Positive and negative numbers, definition, examples. Positive and negative numbers: definition, examples
As a special number, it has no sign.
Examples of writing numbers: + 36, 6; − 273 ; 142. (\displaystyle +36(,)6;\ -273;\ 142.) The last number has no sign and is therefore positive.
It should be noted that plus and minus indicate sign for numbers, but not for literal variables or algebraic expressions. For example, in formulas − t ; a+b; − (a 2 + b 2) (\displaystyle -t;\ a+b;\ -(a^(2)+b^(2))) The plus and minus symbols do not specify the sign of the expression they precede, but the sign of the arithmetic operation, so the sign of the result can be anything; it is determined only after the expression has been evaluated.
In addition to arithmetic, the concept of a sign is used in other branches of mathematics, including for non-numerical mathematical objects (see below). The concept of sign is also important in those branches of physics where physical quantities are divided into two classes, conventionally called positive and negative - for example, electric charges, positive and negative feedback, various forces of attraction and repulsion.
Number sign
Positive and negative numbers
Zero is not assigned any sign, that is + 0 (\displaystyle +0) And − 0 (\displaystyle -0)- this is the same number in arithmetic. In mathematical analysis, the meaning of symbols + 0 (\displaystyle +0) And − 0 (\displaystyle -0) may vary, see about this Negative and positive zero; in computer science, computer encoding of two zeros (integer type) may differ, see Direct code.
In connection with the above, several more useful terms are introduced:
- Number non-negative, if it is greater than or equal to zero.
- Number negative, if it is less than or equal to zero.
- Positive numbers without zero and negative numbers without zero are sometimes (to emphasize that they are non-zero) called "strictly positive" and "strictly negative", respectively.
The same terminology is sometimes used for real functions. For example, the function is called positive, if all its values are positive, non-negative, if all its values are non-negative, etc. They also say that a function is positive/negative on a given interval of its definition..
For an example of using the function, see the article Square root#Complex numbers.
Modulus (absolute value) of a number
If the number x (\displaystyle x) discard the sign, the resulting value is called module or absolute value numbers x (\displaystyle x), it is designated | x | . (\displaystyle |x|.) Examples: | 3 | = 3 ; | − 3 | = 3. (\displaystyle |3|=3;\ |-3|=3.)
For any real numbers a , b (\displaystyle a,b) the following properties hold.
Sign for non-numeric objects
Angle sign
The value of an angle on a plane is considered positive if it is measured counterclockwise, otherwise negative. Two cases of rotation are classified similarly:
- rotation on a plane - for example, rotation by (–90°) occurs clockwise;
- rotation in space around an oriented axis is generally considered positive if the “gimlet rule” is satisfied, otherwise it is considered negative.
Direction sign
In analytical geometry and physics, advancements along a given straight line or curve are often conventionally divided into positive and negative. Such division may depend on the formulation of the problem or on the chosen coordinate system. For example, when calculating the arc length of a curve, it is often convenient to assign a minus sign to this length in one of two possible directions.
Sign in computing
| most significant bit | |||||||||
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | = | 127 |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | = | 126 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | = | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | = | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | = | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | = | −1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | = | −2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | = | −127 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | = | −128 |
| To represent the sign of an integer, most computers use | |||||||||
NUMBER, one of the basic concepts of mathematics; originated in ancient times and gradually expanded and generalized. In connection with the counting of individual objects, the concept of positive integer (natural) numbers arose, and then the idea of the limitlessness of the natural series of numbers: 1, 2, 3, 4. Problems of measuring lengths, areas, etc., as well as isolating shares of named quantities led to the concept of a rational (fractional) number. The concept of negative numbers arose among Indians in the 6th-11th centuries.
For the first time negative numbers are found in one of the books of the ancient Chinese treatise “Mathematics in Nine Chapters” (Jan Can - 1st century BC). Negative number was understood as debt, and positive - as property. Addition and subtraction of negative numbers was done based on reasoning about debt. For example, the addition rule was formulated as follows: “If you add another debt to one debt, the result is debt, not property.” There was no minus sign then, and in order to distinguish between positive and negative numbers, Can Can wrote them with ink of different colors.
The idea of negative numbers had a hard time gaining a place in mathematics. These numbers seemed incomprehensible and even false to the mathematicians of antiquity, and actions with them were unclear and had no real meaning.
Use of negative numbers by Indian mathematicians.
In the 6th and 7th centuries AD, Indian mathematicians already systematically used negative numbers, still understanding them as a duty. Since the 7th century, Indian mathematicians have used negative numbers. They called positive numbers “dhana” or “sva” (“property”), and negative numbers “rina” or “kshaya” (“debt”). For the first time, all four arithmetic operations with negative numbers were given by the Indian mathematician and astronomer Brahmagupta (598 - 660).
For example, he formulated the division rule as follows: “A positive divided by a positive, or a negative divided by a negative becomes positive. But positive divided by negative, and negative divided by positive, remains negative."
(Brahmagupta (598 - 660) is an Indian mathematician and astronomer. Brahmagupta’s work “Revision of the Brahma System” (628) has reached us, a significant part of which is devoted to arithmetic and algebra. The most important here is the doctrine of arithmetic progression and solution quadratic equations, which Brahmagupta dealt with in all cases where they had valid solutions. Brahmagupta allowed and considered the use of zero in all arithmetic operations. In addition, Brahmagupta solved some indefinite equations in integers; he gave the rule for compiling right triangles with rational sides, etc. Brahmagupta knew the inverse triple rule; he found the approximation P, the earliest interpolation formula of the 2nd order. His interpolation rule for sine and inverse sine at equal intervals is a special case of the Newton–Stirling interpolation formula. In a later work, Brahmagupta gives an interpolation rule for unequal intervals. His works were translated into Arabic in the 8th century.)
Understanding negative numbers by Leonard Fibonacci of Pisa.
Independently of the Indians, the Italian mathematician Leonardo Fibonacci of Pisa (13th century) came to understand negative numbers as the opposite of positive numbers. But it took about 400 more years before “absurd” (meaningless) negative numbers gained full recognition among mathematicians, and negative solutions to problems were no longer rejected as impossible.
(Leonardo Fibonacci of Pisa (c. 1170 - after 1228) - Italian mathematician. Born in Pisa (Italy). Primary education received in Bush (Algeria) under the guidance of a local teacher. Here he mastered the arithmetic and algebra of the Arabs. He visited many countries in Europe and the East and expanded his knowledge of mathematics everywhere.
He published two books: “The Book of the Abacus” (1202), where the abacus was considered not so much as a device, but as calculus in general, and “ Practical geometry"(1220). Based on the first book, many generations of European mathematicians studied the Indian positional number system. The presentation of the material in it was original and elegant. The scientist also made his own discoveries, in particular, he initiated the development of issues related to T.N. Fibonacci numbers, and gave an original method for extracting the cube root. His works only became widespread at the end of the 15th century, when Luca Pacioli revised them and published them in his book Summa.
Consideration of negative numbers by Mikhail Stifel in a new way.
In 1544, the German mathematician Michael Stiefel first considered negative numbers as numbers less than zero (i.e. "less than nothing"). From this point on, negative numbers are no longer viewed as a debt, but in a completely new way. (Mikhail Stiefel (19.04.1487 – 19.06.1567) - famous German mathematician. Michael Stiefel studied in a Catholic monastery, then became interested in the ideas of Luther and became a rural Protestant pastor. While studying the Bible, he tried to find a mathematical interpretation in it. As a result His research predicted the end of the world on October 19, 1533, which, of course, did not happen, and Michael Stiefel was imprisoned in Württemberg prison, from which Luther himself rescued him.
After this, Stiefel devoted his work entirely to mathematics, in which he was a self-taught genius. One of the first in Europe after N. Schuke began to operate with negative numbers; introduced fractional and zero exponents, as well as the term “exponent”; in the work “Complete Arithmetic” (1544) he gave the rule for dividing by a fraction as multiplying by the reciprocal of the divisor; took the first step in the development of techniques that simplify calculations with large numbers, for which he compared two progressions: geometric and arithmetic. Later this helped I. Bürgi and J. Napier create logarithmic tables and develop logarithmic calculations.)
Modern interpretation of negative numbers by Girard and Rene Descartes.
The modern interpretation of negative numbers, based on plotting unit segments on the number line to the left of zero, was given in the 17th century, mainly in the works of the Dutch mathematician Girard (1595–1634) and the famous French mathematician and philosopher René Descartes (1596–1650). ) (Girard Albert (1595 - 1632) - Belgian mathematician. Girard was born in France, but fled to Holland from persecution catholic church because he was a Protestant. Albert Girard made major contributions to the development of algebra. His main work was the book A New Discovery in Algebra. For the first time he expressed the fundamental theorem of algebra about the presence of a root in an algebraic equation with one unknown. Although Gauss was the first to give a rigorous proof. Girard is responsible for the derivation of the formula for the area of a spherical triangle.) Since 1629 in the Netherlands. He laid the foundations of analytical geometry, gave the concepts of variable quantities and functions, and introduced many algebraic notations. He expressed the law of conservation of momentum and gave the concept of impulse of force. Author of a theory that explains the formation and movement of celestial bodies by the vortex motion of matter particles (Descartes vortices). Introduced the concept of reflex (Descartes arc). The basis of Descartes' philosophy is the dualism of soul and body, “thinking” and “extended” substance. He identified matter with extension (or space), and reduced movement to the movement of bodies. Common Cause movement, according to Descartes, is God, who created matter, movement and rest. Man is a connection between a lifeless bodily mechanism and a soul with thinking and will. The unconditional foundation of all knowledge, according to Descartes, is the immediate certainty of consciousness (“I think, therefore I exist”). The existence of God was considered as a source of objective significance of human thinking. In the doctrine of knowledge, Descartes is the founder of rationalism and a supporter of the doctrine of innate ideas. Main works: “Geometry” (1637), “Discourse on the Method. "(1637), "Principles of Philosophy" (1644).
DESCARTES (Descartes) Rene (Latinized - Cartesius; Cartesius) (March 31, 1596, Lae, Touraine, France - February 11, 1650, Stockholm), French philosopher, mathematician, physicist and physiologist, founder of modern European rationalism and one of the most influential metaphysicians of the New Age.
Life and writings
Born into a noble family, Descartes received good education. In 1606, his father sent him to the Jesuit college of La Flèche. Considering Descartes’s not very good health, he was given some concessions in the strict regime of this educational institution, for example. , were allowed to get up later than others. Having acquired a lot of knowledge at the college, Descartes at the same time became imbued with antipathy towards scholastic philosophy, which he retained throughout his life.
After graduating from college, Descartes continued his education. In 1616, at the University of Poitiers, he received a bachelor's degree in law. In 1617, Descartes enlisted in the army and traveled extensively throughout Europe.
The year 1619 turned out to be a key year for Descartes scientifically. It was at this time, as he himself wrote in his diary, that the foundations of a new “most amazing science” were revealed to him. Most likely, Descartes had in mind the discovery of the universal scientific method, which he subsequently fruitfully applied in a variety of disciplines.
In the 1620s, Descartes met the mathematician M. Mersenne, through whom he “kept in touch” with the entire European scientific community for many years.
In 1628, Descartes settled in the Netherlands for more than 15 years, but did not settle in any one place, but changed his place of residence about two dozen times.
In 1633, having learned about the condemnation of Galileo by the church, Descartes refused to publish the natural philosophical work “The World,” which outlined the ideas of the natural origin of the universe according to mechanical laws matter.
In 1637 on French Descartes' work “Discourse on Method” is published, with which, as many believe, modern European philosophy began.
In 1641, Descartes' main philosophical work, “Reflections on First Philosophy,” appeared (on Latin), and in 1644 “Principles of Philosophy,” a work conceived by Descartes as a compendium summing up the author’s most important metaphysical and natural philosophical theories.
Descartes’s last philosophical work, “The Passions of the Soul,” published in 1649, also had a great influence on European thought. In the same year, by invitation Swedish Queen Christina Descartes went to Sweden. The harsh climate and unusual regime (the queen forced Descartes to get up at 5 a.m. to give her lessons and carry out other assignments) undermined Descartes' health, and, having caught a cold, he died of pneumonia.
The philosophy of Descartes clearly illustrates the desire European culture to liberation from old dogmas and the construction of a new science and life itself “with clean slate" The criterion of truth, Descartes believes, can only be the “natural light” of our mind. Descartes does not deny educational value experience, but he sees its function exclusively in coming to the aid of reason where own strength the latter is not enough for knowledge. Reflecting on the conditions for achieving reliable knowledge, Descartes formulates the “rules of method” with the help of which one can arrive at the truth. Initially thought by Descartes to be very numerous, in the “Discourse on Method”, he reduces them to four main provisions that constitute the “quintessence” of European rationalism: 1) start with the undoubted and self-evident, i.e. with that which cannot be thought to be the opposite, 2) divide any problem into as many parts as necessary to solve it effectively, 3) start with the simple and gradually move towards the complex, 4) constantly recheck the correctness of the conclusions. The self-evident is grasped by the mind in intellectual intuition, which cannot be confused with sensory observation and which gives us a “clear and distinct” comprehension of the truth. Dividing a problem into parts makes it possible to identify “absolute” elements in it, that is, self-evident elements from which subsequent deductions can be based. Descartes calls deduction the “movement of thought” in which the cohesion of intuitive truths occurs. The weakness of human intelligence requires checking the correctness of the steps taken to ensure that there are no gaps in reasoning. Descartes calls this verification “enumeration” or “induction.” The result of consistent and ramified deduction should be the construction of a system of universal knowledge, “universal science.” Descartes compares this science to a tree. Its root is metaphysics, its trunk is physics, and its fruitful branches are formed by concrete sciences, ethics, medicine and mechanics, which bring direct benefit. From this diagram it is clear that the key to the effectiveness of all these sciences is correct metaphysics.
What distinguishes Descartes from the method of discovering truths is the method of presenting already developed material. It can be presented “analytically” and “synthetically”. Analytical method problematic, it is less systematic, but more conducive to understanding. Synthetic, as if “geometrizing” material, is more strict. Descartes still prefers the analytical method.
Doubt and certainty
The initial problem of metaphysics as a science about the most general kinds of being is, as in any other disciplines, the question of self-evident foundations. Metaphysics must begin with the undoubted statement of some existence. Descartes “tests” the theses about the existence of the world, God and our “I” for self-evidence. The world can be imagined as non-existent if we imagine that our life is a long dream. One can also doubt the existence of God. But our “I,” Descartes believes, cannot be questioned, since doubt itself in its existence proves the existence of doubt, and therefore of the doubting I. “I doubt, therefore I exist” - this is how Descartes formulates this most important truth, denoting the subjectivist turn of European philosophy New time. In a more general form, this thesis sounds like this: “I think, therefore I exist” - cogito, ergo sum. Doubt is only one of the “modes of thinking”, along with desire, rational comprehension, imagination, memory and even sensation. The basis of thinking is consciousness. Therefore, Descartes denies the existence of unconscious ideas. Thinking is an integral property of the soul. The soul cannot help but think; it is a “thinking thing,” res cogitans. Recognizing the thesis of one's own existence as undoubted does not mean, however, that Descartes considers the non-existence of the soul generally impossible: it cannot but exist only as long as it thinks. Otherwise, the soul is a random thing, that is, it can either be or not be, because it is imperfect. All random things derive their existence from the outside. Descartes states that the soul is maintained in its existence every second by God. Nevertheless, it can be called a substance, since it can exist separately from the body. However, in reality, the soul and body interact closely. However, the fundamental independence of the soul from the body is for Descartes the guarantee of the probable immortality of the soul.
Doctrine of God
From philosophical psychology, Descartes moves on to the doctrine of God. He gives several proofs of the existence of a supreme being. The most famous is the so-called “ontological argument”: God is an all-perfect being, therefore the concept of him cannot lack the predicate of external existence, which means it is impossible to deny the existence of God without falling into contradiction. Another proof offered by Descartes is more original (the first was well known in medieval philosophy): in our mind there is an idea of God, this idea must have a cause, but the cause can only be God himself, since otherwise the idea of a higher reality would be generated by the fact that it does not possess this reality, that is, there would be more reality in the action than in the cause, which is absurd. The third argument is based on the necessity of God's existence to sustain human existence. Descartes believed that God, while not in himself bound by the laws of human truth, is nevertheless the source of man’s “innate knowledge,” which includes the very idea of God, as well as logical and mathematical axioms. Descartes believes that our faith in the existence of the external material world comes from God. God cannot be a deceiver, and therefore this faith is true, and the material world really exists.
Philosophy of nature
Having convinced himself of the existence of the material world, Descartes began to study its properties. The main property of material things is extension, which can appear in various modifications. Descartes denies the existence of empty space on the grounds that wherever there is extension, there is also an “extended thing,” res extensa. Other qualities of matter are vaguely conceived and, perhaps, Descartes believes, exist only in perception, and are absent in the objects themselves. Matter consists of the elements fire, air and earth, the only difference being their size. Elements are not indivisible and can transform into each other. Trying to reconcile the concept of discreteness of matter with the thesis about the absence of emptiness, Descartes puts forward a very interesting thesis about instability and the absence of a definite form in the smallest particles of matter. Descartes recognizes collision as the only way to convey interactions between elements and things consisting of their mixture. It occurs according to the laws of constancy, arising from the unchanging essence of God. In the absence of external influences, things do not change their state and move in a straight line, which is a symbol of constancy. In addition, Descartes talks about the conservation of the original momentum in the world. Movement itself, however, is not initially inherent in matter, but is introduced into it by God. But just one initial push is enough for a correct and harmonious cosmos to gradually assemble independently from the chaos of matter.
Body and soul
Descartes devoted a lot of time to studying the laws of functioning of animal organisms. He considered them to be subtle machines capable of independently adapting to environment and respond appropriately to external influences. The experienced effect is transmitted to the brain, which is a reservoir of “animal spirits”, tiny particles, the entry of which into the muscles through the pores that open due to deviations of the brain “pineal gland” (which is the seat of the soul), leads to contractions of these muscles. The movement of the body is composed of a sequence of such contractions. Animals have no souls and do not need them. Descartes said that he was more surprised by the presence of a soul in humans than by its absence in animals. The presence of a soul in a person, however, is not useless, since the soul can correct the natural reactions of the body.
Descartes the physiologist
Descartes studied the structure of various organs in animals and examined the structure of embryos at various stages of development. His doctrine of “voluntary” and “involuntary” movements laid the foundations modern teaching about reflexes. The works of Descartes presented schemes of reflex reactions with the centripetal and centrifugal parts of the reflex arc.
The significance of Descartes' works in mathematics and physics
Descartes' natural scientific achievements were born as a “by-product” of the unified method of a unified science he developed. Descartes is credited with creating modern systems notations: he introduced signs for variables (x, y, z.), coefficients (a, b, c.), notation for degrees (a2, x-1.).
Descartes is one of the authors of the theory of equations: he formulated the rule of signs for determining the number of positive and negative roots, raised the question of the boundaries of real roots and put forward the problem of reducibility, that is, the representation of an entire rational function with rational coefficients in the form of a product of two functions of this kind. He pointed out that an equation of the 3rd degree is solvable in square radicals (and also indicated a solution using a compass and straightedge if the equation is reducible).
Descartes is one of the creators of analytical geometry (which he developed simultaneously with P. Fermat), which made it possible to algebraize this science using the coordinate method. The coordinate system he proposed received his name. In his work “Geometry” (1637), which opened the interpenetration of algebra and geometry, Descartes first introduced the concepts of a variable quantity and a function. He interprets a variable in two ways: as a segment of variable length and constant direction (the current coordinate of a point that describes a curve with its movement) and as a continuous numerical variable running through a set of numbers expressing this segment. In the field of study of geometry, Descartes included “geometric” lines (later called algebraic by Leibniz) - lines described by hinged mechanisms in motion. He excluded transcendental curves (Descartes himself calls them “mechanical”) from his geometry. In connection with the study of lenses (see below), "Geometry" sets out methods for constructing normals and tangents to plane curves.
"Geometry" had a huge influence on the development of mathematics. In the Cartesian coordinate system, negative numbers received a real interpretation. Real numbers Descartes actually interpreted it as the relation of any segment to a unit (although the formulation itself was given later by I. Newton). Descartes' correspondence also contains his other discoveries.
In optics, he discovered the law of refraction of light rays at the boundary of two different media (set out in Dioptrics, 1637). Descartes made a major contribution to physics by giving a clear formulation of the law of inertia.
Influence of Descartes
Descartes had a tremendous influence on subsequent science and philosophy. European thinkers adopted his calls for the creation of philosophy as an exact science (B. Spinoza) and for the construction of metaphysics on the basis of the doctrine of the soul (J. Locke, D. Hume). Descartes also intensified theological debate on the possibility of proving the existence of God. Descartes' discussion of the question of the interaction of soul and body, to which N. Malebranche, G. Leibniz and others responded, as well as his cosmogonic constructions had a huge resonance. Many thinkers made attempts to formalize Descartes' methodology (A. Arnauld, N. Nicole, B. Pascal). In the 20th century, Descartes' philosophy is often referred to by participants in numerous discussions on the problems of philosophy of mind and cognitive psychology.
In order to develop this approach, which is understandable and natural for us now, it took the efforts of many scientists over eighteen centuries from Can Tsang to Descartes.
The natural numbers, their opposites and the number 0 are called integers. Positive numbers(integers and fractions), negative numbers(integers and fractions) and the number 0 form a group rational numbers.
Rational numbers are designated large Latin letter R. The number 0 refers to rational integers. We learned about natural and fractional positive numbers earlier. Let's take a closer look at negative numbers as part of rational numbers.
Negative number has been associated with the word “debt” since ancient times, while positive number can be associated with the words “availability” or “income”. This means that positive integers and fractional numbers in calculations, this is what we have, and negative integers and fractions are what constitute debt. Accordingly, the result of the calculation is the difference between the available amount and our debts.
Negative integers and fractions are written with a minus sign (“-”) in front of the number. The numerical value of a negative number is its modulus. Respectively, modulus of number is the value of a number (both positive and negative) with a plus sign. Number modulus written like this: |2|; |-2|.
To everyone rational number there is a single point corresponding to the number line. Let's look at the number axis (figure below), mark a point on it ABOUT.
Point ABOUT let's match the number 0. The number 0 serves as the boundary between positive and negative numbers: to the right of 0 - positive numbers, the value of which varies from 0 to plus infinity, and to the left of 0 - negative numbers, the value of which also varies from 0 to minus infinity.
Rule. Any number to the right of the number line is greater than the number to the left.
Based on this rule, positive numbers increase from left to right, and negative numbers decrease from right to left (at the same time, the module of a negative number increases).
Properties of numbers on the number line
Every positive number and 0 are greater than any negative number.
Every positive number is greater than 0. Every negative number is less than 0.
Every negative number is less positive number. A positive or negative number to the right is greater than a positive or negative number to the left on the number line.
Definition. Numbers that differ from each other only in sign are called opposite numbers.
For example, the numbers 2 and -2, 6 and -6. -10 and 10. Opposite numbers located on the number axis in opposite directions from point O, but at the same distance from it.
Fractional numbers representing ordinary or decimal, follow the same rules on the number line as integers. Of two fractions, the one to the right on the number axis is larger; negative fractions are smaller than positive fractions; every positive fraction is greater than 0; every negative fraction is less than 0.
Velmyakina Kristina and Nikolaeva Evgenia
This research work is aimed at studying the use of positive and negative numbers in human life.
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MBOU "Gymnasium No. 1" of the Kovylkinsky municipal district
Application of positive and negative numbers in human life
Research work
Completed:
6B class students
Velmyakina Kristina and Nikolaeva Evgenia
Head: teacher of mathematics and computer science
Sokolova Natalya Sergeevna
Kovylkino 2015
Introduction 2
1.The history of positive and negative numbers 4
2.Use of positive and negative numbers 6
Conclusion 13
List of used literature 14
Introduction
The introduction of positive and negative numbers was associated with the need to develop mathematics as a science that gives general methods solving arithmetic problems, regardless of the specific content and initial numerical data.
Having studied positive and negative numbers in mathematics lessons, we decided to find out where else besides mathematics these numbers are used. And it turned out that positive and negative numbers have quite wide application.
This research work aims to study the use of positive and negative numbers in human life.
The relevance of this topic lies in the study of the use of positive and negative numbers.
Purpose of the work: Explore the use of positive and negative numbers in human life.
Object of study:Areas of application of positive and negative numbers in human life.
Subject of research:Positive and negative numbers.
Research method:reading and analyzing the literature used and observations.
To achieve the goal of the study, the following tasks were set:
1. Study the literature on this topic.
2. Understand the essence of positive and negative numbers in human life.
3. Explore the applications of positive and negative numbers in various fields.
4. Draw conclusions.
- The history of positive and negative numbers
Positive and negative numbers first appeared in Ancient China about 2100 years ago.
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 and positive 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 (- a) + (- a) = - 2a. 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, and negative ones were “fu” and were depicted in black. This method of representation 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 depicted negative numbers were crossed out diagonally from right to left. Although Chinese scientists explained negative quantities as debt, and positive quantities as property, they still avoided the broad one. using them, since these numbers seemed incomprehensible, the 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 V-VI centuries, negative numbers appear and spread very widely in Indian mathematics. 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.”
The "+" and "-" signs were widely used in trade. Winemakers put a “-” sign on empty barrels, indicating decline. If the barrel was filled, the sign was crossed out and a “+” sign was received, meaning profit. These signs were introduced as mathematical ones by Jan Widmann in XV.
In European science, negative and positive numbers finally came into use only since the time of the French mathematician R. Descartes (1596 - 1650), who gave a geometric interpretation of positive and negative numbers as directed segments. In 1637 he introduced the "coordinate line".
In 1831, Gauss fully substantiated that negative numbers are absolutely equivalent in rights to positive ones, and the fact that they cannot be applied in all cases does not matter.
The history of the emergence of negative and positive numbers ends in the 19th century when William Hamilton and Hermann Grassmann created a complete theory of positive and negative numbers. From this moment the history of the development of this mathematical concept begins.
- Using positive and negative numbers
- Medicine
Myopia and farsightedness
Negative numbers express eye pathology. Myopia (myopia) is manifested by decreased visual acuity. In order for the eye to see distant objects clearly in case of myopia, diverging (negative) lenses are used.Myopia (-), farsightedness (+).
Farsightedness (hyperopia) is a type of eye refraction in which the image of an object is focused not on a certain area of the retina, but in the plane behind it. This state of the visual system leads to blurred images perceived by the retina.
The cause of farsightedness can be a shortened eyeball, or a weak refractive power of the optical media of the eye. By increasing it, you can ensure that the rays will focus where they focus during normal vision.
With age, vision, especially near vision, increasingly deteriorates due to a decrease in the accommodative ability of the eye due to age-related changes in the lens - the elasticity of the lens decreases, the muscles that hold it weaken, and as a result, vision decreases. That's whyage-related farsightedness (presbyopia ) is present in almost all people after 40–50 years.
With low degrees of farsightedness, high vision is usually maintained both at distance and near, but there may be complaints about fatigue, headache, dizziness. With moderate hypermetropia, distance vision remains good, but near vision is difficult. With high farsightedness, there is poor vision both far and near, since all the eye’s ability to focus images of even distant objects on the retina has been exhausted.
Farsightedness, including age-related, can only be detected through carefuldiagnostic examination (with medicinal dilation of the pupil, the lens relaxes and the true refraction of the eye appears).
Myopia is an eye disease in which a person has difficulty seeing objects located far away, but sees objects that are close well. Nearsightedness is also called myopia.
It is believed that about eight hundred million people are myopic. Everyone can suffer from myopia: both adults and children.
Our eyes contain a cornea and a lens. These components of the eye are capable of transmitting rays by refracting them. And an image appears on the retina. Then this image becomes nerve impulses and is transmitted along the optic nerve to the brain.
If the cornea and lens refract the rays so that the focus is on the retina, then the image will be clear. Therefore, people without any eye diseases will see well.
With myopia, the image appears blurry and unclear. This may happen for the following reasons:
– if the eye elongates greatly, the retina moves away from the stable focus location. In people with myopia, the eye reaches thirty millimeters. And in a normal healthy person, the size of the eye is twenty-three to twenty-four millimeters; - if the lens and cornea refract the light rays too much.
According to statistics, every third person on earth suffers from myopia, that is, myopia. It is difficult for such people to see objects that are far from them. But at the same time, if a book or notebook is located close to the eyes of a person who is myopic, then he will see these objects well.
2) Thermometers
Let's look at the scale of a regular outdoor 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 (scale 2).
3) Balance on the phone
When checking the balance on your phone or tablet, you can see a number with a sign (-), this means that this subscriber has a debt and cannot make a call until he tops up his account, a number without a sign (-) means that he can call or make any -or other function.
- Sea level
Let's look at physical card peace. The land areas on it are painted in various shades of green and brown colors, and the seas and oceans are painted 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, for example, this:
Scale of depths and heights in meters
Deeper 5000 2000 200 0 200 1000 2000 4000 higher
On this scale we see only positive numbers and zero. The height (and depth too) at which the surface of the water in the World Ocean is located is taken as zero. Using only non-negative numbers in this scale is inconvenient for a mathematician or physicist. The physicist comes up with such a scale.
Height scale in meters
Less -5000 -2000 -200 0 200 1000 2000 4000 more
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.
5) Human qualities
Each person is individual and unique! However, we do not always think about what character traits define us as a person, what attracts people to us and what repels us. Highlight positive and negative qualities person. For example, positive qualities are activity, nobility, dynamism, courage, enterprise, determination, independence, courage, honesty, energy, negative qualities, aggressiveness, hot temper, competitiveness, criticality, stubbornness, selfishness.
6) Physics and comb
Place several small pieces of tissue paper on the table. Take a clean, dry plastic comb and run it through your hair 2-3 times. When combing your hair, you should hear a slight crackling sound. Then slowly move the comb towards the scraps of paper. You will see that they are first attracted to the comb and then repelled from it.
The same comb can attract water. This attraction is easy to observe if you bring a comb to a thin stream of water flowing calmly from a tap. You will see that the stream is noticeably bent.
Now roll up two tubes 2-3 cm long from thin paper (preferably tissue paper). and a diameter of 0.5 cm. Hang them side by side (so that they lightly touch each other) on silk threads. After combing your hair, touch the paper tubes with the comb - they will immediately move apart and remain in this position (that is, the threads will be deflected). We see that the tubes repel each other.
If you have a glass rod (or tube, or test tube) and a piece of silk fabric, then the experiments can be continued.
Rub the stick on the silk and bring it to the scraps of paper - they will begin to “jump” onto the stick in the same way as on the comb, and then slide off it. The stream of water is also deflected by the glass rod, and the paper tubes that you touch with the rod repel each other.
Now take one stick, which you touched with a comb, and the second tube, and bring it to each other. You will see that they are attracted to each other. So, in these experiments, attractive and repulsive forces are manifested. In experiments, we saw that charged objects (physicists say charged bodies) can be attracted to each other, and can also repel each other. This is explained by the fact that there are two types, two types of electric charges, and charges of the same type repel each other, and charges different types are attracted.
7) Counting time
IN different countries differently. 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 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 NE). Our era continues for two thousand years. Consider the “time line” in the figure.
Foundation Beginning First mention of Moscow Birth of A. S. Pushkin
Rome revolt
Spartak
Conclusion
Working with various sources and exploring various phenomena and processes, we found out that negative and positive are used in medicine, physics, geography, history, modern means communication, in the study of human qualities and other areas of human activity. This topic is relevant and is widely used and actively used by people.
This activity can be used in math lessons to motivate students to learn about positive and negative numbers.
List of used literature
- Vigasin A.A., Goder G.I., “History ancient world", textbook 5th grade, 2001.
- Vygovskaya V.V. “Lesson-based developments in Mathematics: 6th grade” - M.: VAKO, 2008.
- Newspaper "Mathematics" No. 4, 2010.
- Gelfman E.G. "Positive and Negative Numbers" training manual in mathematics for the 6th grade, 2001.
Negative numbers are numbers with a minus sign (−), for example −1, −2, −3. Reads like: minus one, minus two, minus three.
Application example negative numbers is a thermometer that shows the temperature of the body, air, soil or water. IN winter time, when it is very cold outside, the temperature can be negative (or, as people say, “minus”).
For example, −10 degrees cold:
The ordinary numbers that we looked at earlier, such as 1, 2, 3, are called positive. Positive numbers are numbers with a plus sign (+).
When writing positive numbers, the + sign is not written down, which is why we see the numbers 1, 2, 3 that are familiar to us. But we should keep in mind that these positive numbers look like this: +1, +2, +3.
Lesson contentThis is a straight line on which all numbers are located: both negative and positive. Looks like this:
The numbers shown here are from −5 to 5. In fact, the coordinate line is infinite. The figure shows only a small fragment of it.
Numbers on the coordinate line are marked as dots. Bold in the picture black dot is the starting point. The countdown starts from zero. Negative numbers are marked to the left of the origin, and positive numbers to the right.
The coordinate line continues indefinitely on both sides. Infinity in mathematics is symbolized by the symbol ∞. The negative direction will be indicated by the symbol −∞, and the positive direction by the symbol +∞. Then we can say that all numbers from minus infinity to plus infinity are located on the coordinate line:
Each point on the coordinate line has its own name and coordinate. Name is any Latin letter. Coordinate is a number that shows the position of a point on this line. Simply put, a coordinate is the very number that we want to mark on the coordinate line.
For example, point A(2) reads as "point A with coordinate 2" and will be denoted on the coordinate line as follows:
Here A is the name of the point, 2 is the coordinate of the point A.
Example 2. Point B(4) reads as "point B with coordinate 4"
Here B is the name of the point, 4 is the coordinate of the point B.
Example 3. Point M(−3) reads as "point M with coordinate minus three" and will be denoted on the coordinate line as follows:
Here M is the name of the point, −3 is the coordinate of point M .
Points can be designated by any letters. But it is generally accepted to denote them in capital Latin letters. Moreover, the beginning of the report, which is otherwise called origin usually denoted by the capital Latin letter O
It is easy to notice that negative numbers lie to the left relative to the origin, and positive numbers lie to the right.
There are phrases such as “the further to the left, the less” And "the further to the right, the more". You probably already guessed what we are talking about. With each step to the left, the number will decrease downward. And with each step to the right the number will increase. An arrow pointing to the right indicates a positive reference direction.
Comparing negative and positive numbers
Rule 1. Any negative number is less than any positive number.
For example, let's compare two numbers: −5 and 3. Minus five less than three, despite the fact that five strikes the eye first of all as a number greater than three.
This is due to the fact that −5 is a negative number, and 3 is positive. On the coordinate line you can see where the numbers −5 and 3 are located
It can be seen that −5 lies to the left, and 3 to the right. And we said that “the further to the left, the less” . And the rule says that any negative number is less than any positive number. It follows that
−5 < 3
"Minus five is less than three"
Rule 2. Of two negative numbers, the one that is located to the left on the coordinate line is smaller.
For example, let's compare the numbers −4 and −1. Minus four less, than minus one.
This is again due to the fact that on the coordinate line −4 is located to the left than −1
It can be seen that −4 lies to the left, and −1 to the right. And we said that “the further to the left, the less” . And the rule says that of two negative numbers, the one that is located to the left on the coordinate line is smaller. It follows that
Minus four is less than minus one
Rule 3. Zero is greater than any negative number.
For example, let's compare 0 and −3. Zero more than minus three. This is due to the fact that on the coordinate line 0 is located more to the right than −3
It can be seen that 0 lies to the right, and −3 to the left. And we said that "the further to the right, the more" . And the rule says that zero is greater than any negative number. It follows that
Zero is greater than minus three
Rule 4. Zero is less than any positive number.
For example, let's compare 0 and 4. Zero less, than 4. This is in principle clear and true. But we will try to see this with our own eyes, again on the coordinate line:
It can be seen that on the coordinate line 0 is located to the left, and 4 to the right. And we said that “the further to the left, the less” . And the rule says that zero is less than any positive number. It follows that
Zero is less than four
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