Diagram for Math Numbers

Date: 10/05/97 at 22:44:33
From: Rina Hallock
Subject: Help with Tree Diagram of math terms....

My daughter is doing a tree diagram using terms related to math 
"numbers" (for a calculus assignment). Could you please explain in lay 
terms what surds are?  I don't remember ever having that term, and she 
can't find it in her resources.  

She has the branch with irrational numbers completed; we're working 
through the rational numbers. She has one branch off rational numbers 
started - whole numbers. Whole nubmers is branched to natural numbers 
and integers. I'm not sure that she has that right. The terms with 
which she is working include Real numbers, Surds, Opposites of
natural numbers, Even numbers, Repeating decimals, Non integers, 
Natural numbers, Irrational numbers, Transcendental numbers, Integers, 
Zero, Rational numers, Whole numbers, Odd numbers, Terminating 
decimals. 

Any help you can give would be greatly appreciated.

Thanks,

Rina Hallock

Date: 10/10/97 at 16:22:03
From: Doctor Chita
Subject: Re: Help with Tree Diagram of math terms....

Dear Rina:

Making a tree diagram of the various number systems is a nice way to 
show their relations - in particular, noting which sets of numbers are 
subsets of other numbers. 

For example, the natural numbers (also called "counting numbers) are a 
subset of the whole numbers. The natural numbers consist of the 
numbers {1, 2, 3, ... }, and the whole numbers consist of the numbers 
{0, 1, 2, 3, ... } The ellipsis (...) in each case indicates that the 
numbers go on forever, without bound. There is no subset of largest 
numbers.

The list that your daughter's teacher gave her includes several 
redundancies. For example, you asked about "surds". A surd is another 
name for an irrational number. Thus, sqrt(2) is a surd. It is also an 
irrational number. This means that when you try to evaluate it as a 
decimal, it neither repeats nor terminates. 

A decimal number that repeats is 0.3333... . A number that terminates 
is 0.75. In the first case, 0.333... = 1/3, and in the second case 
0.75 = 3/4. Both of these numbers are rational numbers because they 
can be expressed as the ratio between two integers.

Your daughter was correct in separating the real numbers into two 
mutually exclusive sets: the set of rational numbers and the set of 
irrational numbers. These two sets of numbers have no numbers in 
common: that is, a number is either rational or irrational. Each set 
is also infinite: that is, you can't count the number of numbers 
within either set.

                        Real Numbers
                        /          \
              Irrational No.       Rational Numbers

A Venn diagram is a good way to "see" the relations among different 
sets of numbers: 

        

The set of rational numbers includes the integers, the whole numbers, 
and the counting (natural) numbers, in that order. Therefore, any 
number that is a natural number is also a whole number, an integer, 
and a rational number.

Thus, starting from the bottom of the tree and proceding up, the 
number 3 is a natural number, a whole number, an integer, and a 
rational number. However, the number 0 is not a natural number - but 
it is a whole number, an integer, and a rational number. The number 
0.5 is rational only (it terminates). It is not an integer, a whole 
number, or a natural number.

The list your daughter was given contains the names of sets (such as 
the integers and the whole numbers) as well as names of elements that 
belong to one or more sets (for example, 0 and transcendental 
numbers). Therefore, to complete this classification tree, first make 
the tree, starting as shown in the text above. Then find examples of 
specific types of numbers and place them in their corresponding sets 
as shown in the Venn diagram.   

Your daughter is on the right track. Keep going.

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