Natural numbers. They serve to designate the number of elements that a certain set has , and it is called the cardinal of said set .
Summary
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- 1 Story
- 2 natural numbers
- 3 Axiomatic of Peano
- 4 Representation of natural numbers
- 5 Adding natural numbers, properties
- 1 Associative
- 2 Commutative
- 3 Cancellation
- 4 Neutral element
- 6 Subtraction of natural numbers, properties
- 1 It is not an internal operation:
- 2 It is not commutative:
- 7 Multiplication of natural numbers and their properties
- 1 Internal:
- 2 Associative:
- 3 Commutative:
- 4 Neutral element:
- 5 Distributive:
- 6 Extract common factor:
- 8 Division of natural numbers
- 1 Types of divisions
- 2 Properties of division of natural numbers
- 9 Powers of natural numbers
- 1 Properties of the powers of natural numbers
- 10 See also
- 11 References
- 12 Sources
History
Before the numbers for the representation of quantities arose , the human being used other methods of counting, using objects such as stones, wooden sticks, rope knots, or simply fingers. Later, graphic symbols began to appear as counting signs, for example marks on a stick or simply specific strokes on the sand. But it was in Mesopotamia around the year 4,000 BC. C. where the first vestiges of the numbers appear, which consisted of engravings of signs in the form of wedges on small clay tablets using a pointed stick. that’s the reason for the name cuneiform script.
This numbering system was later adopted, albeit with different graphic symbols, in ancient Greece and ancient Rome . In ancient Greece , the letters of its alphabet were simply used, while in ancient Rome, in addition to the letters, some symbols were used. The contribution of the Mayans who managed a vigesimal numbering system is an indicator of the scientific advance of pre-Columbian America.
In the introduction to Arithmetic of the Greek mathematician, Nicomache of Gerasa, it is the “natural series”. In a review of this work, the Roman Neoplatonic philosopher, Boethius, used the phrase numeri naturalis for the first time . Loyal to the Greek roots he considered the one “mother of all other numbers”; but without assigning one the status of number. The Gallic mathematician, Cauchy, “proved” that the set of natural numbers was finite and “in this way, science leads us to the same result as faith.” The sense that a natural number currently carries is due to D’Alembert: infinite succession. [one]
Natural numbers
Natural numbers are the first to appear in different civilizations, since the tasks of counting and ordering are the most elementary tasks that can be performed in the treatment of quantities.
Natural numbers are the numbers we use to count; one, two, three, four, etc. We give them a name, “Natural numbers” to distinguish them from other numbers, such as “one half”, “four thirds”, “three point seven”, “minus five”; that is, of the fractional numbers (1/2), the numbers with a decimal point (3.7) and the negative numbers (-5).
The natural numbers are infinite. The set of all of them is designated by N:
N = {0,1, 2, 3, 4, …, 10, 11, 12, …} [2]
As can be seen, zero is not included in the set of natural numbers.
In addition to cardinals (to count), natural numbers are ordinal, since they serve to order the elements of a set :
1st (first), 2nd (second), …, 16th (sixteenth), …
The natural numbers are ordered, allowing us to compare two natural numbers:
5> 3; 5 is greater than 3.
3 <5; 3 is less than 5.
The natural numbers are unlimited, if we add 1 to a natural number, we obtain another natural number.
Axiomatic of Peano
They are considered as primitive or undefined concepts: 1, natural and successive number.
1.- 1 (one) is a natural number.
2.- For each natural number m there is a natural number called successive and is denoted S (m) .
3.- For every natural number m , S (m) is different from 1.
4.- The equation S (m) = S (p) implies m = p.
5.- The set of natural numbers, which contains 1 and for each of m elements, the successive element S (m), contains all the natural numbers. this axiom is
flame: Principle of complete induction . [3] .
The addition and multiplication of natural numbers are defined by the equations
n + 1 = S (n)
m + S (n) = S (n + 1)
n * 1 = n
n * S (m) = n * m + n
Representation of natural numbers
Natural numbers can be represented on a line ordered from least to greatest.
On a line we indicate a point , which we mark with the number zero . To the right of zero , and with the same separations, we place the following natural numbers from smallest to largest: 1, 2, 3 …
Adding natural numbers, properties
a + b = c
The terms of the sum , a and b, are called addends and the result, c, sum .
The result of adding two natural numbers is another natural number .
Associative
The way of grouping the addends does not vary the result.
(a + b) + c = a + (b + c)
Commutative
The order of the addends does not vary the sum .
a + b = b + a
Cancellative
a + c = b + c implies a = b
This property, expanding, can be expressed as
a + c = b + c yes, only if a = b
For strict order relationships
a + c <b + c yes, only if a <b
a + c> b + c yes, only if a> b
In the case of wide-order relationships
a + c ≤ b + c if, only if a ≤ b
a + c ≥ b + c yes, only if a ≥ b
Neutral element
0 is the neutral element in the extended set N 0 = {0,, 2, …, n, …}, with respect to the sum because every number added with it gives the same number .
a + 0 = a
Subtraction of natural numbers, properties
a – b = c, we say that c is the difference from a and b whenever a ≥ b. In turn a- b si, only if a = b + c.
The terms involved in a subtraction are called: a, minuendo and b, subtracting. The result, c, we call difference.
It is not an internal operation:
Subtraction or subtraction is a partially defined operation. It is not always possible to subtract two natural numbers and find another natural number that represents the difference. Therefore, N is extended to the set Z of all integers and identified with positive integers, and subtraction is always possible for any two positive integers.
2 – 5 Does not belong to the set of natural numbers
It is not commutative:
5 – 2 ≠ 2 – 5
Multiplication of natural numbers and their properties
Multiplying two natural numbers consists of adding one of the factors to itself as many times as the other factor indicates.
a · b = c
The terms a and b are called factors and the result, c, product.
Internal:
The result of multiplying two natural numbers is another natural number.
Associative:
The way of grouping the factors does not vary the result.
(a · b) · c = a · (b · c)
Commutative:
“The order of the factors does not vary the product.”
a · b = b · a
Neutral element:
1 is the neutral element of the multiplication of natural numbers, because every number times it gives the same number .
a · 1 = a
Distributive:
The multiplication of a natural number by a sum is equal to the sum of the multiplications of that natural number by each of the addends.
a · (b + c) = a · b + a · c
Take out common factor:
It is the reverse process of the distributive property.
If several addends have a common factor, we can transform the sum into a product by extracting that factor.
a · b + a · c = a · (b + c)
Division of natural numbers
D: d = c
The terms involved in a division are called, D, dividend and, d, divisor. We call the result, c, the quotient.
Types of divisions
- Exact division:
A division is exact when the remainder is zero .
D = d · c
15 = 5 · 3
- Whole division:
A division is an integer when the remainder is other than zero .
D = d · c + r
17 = 5 · 3 + 2
Properties of division of natural numbers
- It is not an internal operation:
The result of dividing two natural numbers is not always another natural number.
2: 6
does not belong to the natives
- It is not commutative:
a: b ≠ b: a
6: 2 ≠ 2: 6
- Zero divided by any number is zero.
0: 5 = 0
- It cannot be divided by 0.
The division by zero is indeterminate, as if to: 0 = c, a = 0 × entoces C holds for all values of c.
Powers of natural numbers
A power is a shorthand way of writing a product made up of several equal factors.
Base : The base of a power is the number that we multiply by itself, in this case the 5.
Exponent : The exponent of a power indicates the number of times we multiply the base, in the example it is 4.
Properties of the powers of natural numbers
–
– Product of powers with the same base: It is another power with the same base and whose exponent is the sum of the exponents.
– Division of powers with the same base: It is another power with the same base and whose exponent is the difference of the exponents.
– Power of a power: It is another power with the same base and whose exponent is the product of the exponents.
– Product of powers with the same exponent: It is another power with the same exponent and whose base is the product of the bases.
– Quotient of powers with the same exponent: It is another power with the same exponent and whose base is the quotient of the bases.
– Polynomial decomposition of a number.
A natural number can be decomposed using powers of base 10.
We can decompose the number 3 658 as follows:
Radication is an inverse operation of potentiation [4] . And it consists in that given two numbers, called radicand and index, find a third, called root, such that, raised to the index, it is equal to the radicand.
In the square root, the index is 2, although in this case it is omitted. It consists of finding a number when its square is known.
The square root of a number, a, is exact when we find a number, b, which squared equals the radicand: b² = a.
√9 = 3
The exact square root has remainder 0.
Perfect squares: They are the numbers that have exact square roots.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, …
Integer square root: If a number is not a perfect square, its root is integer.
Cases and examples
Given the natural number a, we will say that the number r, if it exists. is the square root of a and is written at 0.5 = r, when a = r 2
For example, the square root of 144 is 12, since 144 = 12 2
In this profile, only the numbers that are perfect squares have a square root. To avoid this restriction, the default integer square root is defined .
We will say that r is the default integer square root of a if r 2 <a <(r + 1) 2 . The difference d = a-r 2 is called the remainder of the square root of a.
As an example, the default integer square root of 175 is 13, since 13 2 <175 <(13 +1) 2 , equivalently 169 <175 <196. at 175-169 = 6, the remainder of the integer square root is called default 175.
In the case of 144/529 its square root is 12/23; however when it comes to other fractional numbers or other positive integers, the result is not always a rational number. A case, in fact historical, proved that the square root of 2 is not a rational number. Why the interest in knowing and specifying the square root of 2? For if we know that the side of a square is 1, we have that the length of the diagonal is precisely the square root of 2. This results from applying the Pythagorean theorem to the isosceles right triangle whose legs are two consecutive sides of the square and its hypotenuse is the diagonal of the square. There is no fraction that is the square root of 2. Hence the concept arises that the square root of 2 is an irrational number. Decimal approximations can be made, but the last number is not reached. In the application work tasks it is enough with five or six decimal places. An approximation rc of 2 = 1.414213562 … up to a billionth
