Associated Topics || Dr. Math Home || Search Dr. Math

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.

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
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/
```
Associated Topics:
High School Discrete Mathematics
High School Logic
High School Number Theory
High School Sets
Middle School Logic