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

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]

Flight of a Bird

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 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.  

In the  formulas, for precision (ad nauseum) we say

1. the lowercase Greek letter   p is constant given by 3.1416 (approximately) 
2. the uppercase Roman letter A stands for the area of the circle or rectangle (depending on which one you are looking at), 
3. the lowercase  Roman letter r stands for the radius of the circle, 
4. the uppercase  Roman letter H stands for the height of the rectangle, '
5. the uppercase Roman letter W stands for its width,  
6. the lowercase Roman letter p stands for the perimeter of the rectangle, and
7. 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.  

1. When should we say that a letter or shorthand symbol is variable? 
2. 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.  

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby,  All Rights Reserved.