الأعداد الطبيعية

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Numbers

Date: 12/10/97 at 19:14:39

From: Joanna Greeno

Subject: Numbers

How do integers and whole numbers, rational numbers, transcendental

numbers, and counting numbers relate to each other?

Date: 12/10/97 at 21:41:50

From: Doctor Luis

Subject: Re: Numbers

NATURAL NUMBERS (COUNTING NUMBERS)

First of all, there are the natural numbers. These are the most

fundamental of all and they are sometimes referred as counting numbers

Natural Numbers 1,2,3,4,5,....

Here, the dots are taken to signify that the list goes on forever.

They give us a basic and primitive notion of infinity. That is why the

set of natural numbers is so essential to mathematics.

Although you cannot literally "count" how many natural numbers there

are, mathematicians have given a name to this "infinitely" large

number that represents how many natural numbers there are. It is

usually called "aleph nought" (aleph is the Hebrew letter "A"). Now,

this is a different kind of number from what we are used to, and you

can use it to define the cardinal numbers, which in a way let us speak

about various degrees of infinities. For example, there are more real

numbers than natural numbers, and more irrational numbers than

rational numbers.

Anyway, we now have the natural numbers.

WHOLE NUMBERS

If you add zero to your list then you have the whole numbers,

Whole numbers 0,1,2,3,....

The number zero has an interesting history. The ancient Mayans were

the first to introduce the concept of zero as a number, centuries ago.

Why did we have to introduce a symbol to denote "nothing" or a "zero

quantity" of something? The reason for this is apparent when you start

considering numbers that are too large for us to represent by a single

symbol.

What are some ways to represent numbers? Well, we can start naming

them; that is, representing each number by a symbol (they would all

have to be different). Eventually, we would run out of symbols, or the

list of symbols would be too large for us to remember (imagine trying

to multiply numbers - you'd have to remember ALL the symbols for every

number and memorize an INFINITELY long multiplication table.)

Is there any way around this problem? Fortunately, there is a very

clever way to represent numbers that solves most of our problems. It

is called place value notation.

PLACE VALUE NOTATION

In place value notation, you assign a value to each symbol according

to where it is in the number representation (its place). (The value

you assign is related to the base of your number system.)

Example: 123 (base 10)

In the example, the symbol 3 is worth 3 units BUT the 2 is worth 2*10

units, and the 1 is worth 1*100 units.

This allows us to represent more numbers with a shorter list of

symbols. However, in order to represent the entire set of natural

numbers in place value notation you NEED the number zero (that way you

can say that you DON'T have any units of 100, for example, in the

number 1025).

This completes our representation of the natural numbers very nicely,

AND it gives us a cool new number to work with. Okay. We have the

natural numbers and the whole numbers.

INTEGERS

To get the integers, just add all the negative numbers to your list!

Integers ...,-3,-2,-1,0,1,2,3,....

The dots before the -3 mean that the list continues FROM NEGATIVE

infinity, so that if you go back all the way, you'll see all of the

negative numbers.

Note: for obvious reasons, the natural numbers are often called

positive integers; and the whole numbers are also often

called nonnegative integers.

So far, our use of number has been restricted to mean integers.

However, you can define different kinds of numbers as well

RATIONAL NUMBERS

Now, the rational numbers are simply defined as ratios of integers.

1/2 is a rational number. 2/3 is also a rational number. Note that all

of the integers are rational numbers, because you can think of them as

the ratio of themselves to 1, as in 2 = 2/1 which is certainly the

ratio of two integers, and so 2 is a rational number.

The story of the rational numbers is also interesting. Ancient Greek

mathematicians were very fond of rational numbers. In fact, when they

discovered that there were other numbers which were not rational, they

swore that "terrible" discovery to secrecy.

IRRATIONAL NUMBERS

The square root of 2 is a classic example of an irrational number: you

cannot write it as the ratio of ANY two integers. Here's the square

root of 2 to a few decimal places (it continues forever, and there is

no pattern!):

sqrt(2) = 1.414213562373095048801688724209698078570

TRANSCENDENTAL AND ALGEBRAIC NUMBERS

To understand transcendental numbers, you also need to understand

another type of numbers called algebraic numbers.

A number is called algebraic if it is the root of a polynomial (of any

degree) with rational coefficients. Any number that is not algebraic

is called transcendental.

2 and 2/3 are algebraic (they are also rational) because they are the

roots of the rational polynomial 3x^2 - 8x + 4 (it is rational because

the coefficients are rational numbers)

3x^2 - 8x + 4 = 0

or (3x - 2)(x - 2) = 0 so x = 2 or 2/3

Note that the square root of 2 is also algebraic (it is also

irrational) because it is a solution of the rational polynomial

x^2 - 2 = 0

A classic example of an transcendental number (that is, not algebraic)

is the number pi (shown here to an accuracy of a thousand decimal

digits) It goes on forever, just as the square root of 2.

3.141592653589793238462643383279502884197169399375 10582097494459230781

64062862089986280348253421170679821480865132823066 47093844609550582231

72535940812848111745028410270193852110555964462294 89549303819644288109

75665933446128475648233786783165271201909145648566 92346034861045432664

82133936072602491412737245870066063155881748815209 20962829254091715364

36789259036001133053054882046652138414695194151160 94330572703657595919

53092186117381932611793105118548074462379962749567 35188575272489122793

81830119491298336733624406566430860213949463952247 37190702179860943702

77053921717629317675238467481846766940513200056812 71452635608277857713

42757789609173637178721468440901224953430146549585 37105079227968925892

35420199561121290219608640344181598136297747713099 60518707211349999998

37297804995105973173281609631859502445945534690830 26425223082533446850

35261931188171010003137838752886587533208381420617 17766914730359825349

04287554687311595628638823537875937519577818577805 32171226806613001927

876611195909216420199....

It turns out that pi is also irrational (you cannot write it as the

ratio of two integers).

It is impossible to cover all possible aspects of these numbers in a

single page, so I have kept their description at a bare minimum. If

you are interested and want to find out more, you can find a lot of

information in any good history of mathematics book.

If you have any questions, feel free to reply.

-Doctor Luis, The Math Forum

Check out our web site! http://mathforum.org/dr.math/

بسم الله الرحمن الرحيم
الاخ العزيز والمثابر (نبيل)
السلام عليكم ورحمة الله وبركاته
اولا : نحن اخوة والأختلاف وارد في ما بيننا فاتمنى ان لايضيق صدرك كما نحن لايضيق صدرنا
بالنسبة للاخ الربيعي فهو من حقه الاقتباس (انظر في حاشية كل مشاركة تجد رد مع الاقتباس) وهو يعني تضمين المشاركة السابقة قبل الرد او التعليق عليها وذا كنت قد حذفت المشاركة الاصية وتود حذفها من الاقتباس فنحن والاخ الربيعي كذلك بالخدمة مع العلم ان الاستاذ الربيعي من مشرفي المنتدى ومن حقه حذف او اضافة او تثبيت اي مشاركة

بالنسبة ( لمصيبة الصفر) التي رزءنا به او ان ابين ما يلي

ظهور الصفر قبل ظهور فلسفة الرياضيات التي ظهرت مع مبدا الشك لديكارت(على حد علمي) وبهذا فهو كان قبل هذه الفترة (الصفر) من ألاعداد الطبيعية بلامنازع
وعندما بدء علماء الرياضيات بايجاد تعريف حديثة لكل التعاريف والمسلمات السابقة بدأ الاختلاف وكما قلنا هو امر طبيعي
لكن وأن ذهب بعض العلماء إلى(طرد) الصفر من الاعداد الطبيعية فهذا لايعني ان ليس طبيعيا فأنت تعلم ان (المرحوم) ( أنشتاين) مات ولم يؤمن بنظرية الكم وهذا لايعني بطلان النظرية
فأتمنى لك التوفيق ولاستمرار بهذه الشماركات الجميلة لتضيف للمنتدى هذا النقاش الجميل لكن دون كلمات قد تؤذي البعض تحياتي لك مرة أخرى

بسم الله الرحمن الرحيم
السلام عليكم ورحمة الله وبركاته
اشكر جميع الاساتذه على التوضيح
المقدم من حضارتهم
(فراس الناصري حاجم الربيعي نبيل عبدالحميد السيد الناصري)
وان شاء الله وصلت المعلومه للجميع
وهذا هو الاهم
والسلام عليكم ورحمة الله وبركاته
اخوكم الزبيدي

حدد بيانو العالم الايطالي الذي عاش بين( 1858 - 1932 ) بديهيات سميت باسمه وهي :
ط1 :الصفر عدد طبيعي .
ط 2: لكل عدد طبيعي أ يوجد عدد طبيعي آخر يليه نرمز له أ+
ط3: لا يكون الصفر تاليا لاي عدد .
ط4: اذا تلا عدد طبيعي واحد عددين طبيعين كانا متساويين اذا كان أ+ =ب+ يؤدي الى أ = ب
ط5: اذا كانت ج مجموعة جزئية من ط وحققت الشرطين :
0 ينتمي ج ، هـ ينتمي الى ج يؤدي هـ+ ينتمي الى ج فان ج = ط .

الصفر هو عدد زوجي لانه عند رسم خط الاعداد الصحيحة فأن الصفر يقع بين 1- و 1+