Now we'll figure it out positive and negative numbers. First, we will give definitions, introduce notation, and then give examples of positive and negative numbers. We will also dwell on the semantic load that positive and negative numbers carry.
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Positive and Negative Numbers - Definitions and Examples
Give identifying positive and negative numbers will help us. For convenience, we will assume that it is located horizontally and directed from left to right.
Definition.
Numbers that correspond to points of the coordinate line lying to the right of the origin are called positive.
Definition.
The numbers that correspond to the points of the coordinate line lying to the left of the origin are called negative.
The number zero, which corresponds to the origin, is neither a positive nor a negative number.
From the definition of negative and positive numbers it follows that the set of all negative numbers is the set of numbers opposite all positive numbers (if necessary, see the article opposite numbers). Therefore, negative numbers are always written with a minus sign.
Now, knowing the definitions of positive and negative numbers, we can easily give examples of positive and negative numbers. Examples of positive numbers are the natural numbers 5, 792 and 101,330, and indeed any natural number is positive. Examples of positive rational numbers are the numbers , 4.67 and 0,(12)=0.121212... , and negative ones are the numbers , −11 , −51.51 and −3,(3) . Examples of positive irrational numbers include the number pi, the number e, and the infinite non-periodic decimal fraction 809.030030003..., and examples of negative irrational numbers include the numbers minus pi, minus e, and the number equal to. It should be noted that in the last example it is not at all obvious that the value of the expression is a negative number. To find out for sure, you need to get the value of this expression in the form of a decimal fraction, and we will tell you how to do this in the article comparison of real numbers.
Sometimes positive numbers are preceded by a plus sign, just as negative numbers are preceded by a minus sign. In these cases, you should know that +5=5, 
and so on. That is, +5 and 5, etc. - this is the same number, but designated differently. Moreover, you can come across definitions of positive and negative numbers based on the plus or minus sign.
Definition.
Numbers with a plus sign are called positive, and with a minus sign – negative.
There is another definition of positive and negative numbers based on comparison of numbers. To give this definition, it is enough just to remember that the point on the coordinate line corresponding to the larger number lies to the right of the point corresponding to the smaller number.
Definition.
Positive numbers are numbers that are greater than zero, and negative numbers are numbers less than zero.
Thus, zero sort of separates positive numbers from negative ones.
Of course, we should also dwell on the rules for reading positive and negative numbers. If a number is written with a + or − sign, then pronounce the name of the sign, after which the number is pronounced. For example, +8 is read as plus eight, and - as minus one point two fifths. The names of the signs + and − are not declined by case. An example of correct pronunciation is the phrase “a equals minus three” (not minus three).
Interpretation of positive and negative numbers
We have been describing positive and negative numbers for quite some time. However, it would be nice to know what meaning they carry? Let's look at this issue.
Positive numbers can be interpreted as an arrival, as an increase, as an increase in some value, and the like. Negative numbers, in turn, mean exactly the opposite - expense, deficiency, debt, reduction of some value, etc. Let's understand this with examples.
We can say that we have 3 items. Here the positive number 3 indicates the number of items we have. How can you interpret the negative number −3? For example, the number −3 could mean that we have to give someone 3 items that we don't even have in stock. Similarly, we can say that at the cash register we were given 3.45 thousand rubles. That is, the number 3.45 is associated with our arrival. In turn, a negative number of −3.45 will indicate a decrease in money in the cash register that issued this money to us. That is, −3.45 is the expense. Another example: a temperature increase of 17.3 degrees can be described as a positive number +17.3, and a temperature decrease of 2.4 can be described using a negative number, as a temperature change of -2.4 degrees.
Positive and negative numbers are often used to describe the values of certain quantities in various measuring instruments. The most accessible example is a device for measuring temperatures - a thermometer - with a scale on which both positive and negative numbers are written. Often negative numbers are depicted in blue (it symbolizes snow, ice, and at temperatures below zero degrees Celsius, water begins to freeze), and positive numbers are written in red (the color of fire, the sun, at temperatures above zero degrees Celsius, ice begins to melt). Writing positive and negative numbers in red and blue is also used in other cases when you need to highlight the sign of the numbers.
Bibliography.
- Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
Let's say Denis has a lot of sweets - 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, 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 number of candies 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.
In this way, 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 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 can be forgotten after the final test. 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 about 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 go 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;
and so on.
− 6 + 10 = 10 − 6 = 4.
2.3.5. Find the meaning of expressions:
8 − 20;
34 − 98;
and so on.
8 − 20 = − (20 − 8) = − 12.
2.3.6. Find the meaning of expressions:
− 4 − 13;
− 48 − 53;
and so on.
− 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);
and so on.
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?
Positive and negative numbers
Coordinate line
Let's go straight. Let's mark point 0 (zero) on it and take this point as the starting point.
We indicate with an arrow the direction of movement in a straight line to the right from the origin of coordinates. In this direction from point 0 we will plot positive numbers.
That is, numbers that are already known to us, except zero, are called positive.
Sometimes positive numbers are written with a “+” sign. For example, "+8".
For brevity, the “+” sign before a positive number is usually omitted and instead of “+8” they simply write 8.
Therefore, “+3” and “3” are the same number, only designated differently.
Let's choose some segment whose length we take as one and move it several times to the right from point 0. At the end of the first segment the number 1 is written, at the end of the second - the number 2, etc. 
Putting the unit segment to the left from the origin we get negative numbers: -1; -2; etc.
Negative numbers used to denote various quantities, such as: temperature (below zero), flow - that is, negative income, depth - negative height, and others.
As can be seen from the figure, negative numbers are numbers already known to us, only with a minus sign: -8; -5.25, etc.
- The number 0 is neither positive nor negative.
The number axis is usually positioned horizontally or vertically.
If the coordinate line is located vertically, then the direction up from the origin is usually considered positive, and the direction down from the origin is negative.
The arrow indicates the positive direction.
The straight line marked:
. origin (point 0);
. unit segment;
. the arrow indicates the positive direction;
called coordinate line
or number axis.
Opposite numbers on a coordinate line
Let us mark two points A and B on the coordinate line, which are located at the same distance from point 0 on the right and left, respectively.
In this case, the lengths of the segments OA and OB are the same.
This means that the coordinates of points A and B differ only in sign. 
Points A and B are also said to be symmetrical about the origin.
The coordinate of point A is positive “+2”, the coordinate of point B has a minus sign “-2”.
A (+2), B (-2).
- Numbers that differ only in sign are called opposite numbers. The corresponding points of the numerical (coordinate) axis are symmetrical relative to the origin.
Every number has only one opposite number. Only the number 0 does not have an opposite, but we can say that it is the opposite of itself.
The notation "-a" means the opposite number of "a". Remember that a letter can hide either a positive number or a negative number.
Example:
-3 is the opposite number of 3.
We write it as an expression:
-3 = -(+3)
Example:
-(-6) is the opposite number to the negative number -6. So -(-6) is a positive number 6.
We write it as an expression:
-(-6) = 6
Adding Negative Numbers
The addition of positive and negative numbers can be analyzed using the number line.
It is convenient to perform the addition of small modulo numbers on a coordinate line, mentally imagining how the point denoting the number moves along the number axis.
Let's take some number, for example, 3. Let's denote it on the number axis by point A.
Let's add the positive number 2 to the number. This will mean that point A must be moved two unit segments in the positive direction, that is, to the right. As a result, we get point B with coordinate 5.
3 + (+ 2) = 5
In order to add a negative number (- 5) to a positive number, for example, 3, point A must be moved 5 units of length in the negative direction, that is, to the left.
In this case, the coordinate of point B is - 2.
So, the order of adding rational numbers using the number line will be as follows:
. mark a point A on the coordinate line with a coordinate equal to the first term;
. move it a distance equal to the modulus of the second term in the direction that corresponds to the sign in front of the second number (plus - move to the right, minus - to the left);
. the point B obtained on the axis will have a coordinate that will be equal to the sum of these numbers.
Example.
- 2 + (- 6) =
Moving from point - 2 to the left (since there is a minus sign in front of 6), we get - 8.
- 2 + (- 6) = - 8
Adding numbers with the same signs
Adding rational numbers can be easier if you use the concept of modulus.
Let's say we need to add numbers that have the same signs.
To do this, we discard the signs of the numbers and take the modules of these numbers. Let's add the modules and put the sign in front of the sum that was common to these numbers.
Example. 
An example of adding negative numbers.
(- 3,2) + (- 4,3) = - (3,2 + 4,3) = - 7,5
- To add numbers of the same sign, you need to add their modules and put in front of the sum the sign that was before the terms.
Adding numbers with different signs
If the numbers have different signs, then we act somewhat differently than when adding numbers with the same signs.
. We discard the signs in front of the numbers, that is, we take their modules.
. From the larger module we subtract the smaller one.
. Before the difference we put the sign that was for the number with a larger module.
An example of adding a negative and a positive number.
0,3 + (- 0,8) = - (0,8 - 0,3) = - 0,5
An example of adding mixed numbers.
To add numbers of different signs you need:
. subtract the smaller module from the larger module;
. Before the resulting difference, put the sign of the number with the larger modulus.
Subtracting Negative Numbers
As you know, subtraction is the opposite of addition.
If a and b are positive numbers, then subtracting the number b from the number a means finding a number c that, when added to the number b, gives the number a.
a - b = c or c + b = a
The definition of subtraction holds true for all rational numbers. That is subtracting positive and negative numbers can be replaced by addition.
- To subtract another from one number, you need to add the opposite number to the one being subtracted.
Or, in another way, we can say that subtracting the number b is the same as addition, but with the opposite number to b.
a - b = a + (- b)
Example.
6 - 8 = 6 + (- 8) = - 2
Example.
0 - 2 = 0 + (- 2) = - 2
- It is worth remembering the expressions below.
- 0 - a = - a
- a - 0 = a
- a - a = 0
Rules for subtracting negative numbers
As can be seen from the examples above, subtracting a number b is an addition with the opposite number of b.
This rule holds true not only when subtracting a smaller number from a larger number, but also allows you to subtract a larger number from a smaller number, that is, you can always find the difference of two numbers.
The difference can be a positive number, a negative number, or a zero number.
Examples of subtracting negative and positive numbers.
. - 3 - (+ 4) = - 3 + (- 4) = - 7
. - 6 - (- 7) = - 6 + (+ 7) = 1
. 5 - (- 3) = 5 + (+ 3) = 8
It is convenient to remember the sign rule, which allows you to reduce the number of parentheses.
The plus sign does not change the sign of the number, so if there is a plus in front of the parenthesis, the sign in the parentheses does not change.
+ (+ a) = + a
+ (- a) = - a
The minus sign in front of the parentheses reverses the sign of the number in the parentheses.
- (+ a) = - a
- (- a) = + a
From the equalities it is clear that if there are identical signs before and inside the brackets, then we get “+”, and if the signs are different, then we get “-”.
(- 6) + (+ 2) - (- 10) - (- 1) + (- 7) = - 6 + 2 + 10 + 1 - 7 = - 13 + 13 = 0
The sign rule also applies if the brackets contain not just one number, but an algebraic sum of numbers.
a - (- b + c) + (d - k + n) = a + b - c + d - k + n
Please note that if there are several numbers in brackets and there is a minus sign in front of the brackets, then the signs in front of all the numbers in these brackets must change.
To remember the rule of signs, you can create a table for determining the signs of a number.
Sign rule for numbers
Or learn a simple rule.
- Two negatives make an affirmative,
- Plus times minus equals minus.
Multiplying Negative Numbers
Using the concept of the modulus of a number, we formulate the rules for multiplying positive and negative numbers.
Multiplying numbers with the same signs
The first case that you may encounter is the multiplication of numbers with the same signs.
To multiply two numbers with the same signs:
. multiply the modules of numbers;
. put a “+” sign in front of the resulting product (when writing the answer, the “plus” sign before the first number on the left can be omitted).
Examples of multiplying negative and positive numbers.
. (- 3) . (- 6) = + 18 = 18
. 2 . 3 = 6
Multiplying numbers with different signs
The second possible case is the multiplication of numbers with different signs.
To multiply two numbers with different signs:
. multiply the modules of numbers;
. Place a “-” sign in front of the resulting work.
Examples of multiplying negative and positive numbers.
. (- 0,3) . 0,5 = - 1,5
. 1,2 . (- 7) = - 8,4
Rules for multiplication signs
Remembering the sign rule for multiplication is very simple. This rule coincides with the rule for opening parentheses.
- Two negatives make an affirmative,
- Plus times minus equals minus.
In “long” examples, in which there is only a multiplication action, the sign of the product can be determined by the number of negative factors.
At even number of negative factors, the result will be positive, and with odd quantity - negative.
Example.
(- 6) . (- 3) . (- 4) . (- 2) . 12 . (- 1) =
There are five negative factors in the example. This means that the sign of the result will be “minus”.
Now let's calculate the product of the moduli, not paying attention to the signs.
6 . 3 . 4 . 2 . 12 . 1 = 1728
The end result of multiplying the original numbers will be:
(- 6) . (- 3) . (- 4) . (- 2) . 12 . (- 1) = - 1728
Multiplying by zero and one
If among the factors there is a number zero or positive one, then the multiplication is performed according to known rules.
. 0 . a = 0
. a. 0 = 0
. a. 1 = a
Examples:
. 0 . (- 3) = 0
. 0,4 . 1 = 0,4
Negative unity (- 1) plays a special role when multiplying rational numbers.
- When multiplied by (- 1), the number is reversed.
In literal expression, this property can be written:
a. (- 1) = (- 1) . a = - a
When adding, subtracting and multiplying rational numbers together, the order of operations established for positive numbers and zero is maintained.
An example of multiplying negative and positive numbers.
Dividing negative numbers
It's easy to understand how to divide negative numbers by remembering that division is the inverse of multiplication.
If a and b are positive numbers, then dividing the number a by the number b means finding a number c that, when multiplied by b, gives the number a.
This definition of division applies to any rational numbers as long as the divisors are non-zero.
Therefore, for example, dividing the number (- 15) by the number 5 means finding a number that, when multiplied by the number 5, gives the number (- 15). This number will be (- 3), since
(- 3) . 5 = - 15
Means
(- 15) : 5 = - 3
Examples of dividing rational numbers.
1. 10: 5 = 2, since 2 . 5 = 10
2. (- 4) : (- 2) = 2, since 2 . (- 2) = - 4
3. (- 18) : 3 = - 6, since (- 6) . 3 = - 18
4. 12: (- 4) = - 3, since (- 3) . (- 4) = 12
From the examples it is clear that the quotient of two numbers with the same signs is a positive number (examples 1, 2), and the quotient of two numbers with different signs is a negative number (examples 3,4).
Rules for dividing negative numbers
To find the modulus of a quotient, you need to divide the modulus of the dividend by the modulus of the divisor.
So, to divide two numbers with the same signs, you need to:
. Place a “+” sign in front of the result.
Examples of dividing numbers with the same signs:
. (- 9) : (- 3) = + 3
. 6: 3 = 2
To divide two numbers with different signs, you need to:
. divide the module of the dividend by the module of the divisor;
. Place a “-” sign in front of the result.
Examples of dividing numbers with different signs:
. (- 5) : 2 = - 2,5
. 28: (- 2) = - 14
You can also use the following table to determine the quotient sign.
Rule of signs for division
When calculating “long” expressions in which only multiplication and division appear, it is very convenient to use the sign rule. For example, to calculate a fraction
Please note that the numerator has 2 minus signs, which when multiplied will give a plus. There are also three minus signs in the denominator, which when multiplied will give a minus sign. Therefore, in the end the result will turn out with a minus sign.
Reducing a fraction (further actions with the modules of numbers) is performed in the same way as before:
- The quotient of zero divided by a number other than zero is zero.
- 0: a = 0, a ≠ 0
- You CANNOT divide by zero!
All previously known rules of division by one also apply to the set of rational numbers.
. a: 1 = a
. a: (- 1) = - a
. a: a = 1
, where a is any rational number.
The relationships between the results of multiplication and division, known for positive numbers, remain the same for all rational numbers (except zero):
. if a . b = c; a = c: b; b = c: a;
. if a: b = c; a = c. b; b = a: c
These dependencies are used to find the unknown factor, dividend and divisor (when solving equations), as well as to check the results of multiplication and division.
An example of finding the unknown.
x. (- 5) = 10
x = 10: (- 5)
x = - 2
Minus sign in fractions
Divide the number (- 5) by 6 and the number 5 by (- 6).
We remind you that the line in the notation of an ordinary fraction is the same division sign, and we write the quotient of each of these actions in the form of a negative fraction.
Thus, the minus sign in a fraction can be:
. before a fraction;
. in the numerator;
. in the denominator. 
- When writing negative fractions, the minus sign can be placed in front of the fraction, transferred from the numerator to the denominator, or from the denominator to the numerator.
This is often used when working with fractions, making calculations easier.
Example. Please note that after placing the minus sign in front of the bracket, we subtract the smaller one from the larger module according to the rules for adding numbers with different signs. 
Using the described property of sign transfer in fractions, you can act without finding out which of the given fractions has a greater modulus.
From previous Assembler language lessons we know that the processor works with binary numbers, these numbers can be positive or negative. And today I will tell you in detail what positive (unsigned) and negative (signed) numbers are.
Positive numbers
If the number is positive, then it simply represents the result of converting a decimal number to binary. Special encoding is used to represent positive numbers. The most significant bit in this case indicates the sign of the number. If the sign bit is zero, then the number is positive, otherwise it is negative.
In the Intel family of processors, the basic unit of storage for all types of data is the byte. A byte consists of eight bits. The table below shows the ranges of possible values of positive integers that the processor can work with:
When working with numbers, do not forget that a number with a value of no more than 255 can be written into a byte, a number with a value of no more than 65,535 can be written into a word, etc. For example, if, when working with a byte, you perform the addition operation 255 + 1, then the result should be the number 256. However, if you write the result into a byte, then the result will not be 256, but 0. This situation occurs in cases of “overflow”.
An overflow is when the result of an operation does not fit into the register intended for that result. Also, if there is an overflow, the result may not be zero, but another number.
Negative numbers
Representing negative numbers in computers encounters certain difficulties. A negative number has no numerical meaning; it symbolizes, rather, a future action - the fact that in the future we must subtract a few more from the objects that appear again.
Negative numbers are numbers with a minus sign.
Ranges of possible values of negative numbers:
To indicate the sign of a number, one digit (bit) is sufficient. Typically, the sign bit occupies the most significant bit of the number. If the most significant bit of a number is 0, then the number is considered positive. If the most significant digit of a number is 1, then the number is considered negative.
When programming in assembly language, one important point must be taken into account: “Limiting the range of representation of numbers.”
For example, if the size of a positive variable is 1 byte, then it can take a total of 256 different values. This means that we cannot use it to represent a number greater than 255 (111111112). For the same negative variable, the maximum value will be 127 (011111112), and the minimum -128 (100000002). The range is defined similarly for 2- and 4-byte variables.
History of Negative Numbers
It is known that natural numbers arose when counting objects. The human need to measure quantities and the fact that the result of a measurement is not always expressed as an integer led to the expansion of the set of natural numbers. Zero and fractional numbers were introduced.
The process of historical development of the concept of number did not end there. However, the first impetus for expanding the concept of number was not always purely the practical needs of people. It also happened that the problems of mathematics itself required expanding the concept of number. This is exactly what happened with the emergence of negative numbers. Solving many problems, especially those involving equations, involved subtracting a larger number from a smaller number. This required the introduction of new numbers.
Negative numbers first appeared in Ancient China about 2100 years ago. They also knew how to add and subtract positive and negative numbers; the rules of multiplication and division were not applied.
In the II century. BC e. Chinese scientist Zhang Can wrote the book Arithmetic in Nine Chapters. From the contents of the book it is clear that this is not a completely independent work, but a reworking of other books written long before Zhang Can. In this book, negative quantities are encountered for the first time in science. They are understood differently from the way we understand and apply them. He does not have a complete and clear understanding of the nature of negative quantities and the rules for operating with them. He understood every negative number as a debt, and every positive number as property. He performed operations with negative numbers not the same way as we do, but using reasoning about debt. For example, if you add another debt to one debt, then the result is debt, not property (i.e., according to ours (- x) + (- x) = - 2x. The minus sign was not known then, therefore, in order to distinguish the numbers , expressing debt, Zhan Can wrote them in a different ink than the numbers expressing property (positive).
In Chinese mathematics, positive quantities were called “chen” and were depicted in red, while negative quantities were called “fu” and were depicted in black. This method of depiction was used in China until the middle of the 12th century, until Li Ye proposed a more convenient designation for negative numbers - the numbers that represented negative numbers were crossed out with a line diagonally from right to left. Although Chinese scientists explained negative quantities as debt, and positive quantities as property, they still avoided their widespread use, since these numbers seemed incomprehensible, and actions with them were unclear. If the problem led to a negative solution, then they tried to replace the condition (like the Greeks) so that in the end a positive solution would be obtained.
In the 5th-6th centuries, negative numbers appeared and became very widespread in Indian mathematics. For calculations, mathematicians of that time used a counting board, on which numbers were depicted using counting sticks. Since there were no signs + and – at that time, positive numbers were depicted with red sticks, and negative numbers were depicted with black sticks and were called “debt” and “shortage.” Positive numbers were interpreted as “property.” Unlike China, the rules of multiplication and division were already known in India. In India, negative numbers were used systematically, much as we do now. Already in the work of the outstanding Indian mathematician and astronomer Brahmagupta (598 - about 660) we read: “property and property is property, the sum of two debts is a debt; the sum of property and zero is property; the sum of two zeros is zero... Debt, which is subtracted from zero, becomes property, and property becomes debt. If it is necessary to take away property from debt, and debt from property, then they take their sum.”
Indian mathematicians used negative numbers when solving equations, and subtraction was replaced by addition with an equally opposite number.
Along with negative numbers, Indian mathematicians introduced the concept of zero, which allowed them to create a decimal number system. But for a long time, zero was not recognized as a number; “nullus” in Latin means no, the absence of a number. And only after 10 centuries, in the 17th century, with the introduction of the coordinate system, zero became a number.
The Greeks also did not use signs at first. The ancient Greek scientist Diophantus did not recognize negative numbers at all, and if, when solving an equation, a negative root was obtained, he discarded it as “inaccessible.” And Diophantus tried to formulate problems and compose equations in such a way as to avoid negative roots, but soon Diophantus of Alexandria began to denote subtraction with the sign .
Despite the fact that negative numbers have been used for a long time, they were treated with some distrust, considering them not entirely real, their interpretation as property-debt caused bewilderment: how can one “add” and “subtract” property and debts?
In Europe, recognition came a thousand years later. The idea of a negative quantity was approached quite closely at the beginning of the 13th century by Leonardo of Pisa (Fibonacci), who also introduced it to solve financial problems with debts and came to the idea that negative quantities should be taken in the opposite sense to positive ones. In those years, the so-called mathematical duels were developed. At a problem-solving competition with the court mathematicians of Frederick II, Leonardo of Pisa (Fibonacci) was asked to solve a problem: it was necessary to find the capital of several individuals. Fibonacci received a negative value. “This case,” said Fibonacci, “is impossible, unless we assume that one had not capital, but debt.”
In 1202, he first used negative numbers to calculate his losses. However, negative numbers were used explicitly for the first time at the end of the 15th century by the French mathematician Chuquet.
Nevertheless, until the 17th century, negative numbers were “in the fold” and for a long time they were called “false”, “imaginary” or “absurd”. And even in the 17th century, the famous mathematician Blaise Pascal argued that 0-4 = 0 because there is no number that can be less than nothing, and until the 19th century mathematicians often discarded negative numbers in their calculations, considering them meaningless...
Bombelli and Girard, on the contrary, considered negative numbers to be quite acceptable and useful, in particular for indicating the lack of something. An echo of those times is the fact that in modern arithmetic the operation of subtraction and the sign of negative numbers are denoted by the same symbol (minus), although algebraically these are completely different concepts.
In Italy, when lending money, moneylenders put the amount of the debt and a line in front of the debtor’s name, like our minus, and when the debtor returned the money, they crossed it out, so it looked like our plus. You can consider a plus as a crossed out minus!
Modern notation for positive and negative numbers with signs
“+” and “-” were used by the German mathematician Widmann.
The German mathematician Michael Stiefel, in his book “Complete Arithmetic” (1544), first introduced the concept of negative numbers as numbers less than zero (less than nothing). This was a very big step forward in justifying negative numbers. He made it possible to view negative numbers not as a debt, but in a completely different, new way. But Stiefel called negative numbers absurd; actions with them, in his words, “also go absurdly, topsy-turvy.”
After Stiefel, scientists began to more confidently perform operations with negative numbers.
Negative solutions to problems were increasingly retained and interpreted.
In the 17th century The great French mathematician Rene Descartes proposed putting negative numbers on the number line to the left of zero. Now it all seems so simple and understandable to us, but to reach this idea, it took eighteen centuries of work of scientific thought from the Chinese scientist Zhang Can to Descartes.
In the works of Descartes, negative numbers received, as they say, a real interpretation. Descartes and his followers recognized them on an equal basis with positive ones. But in operations with negative numbers, not everything was clear (for example, multiplication by them), so many scientists did not want to recognize negative numbers as real numbers. A large and long dispute broke out among scientists about the essence of negative numbers and whether to recognize negative numbers as real numbers or not. This dispute after Descartes lasted about 200 years. During this period, mathematics as a science developed very greatly, and negative numbers were encountered at every step. Mathematics has become unthinkable, impossible without negative numbers. It became clear to an increasing number of scientists that negative numbers are real numbers, just as real, actually existing numbers as positive numbers.
Negative numbers have hardly won their place in mathematics. No matter how hard scientists try to avoid them. However, they did not always succeed in this. Life presented science with new and new tasks, and more and more often these tasks led to negative solutions in China, India, and Europe. Only at the beginning of the 19th century. the theory of negative numbers completed its development, and “absurd numbers” received universal recognition.
Every physicist constantly deals with numbers: he always measures, calculates, calculates something. Everywhere in his papers there are numbers, numbers and numbers. If you look closely at the physicist’s notes, you will find that when writing numbers, he often uses the signs “+” and “-”.
How do positive, and especially negative numbers arise in physics?
A physicist deals with various physical quantities that describe the various properties of objects and phenomena around us. The height of a building, the distance from school to home, the mass and temperature of the human body, the speed of a car, the volume of a can, the strength of an electric current, the refractive index of water, the power of a nuclear explosion, the voltage between electrodes, the duration of a lesson or recess, the electric charge of a metal ball - these are all examples. physical quantities. A physical quantity can be measured.
One should not think that any characteristic of an object or natural phenomenon can be measured and, therefore, is a physical quantity. It's not like that at all. For example, we say: “What beautiful mountains around! And what a beautiful lake there, below! And what a beautiful spruce tree there is on that rock! But we cannot measure the beauty of the mountains, the lake, or this lonely spruce!” This means that a characteristic such as beauty is not a physical quantity.
Measurements of physical quantities are carried out using measuring instruments such as a ruler, watch, scales, etc.
So, numbers in physics arise as a result of measuring physical quantities, and the numerical value of a physical quantity obtained as a result of measurement depends: on how this physical quantity is defined; from the units of measurement used.
Let's look at the scale of a regular street thermometer.
It has the form shown on scale 1. Only positive numbers are printed on it, and therefore, when indicating the numerical value of the temperature, it is necessary to additionally explain 20 degrees Celsius (above zero). This is inconvenient for physicists - after all, you can’t put words into a formula! Therefore, in physics a scale with negative numbers is used.
Let's look at the physical map of the world. The land areas on it are painted in various shades of green and brown, and the seas and oceans are painted in blue and blue. Each color has its own height (for land) or depth (for seas and oceans). A scale of depths and heights is drawn on the map, which shows what height (depth) a particular color means,
Using such a scale, it is enough to indicate the number without any additional words: positive numbers correspond to various places on land located above the surface of the sea; negative numbers correspond to points below the sea surface.
In the height scale we considered, the height of the water surface in the World Ocean is taken as zero. This scale is used in geodesy and cartography.
In contrast, in everyday life we usually take the height of the earth’s surface (in the place where we are) as zero height.
3.1 How were years counted in ancient times?
It's different in different countries. For example, in Ancient Egypt, every time a new king began to rule, the counting of years began anew. The first year of the king's reign was considered the first year, the second - the second, and so on. When this king died and a new one came to power, the first year began again, then the second, the third. The counting of years used by the inhabitants of one of the most ancient cities in the world, Rome, was different. The Romans considered the year the city was founded to be the first, the next year to be the second, and so on.
The counting of years that we use arose a long time ago and is associated with the veneration of Jesus Christ, the founder of the Christian religion. Counting the years from the birth of Jesus Christ was gradually adopted in different countries. In our country, it was introduced by Tsar Peter the Great three hundred years ago. We call the time calculated from the Nativity of Christ OUR ERA (and we write it in abbreviated form N.E.). Our era continues for two thousand years.
Conclusion
Most people know negative numbers, but there are some whose representation of negative numbers is incorrect.
Negative numbers are most common in the exact sciences, mathematics and physics.
In physics, negative numbers arise as a result of measurements and calculations of physical quantities. Negative number - shows the amount of electric charge. In other sciences, such as geography and history, a negative number can be replaced with words, for example, below sea level, and in history - 157 BC. e.
Literature
1. Great scientific encyclopedia, 2005.
2. Vigasin A. A., “History of the Ancient World,” 5th grade textbook, 2001.
3. Vygovskaya V.V. “Lesson-based developments in Mathematics: 6th grade” - M.: VAKO, 2008
4. “Positive and negative numbers”, textbook on mathematics for the 6th grade, 2001.
5. Children's encyclopedia “I know the world”, Moscow, “Enlightenment”, 1995.
6.. “Studying Mathematics”, educational publication, 1994.
7. “Elements of historicism in teaching mathematics in secondary school”, Moscow, “Prosveshchenie”, 1982
8. Nurk E.R., Telgmaa A.E. “Mathematics 6th grade”, Moscow, “Enlightenment”, 1989
9. “History of mathematics at school”, Moscow, “Prosveshchenie”, 1981.
