and multiplication together opens up additional possibilities that we would miss if we considered addition separately from multiplication.
So, the distributive property of multiplication relative to addition states that the product of an integer a and the sum of two integers a and b is equal to the sum of the products a b and a c, that is, a·(b+c)=a·b+a·c. The same property can be written in another form: (a+b)c=ac+bc .
The distributive property of multiplying integers relative to addition, together with the combinatory property of addition, allows us to determine the multiplication of an integer by the sum of three or more integers, and then the multiplication of the sum of integers by the sum.
Also note that all other properties of addition and multiplication of integers can be obtained from the properties we have indicated, that is, they are consequences of the properties indicated above.
Properties of subtracting integers
From the resulting equality, as well as from the properties of addition and multiplication of integers, the following properties of subtraction of integers follow (a, b and c are arbitrary integers):
- Subtraction of integers in general does NOT have the commutative property: a−b≠b−a.
- The difference of equal integers is zero: a−a=0.
- The property of subtracting the sum of two integers from a given integer: a−(b+c)=(a−b)−c .
- The property of subtracting an integer from the sum of two integers: (a+b)−c=(a−c)+b=a+(b−c) .
- Distributive property of multiplication relative to subtraction: a·(b−c)=a·b−a·c and (a−b)·c=a·c−b·c.
- And all other properties of subtracting integers.
Properties of division of integers
While discussing the meaning of dividing integers, we found out that dividing integers is the inverse action of multiplication. We gave the following definition: dividing integers is finding an unknown factor from a known product and a known factor. That is, we call the integer c the quotient of the division of the integer a by the integer b, when the product c·b is equal to a.
This definition, as well as all the properties of operations on integers discussed above, make it possible to establish the validity of the following properties of dividing integers:
- No integer can be divided by zero.
- The property of dividing zero by an arbitrary integer a other than zero: 0:a=0.
- Property of dividing equal integers: a:a=1, where a is any integer other than zero.
- The property of dividing an arbitrary integer a by one: a:1=a.
- In general, division of integers does NOT have the commutative property: a:b≠b:a .
- Properties of dividing the sum and difference of two integers by an integer: (a+b):c=a:c+b:c and (a−b):c=a:c−b:c, where a, b, and c are integers such that both a and b are divisible by c and c is nonzero.
- The property of dividing the product of two integers a and b by an integer c other than zero: (a·b):c=(a:c)·b, if a is divisible by c; (a·b):c=a·(b:c) , if b is divisible by c ; (a·b):c=(a:c)·b=a·(b:c) if both a and b are divisible by c .
- The property of dividing an integer a by the product of two integers b and c (the numbers a , b and c are such that dividing a by b c is possible): a:(b c)=(a:b)c=(a :c)·b .
- Any other properties of dividing integers.
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 this in 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.
The topic that this lesson is devoted to is “Properties of Addition.” In it, you will become familiar with the commutative and associative properties of addition, examining them with specific examples. Find out in what cases you can use them to make the calculation process easier. Test examples will help determine how well you have mastered the material studied.
Lesson: Properties of Addition
Look carefully at the expression:
9 + 6 + 8 + 7 + 2 + 4 + 1 + 3
We need to find its value. Let's do it.
9 + 6 = 15
15 + 8 = 23
23 + 7 = 30
30 + 2 = 32
32 + 4 = 36
36 + 1 = 37
37 + 3 = 40
The result of the expression is 9 + 6 + 8 + 7 + 2 + 4 + 1 + 3 = 40.
Tell me, was it convenient to calculate? It was not very convenient to calculate. Look again at the numbers in this expression. Is it possible to swap them so that the calculations are more convenient?
If we rearrange the numbers differently:
9 + 1 + 8 + 2 + 7 + 3 + 6 + 4 = …
9 + 1 = 10
10 + 8 = 18
18 + 2 = 20
20 + 7 = 27
27 + 3 = 30
30 + 6 = 36
36 + 4 = 40
The final result of the expression is 9 + 1 + 8 + 2 + 7 + 3 + 6 + 4 = 40.
We see that the results of the expressions are the same.
The terms can be swapped if it is convenient for calculations, and the value of the sum will not change.
There is a law in mathematics: Commutative law of addition. It states that rearranging the terms does not change the sum.
Uncle Fyodor and Sharik argued. Sharik found the meaning of the expression as it was written, and Uncle Fyodor said that he knew another, more convenient way of calculation. Do you see a better way to calculate?
Sharik solved the expression as it was written. And Uncle Fyodor said that he knew the law that allows terms to be swapped, and swapped the numbers 25 and 3.
37 + 25 + 3 = 65 37 + 25 = 62
37 + 3 + 25 = 65 37 + 3 = 40
We see that the result remains the same, but the calculation has become much easier.
Look at the following expressions and read them.
6 + (24 + 51) = 81 (to 6 add the sum of 24 and 51)
Is there a convenient way to calculate?
We see that if we add 6 and 24, we get a round number. It is always easier to add something to a round number. Let's put the sum of the numbers 6 and 24 in brackets.
(6 + 24) + 51 = …
(add 51 to the sum of numbers 6 and 24)
Let's calculate the value of the expression and see if the value of the expression has changed?
6 + 24 = 30
30 + 51 = 81
We see that the meaning of the expression remains the same.
Let's practice with one more example.
(27 + 19) + 1 = 47 (add 1 to the sum of numbers 27 and 19)
What numbers are convenient to group to form a convenient method?
You guessed that these are the numbers 19 and 1. Let's put the sum of the numbers 19 and 1 in brackets.
27 + (19 + 1) = …
(to 27 add the sum of numbers 19 and 1)
Let's find the meaning of this expression. We remember that the action in parentheses is performed first.
19 + 1 = 20
27 + 20 = 47
The meaning of our expression remains the same.
Combination law of addition: two adjacent terms can be replaced by their sum.
Now let's practice using both laws. We need to calculate the value of the expression:
38 + 14 + 2 + 6 = …
First, let's use the commutative property of addition, which allows us to swap addends. Let's swap terms 14 and 2.
38 + 14 + 2 + 6 = 38 + 2 + 14 + 6 = …
Now let's use the combination property, which allows us to replace two adjacent terms with their sum.
38 + 14 + 2 + 6 = 38 + 2 + 14 + 6 = (38 + 2) + (14 + 6) =…
First we find out the value of the sum of 38 and 2.
Now the sum is 14 and 6.
3. Festival of pedagogical ideas “Open Lesson” ().
Make it at home
1. Calculate the sum of the terms in different ways:
a) 5 + 3 + 5 b) 7 + 8 + 13 c) 24 + 9 + 16
2. Evaluate the results of the expressions:
a) 19 + 4 + 16 + 1 b) 8 + 15 + 12 + 5 c) 20 + 9 + 30 + 1
3. Calculate the amount in a convenient way:
a) 10 + 12 + 8 + 20 b) 17 + 4 + 3 + 16 c) 9 + 7 + 21 + 13
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 the 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 for 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