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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.

3.14159265358979323846264338327950288419716939937510582097494459230781
6406286208998628034825342117067982148086513282306647093844609550582231
7253594081284811174502841027019385211055596446229489549303819644288109
7566593344612847564823378678316527120190914564856692346034861045432664
8213393607260249141273724587006606315588174881520920962829254091715364
3678925903600113305305488204665213841469519415116094330572703657595919
5309218611738193261179310511854807446237996274956735188575272489122793
8183011949129833673362440656643086021394946395224737190702179860943702
7705392171762931767523846748184676694051320005681271452635608277857713
4275778960917363717872146844090122495343014654958537105079227968925892
3542019956112129021960864034418159813629774771309960518707211349999998
3729780499510597317328160963185950244594553469083026425223082533446850
3526193118817101000313783875288658753320838142061717766914730359825349
0428755468731159562863882353787593751957781857780532171226806613001927
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/   
    
Associated Topics:
High School Number Theory
High School Transcendental Numbers
Middle School Number Sense/About Numbers

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