Zero itself is a very interesting number. By itself it means emptiness, lack of meaning, and next to another number it increases its significance 10 times. Any numbers to the zero power always give 1. This sign was used in the Mayan civilization, and it also denoted the concept of “beginning, cause.” Even the calendar began with day zero. This figure is also associated with a strict ban.
Since our elementary school years, we have all clearly learned the rule “you cannot divide by zero.” But if in childhood you take a lot of things on faith and the words of an adult rarely raise doubts, then over time sometimes you still want to understand the reasons, to understand why certain rules were established.
Why can't you divide by zero? I would like to get a clear logical explanation for this question. In the first grade, teachers could not do this, because in mathematics the rules are explained using equations, and at that age we had no idea what it was. And now it’s time to figure it out and get a clear logical explanation of why you can’t divide by zero.
The fact is that in mathematics, only two of the four basic operations (+, -, x, /) with numbers are recognized as independent: multiplication and addition. The remaining operations are considered to be derivatives. Let's look at a simple example.
Tell me, how much do you get if you subtract 18 from 20? Naturally, the answer immediately arises in our head: it will be 2. How did we come to this result? This question will seem strange to some - after all, everything is clear that the result will be 2, someone will explain that he took 18 from 20 kopecks and got two kopecks. Logically, all these answers are not in doubt, but from a mathematical point of view, this problem should be solved differently. Let us recall once again that the main operations in mathematics are multiplication and addition, and therefore in our case the answer lies in solving the following equation: x + 18 = 20. From which it follows that x = 20 - 18, x = 2. It would seem, why describe everything in such detail? After all, everything is so simple. However, without this it is difficult to explain why you cannot divide by zero.
Now let's see what happens if we want to divide 18 by zero. Let's create the equation again: 18: 0 = x. Since the division operation is a derivative of the multiplication procedure, transforming our equation we get x * 0 = 18. This is where the dead end begins. Any number in place of X when multiplied by zero will give 0 and we will not be able to get 18. Now it becomes extremely clear why you cannot divide by zero. Zero itself can be divided by any number, but vice versa - alas, it’s impossible.
What happens if you divide zero by itself? This can be written as follows: 0: 0 = x, or x * 0 = 0. This equation has an infinite number of solutions. Therefore, the end result is infinity. Therefore, the operation in this case also does not make sense.
Division by 0 is at the root of many imaginary mathematical jokes that can be used to puzzle any ignorant person if desired. For example, consider the equation: 4*x - 20 = 7*x - 35. Let's take 4 out of brackets on the left side and 7 on the right. We get: 4*(x - 5) = 7*(x - 5). Now let's multiply the left and right sides of the equation by the fraction 1 / (x - 5). The equation will take the following form: 4*(x - 5)/(x - 5) = 7*(x - 5)/ (x - 5). Let's reduce the fractions by (x - 5) and it turns out that 4 = 7. From this we can conclude that 2*2 = 7! Of course, the catch here is that it is equal to 5 and it was impossible to cancel fractions, since this led to division by zero. Therefore, when reducing fractions, you must always check that a zero does not accidentally end up in the denominator, otherwise the result will be completely unpredictable.
Class: 3
Presentation for the lesson
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Target:
- Introduce special cases of multiplication with 0 and 1.
- Reinforce the meaning of multiplication and the commutative property of multiplication, practice computational skills.
- Develop attention, memory, mental operations, speech, creativity, interest in mathematics.
Equipment: Slide presentation: Appendix 1.
During the classes
1. Organizational moment.
Today is an unusual day for us. Guests are present at the lesson. Make me, your friends, and your guests happy with your successes. Open your notebooks, write down the number, great job. In the margin, note your mood at the beginning of the lesson. Slide 2.
The whole class orally repeats the multiplication table on cards, saying it out loud. (children mark incorrect answers with clapping).
Physical education lesson (“Brain gymnastics”, “Cap for thinking”, breathing).
2. Statement of the educational task.
2.1. Tasks for the development of attention.
On the board and on the table the children have a two-color picture with numbers:
– What is interesting about the written numbers? (Write in different colors; all “red” numbers are even, and “blue” numbers are odd.)
– Which number is the odd one out? (10 is round, and the rest are not; 10 is two-digit, and the rest are single-digit; 5 is repeated twice, and the rest - one at a time.)
– I’ll close the number 10. Is there an extra one among the other numbers? (3 – he doesn’t have a pair until 10, but the rest do.)
– Find the sum of all the “red” numbers and write it in the red square.
(30.)
– Find the sum of all the “blue” numbers and write it in the blue square. (23.)
– How much more is 30 than 23? (On 7.)
– How much is 23 less than 30? (Also at 7.)
– What action did you use to search for? (Subtraction.) Slide 3.
2.2. Tasks for the development of memory and speech. Updating knowledge.
a) – Repeat in order the words that I will name: addend, addend, sum, minuend, subtrahend, difference. (Children try to reproduce the order of words.)
– What components of actions were named? (Addition and subtraction.)
– What action are you still familiar with? (Multiplication, division.)
– Name the components of multiplication. (Multiplier, multiplier, product.)
– What does the first factor mean? (Equal terms in the sum.)
– What does the second factor mean? (The number of such terms.)
Write down the definition of multiplication.
a+ a+… + a= an
b) – Look at the notes. What task will you be doing?
12 + 12 + 12 + 12 + 12
33 + 33 + 33 + 33
a + a + a
(Replace the sum with the product.)
What will happen? (The first expression has 5 terms, each of which is equal to 12, so it is equal to 12 5. Similarly - 33 4, and 3)
c) – Name the inverse operation. (Replace the product with the sum.)
– Replace the product with the sum in the expressions: 99 2. 8 4. b 3.(99 + 99, 8 + 8 + 8 + 8, b + b + b). Slide 4.
d) Equalities are written on the board:
81 + 81 = 81 – 2
21 3 = 21 + 22 + 23
44 + 44 + 44 + 44 = 44 + 4
17 + 17 – 17 + 17 – 17 = 17 5
Pictures are placed next to each equality.
– The animals of the forest school were completing a task. Did they do it correctly?
Children establish that the elephant, tiger, hare and squirrel were mistaken, and explain what their mistakes were. Slide 5.
e) Compare the expressions:
8 5... 5 8
5 6... 3 6
34 9… 31 2
a 3... a 2 + a
(8 5 = 5 8, since the sum does not change from rearranging the terms;
5 6 > 3 6, since there are 6 terms on the left and right, but there are more terms on the left;
34 9 > 31 2. since there are more terms on the left and the terms themselves are larger;
a 3 = a 2 + a, since on the left and right there are 3 terms equal to a.)
– What property of multiplication was used in the first example? (Commutative.) Slide 6.
2.3. Formulation of the problem. Goal setting.
Are the equalities true? Why? (Correct, since the sum is 5 + 5 + 5 = 15. Then the sum becomes one more term 5, and the sum increases by 5.)
5 3 = 15
5 4 = 20
5 5 = 25
5 6 = 30
– Continue this pattern to the right. (5 7 = 35; 5 8 = 40...)
– Continue it now to the left. (5 2 = 10; 5 1=5; 5 0 = 0.)
– What does the expression 5 1 mean? 50? (? Problem!)
Summary of the discussion:
However, the expressions 5 1 and 5 0 do not make sense. We can agree to consider these equalities true. But to do this, we need to check whether we will violate the commutative property of multiplication.
So, the goal of our lesson is determine whether we can count equalities 5 1 = 5 and 5 0 = 0 true?
- Lesson problem! Slide 7.
3. “Discovery” of new knowledge by children.
a) – Follow steps: 1 7, 1 4, 1 5.
Children solve examples with comments in their notebooks and on the board:
1 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7
1 4 = 1 + 1 + 1 + 1 = 4
1 5 = 1 + 1 + 1 + 1 +1 = 5
– Draw a conclusion: 1 a – ? (1 a = a.) The card is displayed: 1 a = a
b) – Do the expressions 7 1, 4 1, 5 1 make sense? Why? (No, because the sum cannot have one term.)
– What should they be equal to so that the commutative property of multiplication is not violated? (7 1 must also equal 7, so 7 1 = 7.)
4 1 = 4 are considered similarly. 5 1 = 5.
– Conclude: a 1 = ? (a 1 = a.)
The card is displayed: a 1 = a. The first card is superimposed on the second: a 1 = 1 a = a.
– Does our conclusion coincide with what we got on the number line? (Yes.)
– Translate this equality into Russian. (When you multiply a number by 1 or 1 by a number, you get the same number.)
- Well done! So, we will assume: a 1 = 1 a = a. Slide 8.
2) The case of multiplication with 0 is studied in a similar way. Conclusion:
– when multiplying a number by 0 or 0 by a number, zero is obtained: a 0 = 0 a = 0. Slide 9.
– Compare both equalities: what do 0 and 1 remind you of?
Children express their versions. You can draw their attention to the images:
1 – “mirror”, 0 – “terrible beast” or “invisible hat”.
Well done! So, multiplying by 1 gives the same number (1 – “mirror”), and when multiplied by 0 it turns out 0 ( 0 – “invisibility cap”).
4. Physical education (for the eyes – “circle”, “up and down”, for the hands – “lock”, “fists”).
5. Primary consolidation.
Examples written on the board:
23 1 =
1 89 =
0 925 =
364 1 =
156 0 =
0 1 =
Children solve them in a notebook and on the board, pronouncing the resulting rules out loud, for example:
3 1 = 3, since when a number is multiplied by 1, the same number is obtained (1 is a “mirror”), etc.
a) 145 x = 145; b) x 437 = 437.
– When multiplying 145 by an unknown number, it turned out to be 145. So, they multiplied by 1 x = 1. Etc.
a) 8 x = 0; b) x 1= 0.
– When multiplying 8 by an unknown number, the result was 0. So, multiplied by 0 x = 0. Etc.
6. Independent work with testing in class. Slide 10.
Children independently solve written examples. Then according to the finished
Following the example, they check their answers by pronouncing them out loud, mark correctly solved examples with a plus, and correct any mistakes made. Those who made mistakes receive a similar task on a card and work on it individually while the class solves repetition problems.
7. Repetition tasks. (Work in pairs). Slide 11.
a) – Do you want to know what awaits you in the future? You will find out by deciphering the recording:
G – 49:7 O – 9 8 n – 9 9 V – 45:5 th – 6 6 d – 7 8 s – 24:3
| 81 | 72 | 5 | 8 | 36 | 7 | 72 | 56 |
-So what awaits us? (New Year.)
b) - “I thought of a number, subtracted 7 from it, added 15, then added 4 and got 45. What number did I think of?”
Reverse operations must be done in reverse order: 45 – 4 – 15 + 7 = 31.
8. Lesson summary.Slide 12.
What new rules have you met?
What did you like? What was difficult?
Can this knowledge be applied in life?
In the margins you can express your mood at the end of the lesson.
Fill out the self-assessment table:
I want to know more
Okay, but I can do better
I'm still experiencing difficulties
Thanks for your work, you did a great job!
9. Homework
pp. 72–73 Rule, No. 6.
Which of these sums do you think can be replaced by a product?
Let's think like this. In the first sum, the terms are the same, the number five is repeated four times. This means we can replace addition with multiplication. The first factor shows which term is repeated, the second factor shows how many times this term is repeated. We replace the sum with the product.
Let's write down the solution.
In the second sum, the terms are different, so it cannot be replaced by a product. We add the terms and get the answer 17.
Let's write down the solution.
Can a product be replaced by a sum of identical terms?
Let's look at the works.
Let's carry out the actions and draw a conclusion.
1*2=1+1=2
1*4=1+1+1+1=4
1*5=1+1+1+1+1=5
We can conclude: The number of unit terms is always equal to the number by which the unit is multiplied.
Means, When you multiply the number one by any number, you get the same number.
1 * a = a
Let's look at the works.
These products cannot be replaced by a sum, since a sum cannot have one term.
The products in the second column differ from the products in the first column only in the order of the factors.
This means that in order not to violate the commutative property of multiplication, their values must also be equal to the first factor, respectively.
Let's conclude: When you multiply any number by the number one, you get the number that was multiplied.
Let's write this conclusion as an equality.
a * 1= a
Solve examples.
Hint: Don't forget the conclusions we made in the lesson.
Test yourself.
Now let's observe products where one of the factors is zero.
Let's consider products where the first factor is zero.
Let us replace the products with the sum of identical terms. Let's carry out the actions and draw a conclusion.
0*3=0+0+0=0
0*6=0+0+0+0+0+0=0
0*4=0+0+0+0=0
The number of zero terms is always equal to the number by which zero is multiplied.
Means, When you multiply zero by a number, you get zero.
Let's write this conclusion as an equality.
0 * a = 0
Let's consider products where the second factor is zero.
These products cannot be replaced by a sum, since a sum cannot have zero terms.
Let's compare the works and their meanings.
0*4=0
The products of the second column differ from the products of the first column only in the order of the factors.
This means that in order not to violate the commutative property of multiplication, their values must also be equal to zero.
Let's conclude: When any number is multiplied by zero, the result is zero.
Let's write this conclusion as an equality.
a * 0 = 0
But you can't divide by zero.
Solve examples.
Hint: Don't forget the conclusions you made in the lesson. When calculating the values of the second column, be careful when determining the order of actions.
Test yourself.
Today in the lesson we learned about special cases of multiplication by 0 and 1, and practiced multiplying by 0 and 1.
Bibliography
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 1. - M.: “Enlightenment”, 2012.
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. - M.: “Enlightenment”, 2012.
- M.I. Moro. Mathematics lessons: Methodological recommendations for teachers. 3rd grade. - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: “Enlightenment”, 2011.
- “School of Russia”: Programs for primary school. - M.: “Enlightenment”, 2011.
- S.I. Volkova. Mathematics: Test papers. 3rd grade. - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: “Exam”, 2012.
- Nsportal.ru ().
- Prosv.ru ().
- Do.gendocs.ru ().
Homework
1. Find the meanings of the expressions.
2. Find the meanings of the expressions.
3. Compare the meanings of the expressions.
(56-54)*1 … (78-70)*1
4. Create an assignment on the topic of the lesson for your friends.
Even at school, teachers tried to hammer into our heads the simplest rule: “Any number multiplied by zero equals zero!”, - but still a lot of controversy constantly arises around him. Some people just remember the rule and don’t bother themselves with the question “why?” “You can’t and that’s it, because they said so at school, the rule is the rule!” Someone can fill half a notebook with formulas, proving this rule or, conversely, its illogicality.
In contact with
Who's right in the end?
During these disputes, both people with opposing points of view look at each other like a ram and prove with all their might that they are right. Although, if you look at them from the side, you can see not one, but two rams, resting their horns on each other. The only difference between them is that one is slightly less educated than the other.
Most often, those who consider this rule to be incorrect try to appeal to logic in this way:
I have two apples on my table, if I put zero apples on them, that is, I don’t put a single one, then my two apples will not disappear! The rule is illogical!
Indeed, apples will not disappear anywhere, but not because the rule is illogical, but because a slightly different equation is used here: 2 + 0 = 2. So let’s discard this conclusion right away - it is illogical, although it has the opposite goal - to call to logic.
What is multiplication
Originally the multiplication rule was defined only for natural numbers: multiplication is a number added to itself a certain number of times, which implies that the number is natural. Thus, any number with multiplication can be reduced to this equation:
- 25×3 = 75
- 25 + 25 + 25 = 75
- 25×3 = 25 + 25 + 25
From this equation it follows that that multiplication is a simplified addition.
What is zero
Any person knows from childhood: zero is emptiness. Despite the fact that this emptiness has a designation, it does not carry anything at all. Ancient Eastern scientists thought differently - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw a deep meaning in this number. After all, zero, which has the meaning of emptiness, standing next to any natural number, multiplies it ten times. Hence all the controversy about multiplication - this number carries so much inconsistency that it becomes difficult not to get confused. In addition, zero is constantly used to define empty digits in decimal fractions, this is done both before and after the decimal point.
Is it possible to multiply by emptiness?
You can multiply by zero, but it is useless, because, whatever one may say, even when multiplying negative numbers, you will still get zero. It’s enough just to remember this simple rule and never ask this question again. In fact, everything is simpler than it seems at first glance. There are no hidden meanings and secrets, as ancient scientists believed. Below we will give the most logical explanation that this multiplication is useless, because when you multiply a number by it, you will still get the same thing - zero.
Returning to the very beginning, to the argument about two apples, 2 times 0 looks like this:
- If you eat two apples five times, then you eat 2×5 = 2+2+2+2+2 = 10 apples
- If you eat two of them three times, then you eat 2×3 = 2+2+2 = 6 apples
- If you eat two apples zero times, then nothing will be eaten - 2×0 = 0×2 = 0+0 = 0
After all, eating an apple 0 times means not eating a single one. This will be clear to even the smallest child. Whatever one may say, the result will be 0, two or three can be replaced with absolutely any number and the result will be absolutely the same. And to put it simply, then zero is nothing, and when do you have there is nothing, then no matter how much you multiply, it’s still the same will be zero. There is no such thing as magic, and nothing will make an apple, even if you multiply 0 by a million. This is the simplest, most understandable and logical explanation of the rule of multiplication by zero. For a person who is far from all formulas and mathematics, such an explanation will be enough for the dissonance in the head to resolve and everything to fall into place.
Division
From all of the above, another important rule follows:
You can't divide by zero!
This rule has also been persistently hammered into our heads since childhood. We just know that it’s impossible to do everything without filling our heads with unnecessary information. If you are unexpectedly asked the question why it is forbidden to divide by zero, then most will be confused and will not be able to clearly answer the simplest question from the school curriculum, because there are not so many disputes and contradictions surrounding this rule.
Everyone simply memorized the rule and did not divide by zero, not suspecting that the answer was hidden on the surface. Addition, multiplication, division and subtraction are unequal; of the above, only multiplication and addition are valid, and all other manipulations with numbers are built from them. That is, the notation 10: 2 is an abbreviation of the equation 2 * x = 10. This means that the notation 10: 0 is the same abbreviation for 0 * x = 10. It turns out that division by zero is a task to find a number, multiplying by 0, you get 10 And we have already figured out that such a number does not exist, which means that this equation has no solution, and it will be a priori incorrect.
Let me tell you,
So as not to divide by 0!
Cut 1 as you want, lengthwise,
Just don't divide by 0!
Let's look at an example of multiplying an integer by zero. How much will it be if 2 (two) is multiplied by 0 (zero)? Any number multiplied by zero equals zero. And it doesn’t matter whether we know this number or not.
According to the generally accepted definition, zero is the number that separates positive numbers from negative numbers on the number line. Zero is the most problematic place in mathematics, which does not obey logic, and all mathematical operations with zero are based not on logic, but on generally accepted definitions.
Zero is the first digit in all standard number systems. Each month began with day zero in the Mayan calendar. It is interesting that the Mayan mathematicians used the same sign for zero to denote infinity, the second problem of modern mathematics. Zero without a stick. Absolute zero. Zero point five. Five multiplied by zero equals zero 5 x 0 = 0 See the rule for multiplying by zero above in the text. Chatyri multiply by zero for free - I answer for free that it will be zero. Free help included - the word “four” is spelled a little differently than what you write in your search query.
https://youtu.be/EGpr23Tc8iY
Where there is a zero in mathematics, logic is powerless
If you liked the post and want to know more, please help me work on other materials. It appeared in the comments and somehow caught my attention. Student Question: And now, dear author, please multiply zero by zero and tell me how much is the result?
In my article “What is zero” I have already explained where it can be used. You just need to take the answers that are written in textbooks: zero multiplied by zero equals zero; Dividing by zero is prohibited. Of all the foreseeable options for multiplying and dividing by zero, ignorant scientists chose the most acceptable and digestible option.
I personally have no problems with division by zero. This is the first time I’ve heard about the connection between Heron’s formula and 0/0=1. However, there is something unclean about mathematics. Problems with raising zero to zero and negative powers. But we can just as well say that 0^2 also makes no sense, because 0^2=0^5/0^3=0/0, and you can’t divide by zero.
Zero to the zero power is an expression that has no meaning. Zero to the zeroth power equals one - this is what the formulas show. This amount of anything, some real, material things, can be multiplied by a number. In this case, the quantity of something is expressed only by zero or a positive number.
Everything about units and math is fine at this level. This is a convention; degrees cannot be expressed in quantity, so you cannot multiply them by a number. Somewhere on this site there is Durnev with his questions about the school curriculum, including mathematics. Maybe it was invented in the same way as zero? To impose certain rules and subject all other people to them. What a person will not do for himself, his beloved.
It is enough that in textbooks they often write “belongs to the set of natural numbers” even when this is true for all numbers except complex ones. The infinite number of zeros in zero is an invention of shamans for cavemen :) If you close your eyes, then everything we look at will look equally black. Multiplication by zero must be considered from a completely different end. What is multiplication?
It is enough to understand what multiplication is, then the issue with the result of multiplication by zero will be resolved by itself. 2 apples, and trying to multiply them by 0 apples, as a result we lose our 2 apples. Apparently, those asking this have lost at least one digit at the beginning of each number. 10 and 11 - it is appropriate to talk about percentages here.
And it’s interesting how when dividing 0 by any number, you can subtract this number at all (even if it’s zero times)..
It can’t just become zero from multiplication! So mathematics is not an exact science? Someone once came up with this “rule”, it is not known why. Your math is wrong. In practice, this whole mathematical topic with multiplication by 0 cannot happen!!! How does 10 want to multiply something, even by 0, but it turns out to be 0?? Unless, of course, 0 is a black hole, or 0 is like losing, to nowhere, zero is like emptiness, nothing, but this cannot be….
If you can’t divide something (the same 5 apples into 0 imaginary baskets), then write down the result of the integer and the remainder of this division... 0 can be multiplied many times (like I went to the forest 15 times and didn’t find any mushrooms...
For example, we divide 5 apples by zero people; We calculate how many times 5 degrees Celsius is greater than zero degrees Celsius. From this, most likely you cannot multiply by 0 (since by the definition of multiplication this CANNOT be written using the addition operation) and divide 0 itself by something... since the answer cannot be determined...
Substitution of concepts occurs during multiplication by zero... Remember, any number or operation with numbers multiplied by zero is ANNIHILATED... In other words, the operation itself does not occur when multiplied by zero and it can simply be “ignored”... So, you stole my idea!))) For the first time I come across a more or less clear understanding of multiplication and division by zero. Whether we consider this as mathematical operations or not, mathematics doesn’t care at all.
The first example of why zero is problematic is the natural numbers. In Russian schools, zero is not a natural number; in other schools, zero is a natural number. For those who are interested in the question of the origin of zero, I suggest reading the article “The History of Zero” by J. J. O’Connor and E. F. Robertson, translated by I. Yu. Osmolovsky.
At what values of X is the following equation true: zero multiplied by X equals zero? - this equality is true for any values of x. They say that this equality has an infinite number of solutions. The math was a little easier. In the most natural way, my natural illiteracy is superimposed on trivial typos when typing.
I am an opponent of those sermons that mathematicians read to us and to which we all))) refer. This equation was a completely different story. Can this happen or can it not? After thinking a little, I “conducted a thought experiment”))) and imagined this situation. Somewhere in the drafts there are all the calculations on this matter. You are disingenuous. What is not accepted in wide circles is not necessarily untrue.
What is the correct spelling: zero or zero? The words zero and zero have the same meaning, but differ in usage. Who said zero is a number? Mathematicians? 0 + 5/0... zero and five (zeros) in the remainder... and then everything comes together and everyone is happy... Yes, in fact, there are not many difficulties. The problem is how to perceive Zero (as a number or as something empty) and what is meant by multiplication...