Integers– natural numbers are numbers that are used to count objects. The set of all natural numbers is sometimes called the natural series: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, etc.
To write natural numbers, ten digits are used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using them, you can write any natural number. This notation of numbers is called decimal.
The natural series of numbers can be continued indefinitely. There is no such number that would be the last, because you can always add one to the last number and you will get a number that is already greater than the one you are looking for. In this case, they say that there is no greatest number in the natural series.
Places of natural numbers
When writing any number using digits, the place in which the digit appears in the number is critical. For example, the number 3 means: 3 units, if it appears in the last place in the number; 3 tens, if she is in the penultimate place in the number; 4 hundred if she is in third place from the end.
The last digit means the units place, the penultimate digit means the tens place, and the 3 from the end means the hundreds place.
Single and multi-digit numbers
If any digit of a number contains the digit 0, this means that there are no units in this digit.
The number 0 is used to denote the number zero. Zero is “not one”.
Zero is not a natural number. Although some mathematicians think differently.
If a number consists of one digit it is called single-digit, if it consists of two it is called two-digit, if it consists of three it is called three-digit, etc.
Numbers that are not single-digit are also called multi-digit.
Digit classes for reading large natural numbers
To read large natural numbers, the number is divided into groups of three digits, starting from the right edge. These groups are called classes.
The first three digits on the right edge make up the units class, the next three are the thousands class, and the next three are the millions class.
Million – one thousand thousand; the abbreviation million is used for recording. 1 million = 1,000,000.
A billion = a thousand million. For recording, use the abbreviation billion. 1 billion = 1,000,000,000.
Example of writing and reading
This number has 15 units in the class of billions, 389 units in the class of millions, zero units in the class of thousands, and 286 units in the class of units.
This number reads like this: 15 billion 389 million 286.
Read numbers from left to right. Take turns calling the number of units of each class and then adding the name of the class.
What are natural and non-natural numbers? How to explain to a child, or maybe not a child, what are the differences between them? Let's figure it out. As far as we know, non-natural and natural numbers are studied in the 5th grade, and our goal is to explain to students so that they really understand and learn what and how.
Story
Natural numbers are one of the old concepts. A long time ago, when people did not yet know how to count and had no idea about numbers, when they needed to count something, for example, fish, animals, they beat out various subjects dots or dashes, as archaeologists later found out. Life was very difficult for them at that time, but civilization developed first to the Roman number system and then to the decimal number system. Nowadays almost everyone uses Arabic numerals
All about natural numbers
Natural numbers are prime numbers that we use in our daily lives to count objects in order to determine quantity and order. Currently, to write numbers we use decimal system Reckoning. In order to write down any number, we use ten digits - from zero to nine.
Natural numbers are those numbers that we use when counting objects or indicating the serial number of something. Example: 5, 368, 99, 3684.
A number series refers to natural numbers that are arranged in ascending order, i.e. from one to infinity. This series starts with smallest number- 1, and there is no largest natural number, since the series of numbers is simply infinite.
In general, zero - natural number is not counted, since it means the absence of something, and there is also no counting of objects
The Arabic number system is a modern system that we use every day. It is a variant of Indian (decimal).
This number system became modern because of the number 0, which was invented by the Arabs. Before this, it was not available in the Indian system.
Unnatural numbers. What is this?
Natural numbers do not include negative numbers or non-integers. This means that they are - unnatural numbers
Below are examples.
Non-natural numbers are:
- Negative numbers, for example: -1, -5, -36.. and so on.
- Rational numbers, which are expressed in decimal fractions: 4.5, -67, 44.6.
- In the form of a simple fraction: 1 / 2, 40 2 /7, etc.
Irrational numbers such as e = 2.71828, √2 = 1.41421 and the like.
We hope that we have greatly helped you understand non-natural and natural numbers. Now it will be easier for you to explain to your baby this topic, and he will master it as well as the great mathematicians!
Mathematics emerged from general philosophy around the sixth century BC. e., and from that moment her victorious march around the world began. Each stage of development introduced something new - elementary counting evolved, transformed into differential and integral calculus, centuries passed, formulas became more and more confusing, and the moment came when “the most complex mathematics began - all numbers disappeared from it.” But what was the basis?
The beginning of time
Natural numbers appeared along with the first ones mathematical operations. One spine, two spines, three spines... They appeared thanks to Indian scientists who developed the first positional
The word “positionality” means that the location of each digit in a number is strictly defined and corresponds to its rank. For example, the numbers 784 and 487 are the same numbers, but the numbers are not equivalent, since the first includes 7 hundreds, while the second only 4. The Indian innovation was picked up by the Arabs, who brought the numbers to the form that we know Now.
In ancient times, numbers were given a mystical meaning; Pythagoras believed that number underlies the creation of the world along with the basic elements - fire, water, earth, air. If we consider everything only from the mathematical side, then what is a natural number? The field of natural numbers is denoted as N and is an infinite series of numbers that are integers and positive: 1, 2, 3, … + ∞. Zero is excluded. Used primarily to count items and indicate order.
What is it in mathematics? Peano's axioms
Field N is the basic one on which elementary mathematics is based. Over time, fields of integer, rational,
The work of the Italian mathematician Giuseppe Peano made possible the further structuring of arithmetic, achieved its formality and prepared the way for further conclusions that went beyond the field area N.
What is a natural number was clarified earlier in simple language, below we will consider a mathematical definition based on Peano’s axioms.
- One is considered a natural number.
- The number that follows a natural number is a natural number.
- There is no natural number before one.
- If the number b follows both the number c and the number d, then c=d.
- An axiom of induction, which in turn shows what a natural number is: if some statement that depends on a parameter is true for the number 1, then we assume that it also works for the number n from the field of natural numbers N. Then the statement is also true for n =1 from the field of natural numbers N.
Basic operations for the field of natural numbers
Since field N was the first for mathematical calculations, both the domains of definition and the ranges of values of a number of operations below belong to it. They are closed and not. The main difference is that closed operations are guaranteed to leave the result within the set N, regardless of what numbers are involved. It is enough that they are natural. The outcome of other numerical interactions is no longer so clear and directly depends on what kind of numbers are involved in the expression, since it may contradict the main definition. So, closed operations:
- addition - x + y = z, where x, y, z are included in the N field;
- multiplication - x * y = z, where x, y, z are included in the N field;
- exponentiation - x y, where x, y are included in the N field.
The remaining operations, the result of which may not exist in the context of the definition of “what is a natural number,” are as follows:
Properties of numbers belonging to the field N
All further mathematical reasoning will be based on the following properties, the most trivial, but no less important.
- The commutative property of addition is x + y = y + x, where the numbers x, y are included in the field N. Or the well-known “the sum does not change by changing the places of the terms.”
- The commutative property of multiplication is x * y = y * x, where the numbers x, y are included in the N field.
- The combinational property of addition is (x + y) + z = x + (y + z), where x, y, z are included in the N field.
- The matching property of multiplication is (x * y) * z = x * (y * z), where the numbers x, y, z are included in the N field.
- distributive property - x (y + z) = x * y + x * z, where the numbers x, y, z are included in the N field.
Pythagorean table
One of the first steps in students’ knowledge of the entire structure of elementary mathematics after they have understood for themselves which numbers are called natural numbers is the Pythagorean table. It can be considered not only from the point of view of science, but also as a most valuable scientific monument.
This multiplication table has undergone a number of changes over time: zero has been removed from it, and numbers from 1 to 10 represent themselves, without taking into account orders (hundreds, thousands...). It is a table in which the row and column headings are numbers, and the contents of the cells where they intersect are equal to their product.
In teaching practice last decades there was a need to memorize the Pythagorean table “in order,” that is, memorization came first. Multiplication by 1 was excluded because the result was a multiplier of 1 or greater. Meanwhile, in the table with the naked eye you can notice a pattern: the product of numbers increases by one step, which is equal to the title of the line. Thus, the second factor shows us how many times we need to take the first one in order to obtain the desired product. This system is much more convenient than the one that was practiced in the Middle Ages: even understanding what a natural number is and how trivial it is, people managed to complicate their everyday counting by using a system that was based on powers of two.
Subset as the cradle of mathematics
On this moment the field of natural numbers N is considered only as one of the subsets complex numbers, but this does not make them less valuable in science. Natural number is the first thing a child learns when studying himself and the world. One finger, two fingers... Thanks to him, a person develops logical thinking, as well as the ability to determine cause and deduce effect, paving the way for great discoveries.
Where does learning mathematics begin? Yes, that's right, from studying natural numbers and operations with them.Integers (fromlat. naturalis- natural; natural numbers) -numbers that occur naturally when counting (for example, 1, 2, 3, 4, 5, 6, 7, 8, 9...). The sequence of all natural numbers arranged in ascending order is called a natural series.
There are two approaches to defining natural numbers:
- counting (numbering) items ( first, second, third, fourth, fifth"…);
- natural numbers are numbers that arise when quantity designation items ( 0 items, 1 item, 2 items, 3 items, 4 items, 5 items ).
In the first case, the series of natural numbers begins with one, in the second - with zero. There is no consensus among most mathematicians on whether the first or second approach is preferable (that is, whether zero should be considered a natural number or not). The overwhelming majority of Russian sources traditionally adopt the first approach. The second approach, for example, is used in the worksNicolas Bourbaki , where the natural numbers are defined aspower finite sets .
Negative and integer (rational , real ,...) numbers are not considered natural numbers.
The set of all natural numbers usually denoted by the symbol N (fromlat. naturalis- natural). The set of natural numbers is infinite, since for any natural number n there is a natural number greater than n.
The presence of zero makes it easier to formulate and prove many theorems in the arithmetic of natural numbers, so the first approach introduces useful concept extended natural range , including zero. The extended series is designated N 0 or Z 0 .
TOclosed operations (operations that do not derive a result from the set of natural numbers) on natural numbers include the following arithmetic operations:
- addition: term + term = sum;
- multiplication: factor × factor = product;
- exponentiation: a b , where a is the base of the degree, b is the exponent. If a and b are natural numbers, then the result will be a natural number.
Additionally, two more operations are considered (from a formal point of view, they are not operations on natural numbers, since they are not defined for allpairs of numbers (sometimes exist, sometimes not)):
- subtraction: minuend - subtrahend = difference. In this case, the minuend must be greater than the subtrahend (or equal to it, if we consider zero to be a natural number)
- division with remainder: dividend / divisor = (quotient, remainder). The quotient p and the remainder r from dividing a by b are defined as follows: a=p*r+b, with 0<=r
It should be noted that the operations of addition and multiplication are fundamental. In particular,