What is a Variable?
©Alan Selby, August 2000.
www.whyslopes.com/freeAccess/sample_problem_set.html
This webpage presents the text of sample chapter
for a problem book that was solicited, but not written.
problems remain to be added.
Goal: Master the mathematical use of the word variable
Introduction
Look in a dictionary, encyclopedia and a mathematics text for a definition of
what is a variable, an introduction that is understandable to you and easily
explained to others. If you find such a definition or introduction clear enough
to help in mathematics after arithmetic, the rest of this essay need not be
read.
Alice in Wonderland if she could speak today, would observe
that the view of a variable as a function begs the question of how
to explain the notion of a function without using the concept of a variable.
The essay or chapter before put the concepts of what is a variable first and
before the use of symbols and notation in mathematics for numbers, amounts,
quantities and functions.
Variation in a Single Example
variation = amount of change
The next diagram shows the height of a bird during its journey from one tree
to another. The flight is over the ground intervals
[a,b], [b,c], [c,d], [d,e], [e,f]
Letters on horizontal axis end ground intervals where the height
behavior changes. If height is measured above or below sea level, and the
tops of both trees were below sea level, then increasing height would
correspond to make the height relative to sea level less negative.
Identify the intervals where the height of the bird is constant, where this
height is increasing (becoming more positive or less negative) and where this
height is decreasing (becoming less positive or more negative). The height may
have different behaviors on different ground or time intervals. This exercise
could be redone on a graph of height versus time. In this case, the ground
intervals would correspond to time intervals.
To vary means to change. Identify the ground intervals where the height of
the bird is constant (not variable) and where it is variable.
Conclusion: Whether or not a number or quantity is constant or not,
variable may depend on the interval in which is observed or examined or
remembered. We can talk about numbers and quantities being variable without or
before the use of letters to represent them.
The following diagram shows the speed of a car along a straight road.
Piecewise linear graph of speed versus time
Identify the time intervals where the speed of the car is constant and where
it is variable.
Challenge (a hard exercise): From the above
diagram, how would you find the distance traveled by the car in a constant-speed
interval and in the variable speed intervals. Find a solution without the use of
calculus. Hint: See an old high school physic text.
Variation between Examples
In the following diagram are rectangles with
different areas, heights and width.
Rectangles B, C and D
For each rectangle, its area, its height
and its width is constant, at least while the rectangle is not being
stretched. But each of the three quantities area, height and
width change or vary when we shift our attention from one rectangle to
another. So while our attention is fixed on one rectangle, these three
quantities are constant. Yet these three quantities change, are
variable, when we shift our attention from one rectangle to another. These
three quantities do not have the same value for each rectangle shown in the
diagram.
Conclusion: A
number or quantity may have a constant or fixed value in a single situation or a
single circumstance, but the number or quantity in question may vary or be
variable between different circumstances.
The next diagram shows or indicates the
number of people in a home during a day
![[Diagram showing 4 people from midnight to 8 am., 2 people from 8 am to 9 am, 1 from 9 am to 4 pm, 3 from 4 pm to 7 pm and 4 again from 7 pm to midnight.]](images/sample4.gif)
Diagram showing 4 people from midnight to
8 am, 2 people from 8 am to 9 am, 1 person from 9 am to 4 pm, 3 from 4 pm to 7
and 4 again from 7 pm to midnight.
During each hour the number of people is
constant. But the number of people is not constant for a full day because of
departures and arrival at 8 am, 9 am, 4pm and 7pm. So the number of people is
variable. During the small time intervals where people are leaving or entering,
you may have a person not fully in the house. During these small time intervals,
how to count or define the number of people is a matter of taste.
Food for thought: How would you count or define the number of people in the
house during these small transitions, time intervals? When you have 4 people in
the house, and 1 is leaving, my thought is that you should say there are 3 to 4
people in the house, but it may impolite to talk about fractions when speaking
of people. Saying you had 3.45 people to a party might lead to a criminal
investigation :)
Variation of Letters
Letters have not been used in the above
discussions of what numbers and quantities are variable, including when and in
what sense.
In the next diagram, letters and symbols
appear in formulas for the calculation of areas and of perimeters for a circle
and a rectangle.
Correction: For the circle: Area A = p r2
and Perimeter s = 2 p r
In the formulas, for precision (ad
nauseum) we say
- the lowercase Greek letter p
is constant given by 3.1416 (approximately)
- the uppercase Roman letter A stands for
the area of the circle or rectangle (depending on which one you are looking
at),
- the lowercase Roman letter r stands
for the radius of the circle,
- the uppercase Roman letter H stands
for the height of the rectangle, '
- the uppercase Roman letter W stands for
its width,
- the lowercase Roman letter p stands for
the perimeter of the rectangle, and
- the lowercase Roman letter s stands for
the perimeter of the circle.
The phrase "stands for" could be
replaced by the phrase "is shorthand for" or "is placeholder
for" or "stand-in for", or by the word "represents" or
"denotes". Some help with the English language follows.
- denotes
:
to mark, signify, mean, indicate, to be the name of.
- placeholder
:
keeper of a portion of space for an number or quantity or object in general.
- represents
:
stand for, symbolize, act as the embodiment of,
- shorthand:
a method for rapid writing and abbreviation
- stand for
:
act in the place of another.
- stand-in for
:
a deputy or substitute, for another actor.
You may meet other phrases that indicate the
shorthand role of letters as placeholders or notation or
abbreviations for numbers and quantities in calculations.
When does a letter denote a variable?
Letter as shorthand symbols for numbers and
quantities appear in the above formulas.
- When should we say that a letter or
shorthand symbol is variable?
- When should we call a letter or symbol a
variable.
Answers for both questions follow.
In the case of variation in a single example,
when a symbol or letter represents or stands for a number or quantity that may
vary, we will say that that symbol or letter is a variable, and we will call
it a variable as well. Think here of the height h of a bird or the
number n of people in the house in the diagrams given above and
reproduced below.
In the case of variation between examples,
when when a symbol or letter represents or stands for a number or quantity
that may vary, we will also say that that symbol or letter is a variable, and
we will call it a variable as well. Think here of the area A, height H
and width L of the rectangles in the next diagram.
For each rectangle, the numbers or quantities
denoted by A, L and W are constant, but between the rectangles, these three
quantities vary. So we say the symbols or placeholders A, L and W are
constant or variable, according to whether or not we are thinking about their
lack of variation for a single rectangle or their variation between
rectangles.
Old dictionaries and old algebra texts may
be half-right when they indicate without further explanation that variable is
letter used in mathematics, at least when we add the thought that a letter
denotes a number or quantity that may vary. Beyond this, the number or
quantity need not have a physical meaning. Think for instance of a number that
may be written by someone else and placed in an envelope for safe keeping or
privacy. Denoting that number by x allows us to describe calculations with that
number hidden in the envelope, with x as shorthand for it. Calculations
with a number placed in an envelope could also be described with the
abbreviation x before the number is actually placed in the envelope.
Cases of Double Variation
Ten people have ten piggy banks to which they
add and subtract spare coins. The value V of coins in each piggy bank depends
on the person and on time. So there here is an example of double
variation: variation over time for each piggy bank, and variation between
piggy banks at each moment.
Postscript for essay
What
is a Variable.
Diagram of rectangles with width constant over columns, but varying
along rows.
Height too varies
in one direction but not another. The notion of varying or not can be
understood before the use of symbols.
- Width is a
constant for each column, a constant that differs or varies between
columns. That may give a variable constant.
- Height is
variable for each column, but this variable is constant along rows.
That may give a constant variable :)
If you change the width of this page
(resize your browser window), the width may also vary over time.
Conclusion or recapitulation
Numbers and
quantities may vary
- in one or more
spatial directions
- over time
- between
examples
all at once or
separately.
Numbers and
quantities may vary in different directions (spatial or temporal) and
between discrete instances
|
To learn more, see
Three Notions of a Variable
Constants,Parameters,Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
| |
Algebra, Odds & Ends,
1. Hints for Exams
2A. Exact Arithmetic
2B. Fractions Briefly
3. What is a Variable?
4.. Square Roots
5. Straight Lines
6. Problem Solving Methods
7. Trig and Complex No.
8. Complex Applet
9. History of No.s
10. ln(x) and exp(x)
13. Rename the > Sign
14. Problems: Quadratics
15. Problems: Algebra Test
16. Problems: Linear Eqns I
17. Problems: Linear Eqns II
18. Problem Solving Hints
20. Independent Variables
21. Why Logic
22. Why Math
23. The 15 Times Table
24. The 20 Times Table
25. Algebra Formulas
26. On Learning Maths
27. Maths in Biology
28. Navigation +Time
29 Quibble-What is Algebra
30. Logic in Maths
Odd and Ends, Essays
Constant Retirement Rate
Road Safety
3 Strikes Law in California.
Math HOW-TOs
9 Steps in Maths
|
|
// _ _ \\
/\
/\
<| (o) (o) |>
|
| | |
\
/
\ = /
|
Caution: Site advice
is approximately correct, for some circumstances, not all.
Site How-TOs are
logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
|
|