Number Terminology

Date: 08/11/98 at 21:21:30
From: mindy
Subject: Algebra II

Hi. I have several questions concerning Algebra II.

First, my book defines a digit as numbers from which the numerals are 
made. But after looking at the definition, I still don't understand it. 
Could you provide some examples? 

Second, my book defines a transcendental number as a number that cannot 
be expressed as roots of integers. I also don't really understand that 
definition. Do you know if square root of 6 times negative 1 is a 
transcendental number? 

Also, how do you explain the difference between a rational and an 
irrational number? My teacher says that a rational number has a 
repeating decimal, and an irrational number doesn't. So, can 7.163 be a 
rational number?  

Finally, is counting number a synonym for natural number? And speaking 
of a natural number, how is it exactly defined? My guess is that a 
natural number is a number that is equal to or greater than one. 

I was wondering if you could explain all of the questions above and 
provide some examples along with the explanations. Thanks!

Date: 08/12/98 at 08:24:50
From: Doctor Jerry
Subject: Re: Algebra II

Hi Mindy,

I don't know why someone wants to define "digit" but I suppose that for 
the number 1294, the digits are 1, 2, 9, and 4. For the number 10.2, 
the digits are 1, 0, and 2. So, given a number x, just express it in 
decimal form. That is, write it as:

   x = d0 + d1*10^1 + d2*10^2 + ... + D1*10^{-1} + D2*10^{-2} + ...

The digits of x are surely d0, d1, ... , and D1, D2, ....  All belong 
to the set {0,1,2,...,9}.

A real number is algebraic if it is a root of a polynomial of the form 
an*x^n + ... + a1*x^1 + a0 = 0, where an, ... ,a1, and a0 are integers. 
If a number is not algebraic, then it is called transcendental. Thus, 
sqrt(2) is algebraic since it is a root of x^2 - 2 = 0. The number pi 
is not the root of any polynomial equation of the type mentioned above. 
This is not easy to prove.

A rational number is one that can be written as p/q where p and q are 
integers (positive or negative or p can be 0). Otherwise it is 
irrational. For example, sqrt(2) is not rational but 3/17 is. Rational 
numbers have periodic decimal representations. 7.163 is the same as 
7.163000...; that is, it has a repeating 0. Also:

   7.173 = 7 + 173/1000

which can be worked out to be a ratio of integers.

Yes, for the most part counting numbers are natural numbers. Some 
people may argue about 0. Is it a natural number or is it a counting 
number? These differences are not significant (in my opinion). Usually, 
the natural numbers are taken as 1, 2, 3, ....   

For more definitions and descriptions of integers and rational and 
irrational numbers, see the Dr. Math FAQ:

  http://mathforum.org/dr.math/faq/faq.integers.html   

- Doctor Jerry, The Math Forum
Check out our web site! http://mathforum.org/dr.math/