Lesson objectives:
Educational: form the concepts of negative and positive numbers
Developmental: develop memory, speech, observation, notice patterns, generalize, make judgments by analogy, ability to work with a textbook development logical thinking.
Educational: instilling discipline, accuracy, perseverance, and a responsible attitude towards learning.
Plato
Progress of the lesson.
1. Org. moment.
2. Lesson motivation.
One, two, three, four, five,
Six, seven, eight, nine, ten.
Having arisen in ancient times from the practical needs of counting and simple measurements, mathematics developed in connection with the complication of economic activity and social relations, monetary calculations, problems of measuring distances, time, areas and the requirements that other sciences placed on it.
Today we will introduce you to new numbers.
What numbers are you familiar with? Give examples.
Solve No. 954.
However the world around us so complex and varied. Natural and fractional numbers are sometimes not enough to measure some quantities and describe many events.
Numbers -5, +3.
Can you name these numbers?
In what cases do we often use them? (when talking about the weather).
Guys, what time of year is it now? How is the weather different in summer and winter? How did you know it was cold outside? Using what device? Let's look at a thermometer. What is shown on the thermometer? How are the numbers arranged?
Solve No. 838 orally.
Working with the textbook.
Positive and negative numbers are used not only in mathematics, but also in geography. By the twentieth century, almost the entire Earth had been explored. Where did scientists and travelers transfer their research? (bottom of the world's oceans)
What did scientists discover? What is the bottom topography like? Are the reliefs of the Earth's surface and the bottom of the World Ocean similar?
If you need to measure the height of a mountain or the depth of the ocean, from what point should you start? (based on ocean water level)
If you imagine this as a vertical scale, then the zero point is the ocean water level.
In what direction will mountain heights be measured?
What numbers? (positive)
What is the largest positive quantity on Earth you know? (peak of Chomolungma +8848 m)
In what direction will the ocean depths be measured?
What numbers? (negative)
- What is the largest negative quantity you know? (Mariana Trench -11034 m)
Solve No. 835.
Historical information
Negative numbers appeared much later than natural numbers and ordinary fractions. The first information about negative numbers was found among Chinese mathematicians in the 2nd century. BC Positive numbers were then interpreted as property, and negative numbers as debt, shortage.
But neither the Egyptians, nor the Babylonians, nor the ancient Greeks knew negative numbers.
Only in the 7th century did Indian mathematicians begin to widely use negative numbers, but treated them with some distrust.
In Europe, negative numbers began to be used from the 12th-13th centuries, but before the 10th century. as in ancient times, they were understood as debts; most scientists considered them “false” in contrast to positive numbers - “true”.
The recognition of negative numbers was facilitated by the work of the French mathematician, physicist and philosopher René Descartes (1596-1650). He proposed a geometric interpretation of positive and negative numbers - he introduced the coordinate line (1637).
Negative numbers received final and general recognition as truly existing only in the first half of the 18th century. At the same time, the modern notation for negative numbers was established.
Solve No. 833, 834, 836, 839.
Positive and negative numbers and history.
Familiar phrases from history:
“Pythagoras lived in the 6th century BC”;
“Rus was under the yoke of the Mongol-Tatars during the XIII-XV centuries AD”;
“The Olympics took place in Moscow in 1980”;
These dates are marked on the timeline:
|
Answer the questions:
b) How many years did Archimedes live?
b) What number can be used to replace the year? "Christmas Day"?
At what age did the emperor die? |
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Time line
In ancient times different countries believed differently. For example, in Ancient Egypt, every time a new king began to rule, the counting of years began to be ruled by a new king, the counting of years began anew, the Romans considered the first year to be the year the foundation of their city was founded. Such an account of the past years was inconvenient for determining important historical events. There was a need in all countries to start keeping track of time from this event. At this time, the Christian religion, faith in Jesus Christ, spread to many countries. One of the believers suggested counting the years from the birth of Jesus. The time calculated from the Nativity of Christ began to be called our era. Our era continues for two thousand years. Time calculated before the Nativity of Christ - BC.
6. Independent work.
solve No. 841.
7. Lesson summary. D/z.
learn item 28, solve No. 837, 840, 843.
repeat p.281, solve No. 847.
Complete your statements with the following sentence:
I learned in class today………
learned......
Show your ingenuity:
Count, draw, draw!
Well done to all of you! You are all daredevils!
And may your beloved always last for years
There will be math for you!
Topic: Coordinate line.
Lesson objectives:
educational - teach how to build a coordinate line and find negative and positive coordinates on it,
developmental - develop logic of thinking, attention;
educational – to cultivate tolerance and interest in the subject.
1. Org. moment.
2. Lesson motivation.
“You have to have fun learning... To digest knowledge, you have to absorb it with appetite.” (Anatole France). What do these words mean? Let's follow the writer's advice: we will be active, attentive in class, we will absorb knowledge with great desire, because we will soon need it.
3. Updating of basic knowledge.
What numbers are called positive? negative?
Which number is neither positive nor negative?
The greatest ancient Greek mathematician and physicist came up with a way to describe huge numbers. The largest number he could name was so large that to record it digitally would require a tape two thousand times longer than the distance from the Earth to the Sun.
But they had not yet been able to write down such huge numbers. This became possible only after Indian mathematicians in the 6th century. The number 0 was invented and began to denote the absence of units in the decimal places of a number.
Who are we talking about?
1. 8 * 1.2 = 9.6 A
2. 7.2: 2.4 = 3 R
3. 720:18 = 40 X
4. 3*1.6 = 4.8 I
5. 5/8: 1/2=1 M
6. 1:4 = 0.25 E
7. 900:15 = 60 D
So, the name of the scientist is Archimedes.
Writing positive and negative numbers from dictation:
The lowest place on the land surface is the Dead Sea coast 402 m.
The highest active volcano is Kilimanjaro 5895 m.
The oldest and deepest lake is Baikal 1620 m.
The lowest elevation in Russia is the Caspian Lowland 28 m.
They answer questions frontally.
What geometric model is shown in the figure?
Name the components.
What are the numbers corresponding to these points called?
Name the coordinates of the indicated points.
What coordinate will point A have if it is moved:
3 unit segments to the right?
4 unit segments to the left
Let's continue the ray to the left and get a coordinate line.
“Somewhere there is a country called Mathematics. Numbers, signs, expressions live in this country. In the city "+" live - positive numbers, and in the city "-"
live – negative numbers. This state is ruled by King Zero I. One day the straight lady crawled to them and said: “I dream of looking at your beautiful state from above. Help me get up, I can’t do it myself, I’m afraid I’ll break.”
The numbers did not refuse to help. Positive numbers rose and raised the straight line on the right, and negative numbers stood up and raised the straight line on the left. Everything would be fine, but the straight line almost broke, one number was missing. They called the numbers to the king's aid... Zero came to the rescue: he stood between the positive and negative numbers in the middle and said:
I'm on a scale - a number-border,
Where I stand is where the headquarters is.
I allow numbers to be accommodated,
On the selected line:
Oh, the direction and scale.
The numbers were placed as expected from zero, and began to show the position on the straight line (the coordinate of the point), and the straight line chose the direction and scale. But as soon as the straight line rose, out of admiration for the view from above of the beautiful state, it could not resist and fell, crushing the numbers, which were never able to get out and remained to serve the Straight Line forever.
Zero began to be called the origin of reference and was given the title of point “O”, and the Straight Line itself was given the title of coordinate line. To this day she lives in the country of mathematics, but sometimes she visits other countries: history, geography, etc.”
The famous work of the French mathematician, physicist and philosopher René Descartes, “Geometry,” published in 1637, describes the geometric interpretation of positive and negative numbers: “Positive numbers are represented on the number axis by points lying to the right of the beginning 0, negative numbers to the left.”
Solve No. 848, 850, 852.
5. Physical education minute
Game: The teacher calls the numbers, the students must react correctly. If named:
positive number – the student is sitting;
negative number– the student stands up;
positive fraction - the student must stand up and clap his hands;
negative fraction - the student must sit down and clap his hands.
Solve No. 854, 856.
7. Independent work(tasks on cards)
1option
1. Draw a coordinate line, taking five cells of a notebook as a unit segment. Mark on this straight line points A (2), B (-3), C (-1), D (1.2), E (-2/5), F (-2.6), M (-1¼) .
2. Write down the coordinates of points A, M, K and P shown in the figure:
3. Draw a horizontal line and mark point A on it. To the right of point A at a distance of 3 cm, mark point B. Mark point O - the origin if A (- 6), and B (- 3).
2option
1. Draw a coordinate line, taking the length of four notebook cells as a unit segment. Mark on this straight line points A (3), B (-2), C (2.5), D (1.5), E (-2.75), F (-3 2/5), M (- ¼).
2. Write down the coordinates of points M, N, K and D shown in the figure:
3. Draw a horizontal line and mark points C and D on it so that D is to the right of point C and CD = 5 cm. Mark point O - the origin if C (-2) and D (3).
8. Summing up the lesson
Where do you encounter negative and positive numbers?
1. Income – expense
2. Advance - debt
3. Win - lose.
4. Change in air temperature.
5. Changes in water levels in rivers.
6. Calculation in history lessons.
7. Height above sea level - depth of depressions in geography lessons.
The position of a point on the earth's surface located above the water level in the ocean (this level is designated by the number 0) is designated by a positive number, and below the ocean level by a negative number. Any concept discussed at the end of the lesson can be explained in a similar way.
Where else in life do we encounter a coordinate line (scale)? (thermometers, “time line”)
Learn item 29, solve No. 851, 853, 855.
Topic: Coordinate line. Rational numbers.
Lesson objectives:
Educational: repeat and consolidate all the knowledge acquired while studying the paragraph “Positive and negative numbers”. Bring into the system skills and abilities, in particular, the ability to work with a coordinate line,
Educational: to cultivate in students observation skills, the ability to find and correct their mistakes, and respect for classmates.
Developmental: promote the development of logical thinking and correct mathematical speech.
1. Org. moment.
2. Lesson motivation.
3. Updating of basic knowledge.
Questions:
1. What numbers are called positive? negative?
2. Which number is neither positive nor negative?
3. What is a coordinate line?
4. What is the coordinate of a point on a line?
5. What is the coordinate of the origin?
Now we will write mathematical dictation, and you yourself decide which carriage you will travel in. So, open your notebooks and write your answers there. You should only answer “yes” or “no”. I will ask questions according to the options: first the first option, then the second.
Point A(15) is located to the left of zero. / Point B(-7) is located to the left of zero /
The numbers -2.5 and 2.5 are opposites /The numbers 0 and -1 are opposites/
The number 8 is the modulus of the number -8 / The number 0 is the modulus of the number 0.1 /
Number -12 more number-10/The number -16 is less than the number -8/
The length of the spring decreased by 6 mm. The change in its length is equal to -6 mm.
/Spring length increased by 7 mm. The change in its length is equal to 7 mm/
Now, guys, exchange notebooks and rate each other. (Students evaluate the work of their desk neighbors).
Solve No. 858, 861, 863
4. Studying new material.
So, all numbers can be divided into integers and fractions.
All natural numbers, their opposite numbers and 0 are called integers.
Those. Integers are divided into positive integers and negative integers.
Fractions are ordinary and decimal fractions.
By combining whole and fractional numbers we get rational numbers.
Journey through the pages of the dictionary
Rational – reasonably justified, expedient.
Numbers of the form a and –a are called opposite.
Find the opposite word:
Long - ...Thick - ...Right - ...Addition - ...Plus - ...
Answer the questions p.174.
Historical pause.
Back in the 3rd century AD, the ancient Greek mathematician Diophantus actually already used the rule for multiplying positive and negative numbers. But -3 for Diophantus is not independent number, but only “subtracted”, any positive is added. Diophantus did not recognize individual negative numbers, and if, when solving equations, a negative root was obtained, he discarded it as “inadmissible.”
He himself tried to formulate problems and compose equations in such a way as to avoid negative roots.
In India, negative numbers were interpreted as debt, and positive numbers as property. However, despite the widespread use of negative numbers in solving problems using equations, in India negative numbers were viewed with distrust, considering them peculiar and not entirely real.
Bhaskara directly wrote: “People do not approve of abstract negative numbers...”
5. Consolidation of new material.
Solve No. 876, 877, 878, 881.
6. Independent work.
Solve No. 879.
7. Lesson summary. D/z.
Learn paragraph 30, solve No. 880, 882, 896(a), repeat paragraph 11
Students answer teacher questions:
Which line is called the coordinate line?
What numbers are the coordinates of points on the coordinate line to the right of the origin? To the left of the origin?
What is the coordinate of the origin?
Topic: Number module.
Lesson objectives:
educational: study the concept of the modulus of a number and consolidate it when solving exercises, introduce the concept of rational numbers;
developing: development of attention, logical thinking, reasoned mathematical speech; maintaining interest in the subject.
educational: fostering goodwill, tolerance, objectivity.
1. Organizational moment.
I would like to start today's lesson with the words of K.E. Tsiolkovsky: “First I discovered what is known to many, then what is known to some, and then what is unknown to no one.”
In every lesson, you guys acquire new knowledge that was once discovered by great mathematicians. Today, according to scientist K.E. Tsiolkovsky, you will discover something that is known to many. The knowledge gained today will help you in the future when studying many topics, not only in the mathematics course, but also when studying a new course called algebra.
2. Updating of basic knowledge.
Oral counting.
Learn to think accurately
Explore everything to the bottom!
Instead of dots on a piece of paper
The correct number is needed.
I won't give any hints
There are no signs of her.
But it's the same everywhere
Will give us the correct answer.
Solve No. 883, 884.
Among the numbers –(-7); -3; ; -7; 3; ; ; ; 0 indicate pairs of opposite numbers;
What numbers are called opposites?
What is the opposite number of a positive number? Negative?
What number is opposite to itself?
How many opposites does a given number have?
3. Explanation of new material
And now I will tell you a fairy tale, listen and try to hear a word that is still unfamiliar to you.
Gathered for a meeting on a number line different numbers: positive, negative and null. He stood up and began to speak: “Dear numbers, we have gathered here to evaluate our actions. I must note, although perhaps this is not modest, that the score comes from me, so I will give you an assessment. To my right are positive numbers; there is nothing negative to say about them. On the left are negative numbers. In life it is bad to be negative, but in mathematics we often cannot get a positive answer without them. A MODULE that is always non-negative deserves all approval.” The numbers sit and think: how to understand the Zero estimate?
A number greater than zero... Big Encyclopedic Dictionary
positive number- - [L.G. Sumenko. English-Russian dictionary on information technology. M.: State Enterprise TsNIIS, 2003.] Topics information Technology overall EN positive number... Technical Translator's Guide
A number greater than zero. * * * POSITIVE NUMBER POSITIVE NUMBER, a number greater than zero... Encyclopedic Dictionary
A negative number is an element of the set of negative numbers, which (together with zero) appeared in mathematics when expanding the set of natural numbers. The purpose of the extension is to allow the subtraction operation to be performed on any number. As a result... ... Wikipedia
- (Double precision, Double) computer format for representing numbers, occupying two consecutive cells in memory (computer words; in the case of a 32-bit computer, 64 bits or 8 bytes). Typically denotes a floating point number format... ... Wikipedia
- (English half precision) a computer format for representing numbers, occupying half a computer word in memory (in the case of a 32-bit computer, 16 bits or 2 bytes). Value range ± 2−24(5.96E 8) 65504. Approximate... ... Wikipedia
A floating point number is a form of representation of real numbers in which the number is stored in the form of a mantissa and an exponent. In this case, a floating point number has a fixed relative precision and a variable absolute one.... ... Wikipedia
Noun, s., used. very often Morphology: (no) what? numbers, what? number, (see) what? number, what? number, about what? about number; pl. What? numbers, (no) what? numbers, why? numbers, (see) what? numbers, what? numbers, about what? about numbers mathematics 1. By number... ... Dmitriev's Explanatory Dictionary
NUMBER, a, plural. numbers, sat, slam, cf. 1. The basic concept of mathematics is quantity, with the help of which calculation is made. Integer h. Fractional h. Real h. Complex h. Natural h. (positive integer). Prime number (natural number, not... ... Ozhegov's Explanatory Dictionary
E is a mathematical constant, the base of the natural logarithm, an irrational and transcendental number. Sometimes the number e is called the Euler number (not to be confused with the so-called Euler numbers of the first kind) or the Napier number. Indicated by lowercase Latin letter"e".... ...Wikipedia
Books
- Square Root of 2 by Jesse Russell. This book will be produced in accordance with your order using Print-on-Demand technology. High Quality Content by WIKIPEDIA articles! The square root of 2 is positive...
Let's say Denis has a lot of candies - a whole big box. First Denis ate 3 candies. Then dad gave Denis 5 candies. Then Denis gave Matvey 9 candies. Finally, mom gave Denis 6 candies. Question: Did Denis end up with more or less candy than he had at first? If more, how much more? If less, how much less?
In order not to get confused with this task, it is convenient to use one trick. Let's write out all the numbers in a row from the condition. At the same time, we will put a “+” sign in front of the numbers that indicate how much more candy Denis has gained, and a “−” sign in front of the numbers that indicate how much candy Denis has decreased. Then the whole condition will be written out very briefly:
− 3 + 5 − 9 + 6.
This entry can be read, for example, like this: “First Denis received minus three candies. Then plus five candies. Then minus nine candies. And finally, plus six sweets.” The word “minus” changes the meaning of the phrase to the exact opposite. When I say: “Denis received minus three candies,” this actually means that Denis lost three candies. The word “plus,” on the contrary, confirms the meaning of the phrase. “Denis received plus five sweets” means the same thing as simply “Denis received five sweets.”
So, first Denis received minus three candies. This means that Denis now has minus three more candies than he had at the beginning. For brevity, we can say: Denis has minus three candies.
Then Denis received plus five candies. It’s easy to figure out that Denis now has two more candies. Means,
− 3 + 5 = + 2.
Then Denis received minus nine sweets. And this is how many candies he had:
− 3 + 5 − 9 = + 2 − 9 = − 7.
Finally Denis got +6 more candies. And the total amount of candy became:
− 3 + 5 − 9 + 6 = + 2 − 9 + 6 = − 7 + 6 = − 1.
In ordinary language, this means that in the end Denis ended up with one less candy than he had at the beginning. The problem is solved.
The trick with the “+” or “−” signs is used very widely. Numbers with a “+” sign are called positive. Numbers with a “−” sign are called negative. The number 0 (zero) is neither positive nor negative, because +0 is no different from −0. Thus, we are dealing with numbers from the series
..., −5, −4, −3, −2, −1, 0, +1, +2, +3, +4, +5, ...
Such numbers are called integers. And those numbers that have no sign at all and with which we have dealt so far are called natural numbers(only zero does not apply to natural numbers).
Integers can be thought of as rungs on a ladder. Number zero is the landing, which is level with the street. From here you can go up, step by step, to higher floors, or you can go down to the basement. As long as we don't need to go into the basement, just the natural numbers and zero are enough for us. Natural numbers are essentially the same as positive integers.
Strictly speaking, an integer is not a step number, but a command to move up the stairs. For example, the number +3 means that you should go up three steps, and the number −5 means that you should go down five steps. Simply, a command is taken as the number of a step, which moves us to a given step if we start moving from the zero level.
Calculations with integers are easy to do by simply mentally jumping up or down steps - unless, of course, you need to make very large jumps. But what to do when you need to jump a hundred or more steps? After all, we won’t draw such a long staircase!
But why not? We can draw a long staircase from such a great distance that the individual steps are no longer distinguishable. Then our staircase will simply turn into one straight line. And to make it more convenient to place it on the page, let’s draw it without tilting and separately mark the position of step 0.
Let's first learn how to jump along such a straight line using the example of expressions whose values we have long been able to calculate. Let it be required to find
Strictly speaking, since we are dealing with integers, we should write
But a positive number at the beginning of a line usually does not have a “+” sign. Jumping stairs looks something like this:
Instead of two big jumps drawn above the line (+42 and +53), you can make one jump drawn below the line, and the length of this jump, of course, is equal to
In mathematical language, these kinds of drawings are usually called diagrams. This is what the diagram looks like for our usual subtraction example:
First we made a big jump to the right, then a smaller jump to the left. As a result, we remained to the right of zero. But another situation is also possible, as, for example, in the case of the expression
This time the jump to the right turned out to be shorter than the jump to the left: we flew over zero and ended up in the “basement” - where the steps with negative numbers are located. Let's take a closer look at our jump to the left. In total we climbed 95 steps. After we climbed 53 steps, we reached mark 0. The question is, how many steps did we climb after that? Well, of course
Thus, once we were on step 0, we went down another 42 steps, which means that we finally arrived at step number −42. So,
53 − 95 = −(95 − 53) = −42.
Likewise, by drawing diagrams, it is easy to establish that
−42 − 53 = −(42 + 53) = −95;
−95 + 53 = −(95 − 53) = −42;
and finally
−53 + 95 = 95 − 53 = 42.
Thus, we have learned to freely travel through the entire ladder of integers.
Let's now consider this problem. Denis and Matvey exchange candy wrappers. At first Denis gave Matvey 3 candy wrappers, and then took 5 candy wrappers from him. How many candy wrappers did Matvey receive in the end?
But since Denis received 2 candy wrappers, then Matvey received -2 candy wrappers. We added a minus to Denis's profit and got Matvey's profit. Our solution can be written as a single expression
−(−3 + 5) = −2.
Everything is simple here. But let's slightly modify the problem statement. Let Denis first give Matvey 5 candy wrappers, and then take 3 candy wrappers from him. The question is, again, how many candy wrappers did Matvey receive in the end?
Again, first let’s calculate Denis’s “profit”:
−5 + 3 = −2.
This means that Matvey received 2 candy wrappers. But how can we now write down our decision as a single expression? What would you add to the negative number −2 to get the positive number 2? It turns out that this time we need to assign a minus sign. Mathematicians are very fond of uniformity. They strive to ensure that solutions to similar problems are written in the form of similar expressions. IN in this case the solution looks like this:
−(−5 + 3) = −(−2) = +2.
This is how mathematicians agreed: if you add a minus to a positive number, then it turns into a negative one, and if you add a minus to a negative number, then it turns into a positive one. This is very logical. After all, going down minus two steps is the same as going up plus two steps. So,
−(+2) = −2;
−(−2) = +2.
To complete the picture, we also note that
+(+2) = +2;
+(−2) = −2.
This gives us the opportunity to take a fresh look at things that have long been familiar. Let the expression be given
The meaning of this entry can be imagined in different ways. You can, in the old fashioned way, assume that the positive number +3 is subtracted from the positive number +5:
In this case +5 is called reducible, +3 - deductible, and the whole expression is difference. This is exactly what they teach in school. However, the words “reduced” and “subtracted” are not used anywhere except at school and they can be forgotten after the final test work. About this same entry we can say that the negative number −3 is added to the positive number +5:
The numbers +5 and −3 are called terms, and the whole expression is amount. There are only two terms in this sum, but, in general, the sum can consist of as many terms as you like. Likewise, the expression
can with equal right be considered as the sum of two positive numbers:
and as the difference between positive and negative numbers:
(+5) − (−3).
After we have become acquainted with integers, we definitely need to clarify the rules for opening parentheses. If there is a “+” sign in front of the brackets, then such brackets can simply be erased, and all the numbers in them retain their signs, for example:
+(+2) = +2;
+(−2) = −2;
+(−3 + 5) = −3 + 5;
+(−3 − 5) = −3 − 5;
+(5 − 3) = 5 − 3
and so on.
If there is a “−” sign in front of the brackets, then when erasing the bracket, we must also change the signs of all the numbers in it:
−(+2) = −2;
−(−2) = +2;
−(−3 + 5) = +3 − 5 = 3 − 5;
−(−3 − 5) = +3 + 5 = 3 + 5;
−(5 − 3) = −(+5 − 3) = −5 + 3;
and so on.
At the same time, it is useful to keep in mind the problem of the exchange of candy wrappers between Denis and Matvey. For example, the last line can be obtained like this. We believe that Denis first took 5 candy wrappers from Matvey, and then -3 more. In total, Denis received 5 − 3 candy wrappers, and Matvey received the same number, but with the opposite sign, that is, −(5 − 3) candy wrappers. But this same problem can be solved in another way, keeping in mind that every time Denis receives, Matvey gives. This means that at first Matvey received −5 candy wrappers, and then another +3, which ultimately gives −5 + 3.
Like natural numbers, integers can be compared with each other. Let us ask, for example, the question: which number is greater: −3 or −1? Let's look at the ladder with integers, and it immediately becomes clear that −1 is greater than −3, and therefore −3 is less than −1:
−1 > −3;
−3 < −1.
Now let's clarify: how much more is −1 than −3? In other words, how many steps do you need to climb to move from step −3 to step −1? The answer to this question can be written as the difference between the numbers −1 and −3:
− 1 − (−3) = −1 + 3 = 3 − 1 = 2.
Jumping up the steps, it is easy to check that this is so. Here's another interesting question: how much greater is the number 3 than the number 5? Or, which is the same thing: how many steps do you have to climb up to move from step 5 to step 3? Until recently, this question would have puzzled us. But now we can easily write out the answer:
3 − 5 = − 2.
Indeed, if we are on step 5 and go up another −2 steps, we will end up exactly on step 3.
Tasks
2.3.1. What is the meaning of the following phrases?
Denis gave dad minus three candies.
Matvey is minus two years older than Denis.
To get to our apartment, you need to go down minus two floors.
2.3.2. Do such phrases make sense?
Denis has minus three candies.
Minus two cows are grazing in the meadow.
Comment. This problem does not have a unique solution. It would not be a mistake, of course, to say that these statements are meaningless. And at the same time, they can be given a very clear meaning. Let's say Denis has a large box filled to the brim with sweets, but the contents of this box don't count. Or let’s say that two cows from the herd did not go out to graze in the meadow, but for some reason remained in the barn. It is worth keeping in mind that even the most familiar phrases can be ambiguous:
Denis has three candies.
This statement does not exclude the possibility that Denis has a huge box of candies hidden somewhere else, but those candies are simply kept silent. In the same way, when I say: “I have five rubles,” I do not mean that this is my entire fortune.
2.3.3. The grasshopper jumps up the stairs, starting from the floor where Denis's apartment is located. First he jumped 2 steps down, then 5 steps up, and finally 7 steps down. How many steps and in what direction did the grasshopper move?
2.3.4. Find the meaning of expressions:
− 6 + 10;
− 28 + 76;
etc.
− 6 + 10 = 10 − 6 = 4.
2.3.5. Find the meaning of expressions:
8 − 20;
34 − 98;
etc.
8 − 20 = − (20 − 8) = − 12.
2.3.6. Find the meaning of expressions:
− 4 − 13;
− 48 − 53;
etc.
− 4 − 13 = − (4 + 13) = − 17.
2.3.7. For the following expressions, find the values by performing calculations in the order specified by the brackets. Then open the parentheses and make sure that the meanings of the expressions remain the same. Make up problems about candies that can be solved in this way.
25 − (−10 + 4);
25 + (− 4 + 10);
etc.
25 − (− 10 + 4) = 25 − (−(10 − 4)) = 25 − (−6) = 25 + 6 = 31.
25 − (− 10 + 4) = 25 + 10 − 4 = 35 − 4 = 31.
“Denis had 25 candies. He gave dad minus ten candies, and Matvey four candies. How many candies did he have?
