What is a Variable?
©Alan Selby, August 2000.

Volume 2, Three Skills for Algebra

Goal:   Master the mathematical use of the word variable and the allied use of letters and symbols. 

This webpage and its sections on What is a Variable (not part of Three Skills for Algebra) is a postscript originally posted online in August 2000.  Arithmetic and algebraic expressions and formulas are like pictures better read in silence than spoken aloud. That situation has led to a lack of words in mathematics. The discussion of Three Skills for Algebra in Chapters 8 to 14 provide a wordy, or too wordy remedy with the following chapters being recommended.

See how much you swallow today, and return for the rest to perfect your skills and comprehension later. Note too, further skills for algebra, a fourth and/or fifth appear in Chapter 14. That chapter introduces two unifying themes for secondary school mathematics: (i) The forward and backward use of almost all formulas and equations; and (ii) The connection between arithmetic (or numerical) solution and algebraic solution methods in the backward use of formulas and equations. If you understand the algebraic solution, you are halfway to understanding the full strength use of algebra in senior high school mathematics and calculus. 

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. 

This essay  puts the concepts of what is a variable first and before the technical  use of symbols and notation in  mathematics for numbers, amounts, quantities and functions. 

In general, we may talk about and describe numbers and quantities as being variable or constant before and then  besides the use of letters to stand in, represent or denote them or functions.  The order here here  should make the shorthand roles of letters and symbols clearer and easier to learn and teach. .

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]

Flight of a Bird

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 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

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?

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

Three Notions of a Variable

What is a Variable, Sections:  [ Up ] [ Foreword ] [ 7  Arith Skill Check [4 X 2] ] [ 7. The Next Chapters ] [ 8 The Three Skills ] [ PS. What is a Variable [8] ] [ 9. Algebra Talk [7] ] [ 10 Two More Skills[5] ] [ 11 Why Shorthand ] [ 12 Shorthand Usage [10] ] [ 13 What's Next ] [ PS: The 4-th Skill For Algebra ] [ 14 Compound Interest [6] ] [ 15 Linear Equations [5] ] [ 16 Painless Proofs ] [ 17 Pythagoras ] [ 19  Functions & Sets ] [ 20 Degrees & Radians ] [ 21 What's Next ] [ 22. Arith & Geometric Sums [2] ] [ 23 Summation Notation ] [ 24 Your Money [3] ] [ 25 Induction & Recursion [4] ] [ Pathways for Learning ]

The concept of a variable is not simply described in most algebra texts. A clarification follows. This clarification is not for the expert, but for the novice. The specialized use of the term variable should not be the first one given in an algebra text or dictionary, mathematical or not.

A First Notion: Variables Without Symbols. We can talk about numbers and quantities, and among them identify those which are changing or varying, and those which are constant, known, unknown, given, confidential and so on. Here a number and quantity which may vary, or take many values in the circumstances of interest, is called a variable. We can talk about variables without using the shorthand notation, that is, letters and symbols, employed in algebra.Examples follow below. 

Second Notion: Variables with Symbols. Formulas use shorthand notation, symbols or letters, to represent numbers and quantities. This suggests that when a symbol or letter is the shorthand notation for a number or quantity which may vary, we may also call that symbol or letter a variable. 

Remark 1. The association of symbols and letters with numbers and quantities which may vary is so much a taken-for-granted part of the algebraic way of writing and thinking (amongst the mathematical adept) that the observation that we can talk about variables apart from symbols has been overlooked. But this symbol free notion clarifies and refines the concept of a variable in mathematics.

Remark 2. The notion that a variable may be given by a symbol, that is shorthand notation (or a place holder) for a number or quantity which may change, relies on our ability or skill (i) to talk about numbers and quantities and also on our ability or skill (ii) to employ shorthand notation (symbols) for them in and possibly outside calculations.

Third Notion: Variables and Computer Memory Locations:  Computers and calculators  may be used to store the values of numbers, amounts and quantities in named or labeled memory locations.    Computers or calculators may be programmed to use or change the stored values of numbers or quantities, values that may vary.  The values, the memory locations where they are stored, and the names or labels for them may be all be called variables.

Three Skills for Algebra

1. We can talk about numbers and quantities. The words or adjectives used here may be used in mathematics after arithmetic.  There is more to mathematics than just doing arithmetic.
2. We can describe calculations that might be done (or postponed) with words alone or with an (algebraic) shorthand notation. The description of calculations that might be done is also part of mathematics after arithmetic. There is more to mathematics than just doing arithmetic.
3. We can change the way a number or quantity is computed. Some rule-based reason is required here. There is more to mathematics than just doing arithmetic.

Talking about these skills and emphasizing them in examples shows there is more to mathematics than just doing arithmetic.

Constants, Variables, Parameters and Data
added June 23, 2005

When a number or quantity is a constant in one direction of change (over time, in space, between examples)  we say it is constant in that direction. If the direction is understood, we may call it a constant.

When a number or quantity is a variable in one direction of change (over time, in space, between examples)  we say it is variable in that direction.  If the direction is understood, we may call it a variable.

When a number or quantity is a variable in one direction of change (over time, in space, between examples) and constant in another  we say it is parameter in that direction.  So a parameter is a number or variable that may vary or be constant depending on which direction we look. 

If an observed number or quantity may be used in a table or in further calculations, we the number or quantity is question is part of the data of for the table or further calculations. 

Talking about numbers and quantities being constants, variables, parameter or data is part of words before and besides symbols and arithmetic in mathematics.

Talking about numbers, amounts and quantities,  and letters representing them 

Dec 1st, 2002, addition

What is a Variable, Sections:  [ Foreword ] [ 7  Arith Skill Check [4 X 2] ] [ 8 The Three Skills ] [ PS. What is a Variable [8] ] [ 9. Algebra Talk [7] ] [ 10 Two More Skills[5] ] [ 11 Why Shorthand ] [ 12 Shorthand Usage [10] ] [ 13 What's Next ] [ PS: The 4-th Skill For Algebra ] [ 14 Compound Interest [6] ] [ 15 Linear Equations [5] ] [ 16 Painless Proofs ] [ 17 Pythagoras ] [ 19  Functions & Sets ] [ 20 Degrees & Radians ] [ 21 What's Next ] [ 22. Arith & Geometric Sums [2] ] [ 23 Summation Notation ] [ 24 Your Money [3] ] [ 25 Induction & Recursion [4] ] [ Pathways for Learning ]

In mathematics crib sheets or dictionaries, we may see that a letter used in mathematics is a variable, or vice-versa, a letter that appears in an equation is a variable? Now the Greek letter q appears in trigonometry. It is a letter, so that it must be a variable according to above. On the other, what if I said that the letter represents a the measurement of a constant angle - an angle that will not change. Is q then a constant as well? The Greek letter p appears in formulas for perimeters and areas of circles, and in formulas for the volume and surface area of a sphere. Is this letter p to be called a variable?

Now the modern mathematics curriculum may say or define a variable as an element of of a set of numbers or a variable as a function of time or a function defined on a set, discrete or not, continuous or not, in one or more directions. The first definition assumes an understanding of sets and may come after the shorthand use of letters or variables in the development of mathematics. The second approach also comes after the concept of function (computation rules) and so may come after the shorthand use of letters and sets in the explanation of mathematics. Both views represent a mathematical codification of what is a variable in a context too complicated for novices to immediately grasp. A simpler comprehension may be put first.

But before the use of letters and symbols in mathematics - in the statement of formulas and/or the statement or description of properties of real numbers (eg associated law for addition), we can understand the everyday usage of what is a variable. We can talk about the height of a bird or the number of centimeters or meters in the height of its beak above the ground. That height may be constant or unchanging. That height may be increasing or decreasing while the bird is flying or falling. And while the height is changing, we will say it is variable. Here variation is over time, a variation that may later be viewed as a function. However, in the first instance, the common person in the street TCPITs may understand the notion of height varying as the bird moves. If we also talk about two different birds, each will have a beak height above the ground. So height may change or vary between birds. That introduces another sense of what it means for a number or quantity to vary. Variation may give us a set of values over time and position (or bird selection)

I favour the following notion of variable, a notion that can be understood before the use of letters and symbols in mathematics, and a notion that requires no knowledge of sets nor functions to understand and explain. Namely, a number, amount or quantity is a variable in a situation when its values may vary (vary from one situation or moment to another if you wish). This view is precise enough in the first instance to introduce the concept. Beyond this a letter or number that denotes a number, quantity or amount is said to be a 

variable, constant, known, unknown, given

when the value of the number, quantity or amount is respectively

variable, constant, known unknown or given.

This first concept may be codified or described later in terms of sets and functions, if need-be.

Dependent or Independent Variables,
A Matter of Choice

A = WL

That being said, if we choose to calculate A from the values of W and L, then A will be called the dependent quantity or variable while W and L will be called the independent variables or quantities. On the other hand, we choose to compute W in the equation A = WL from the values of A and L, then W = A/L and we will call W the dependent variable and will we call A and L the independent variables. So which variable is dependent or not depends on how we choose to use a formula, or which ones we choose to give or calculate. If we are situation where we know A = WL but we do not know which  will be given and which will be given, we cannot say which one will be dependent, albeit the form A = WL suggests A will be the dependent variable.  What we can say for sure or certain is the the equation A = WL links or relates the area, width and area of a given rectangle.  The set theoretic view or codification of a relation will be given below.

More, Multiple and Advanced Views of What is a Variable 

A. Variation between examples, over time or space View:  (i) In daily life, science, engineering and technology,  a number or quantity is a variable if that number or quantity changes (varies) between examples, problems or different different situation in direction of time or space, or discretely between elements of a list.  A number or quantity  may be constant in one and varying in another direction or sense.  (ii) In algebra,  before we speak of letters denoting variables, we should speak about how numbers and quantities may be variable and in that sense be variables along with any letter that denotes them. 

B. Computer Science View:    A variable is a letter in a formula or expression with a value to be given and used when the formula or expression is evaluated. (So calculators, spreadsheets and programming languages may help develop algebra skills and concepts.

C. PlaceHolder View:  A variable x  is a placeholder for an element of a  set. Here too is an echo of the role of sets in understanding and explaining  modern mathematics - providing a codification. 

D. Function View:  The expression 2x+1 may be identified with the function f where f(x) = 2x+1.  Likewise, the single term expression 3 and x determine two functions, the constant function g(x) = 3 and the the identity function  h(x) = x.  Thus a letter x which denotes an element of a set or a variable may be identified with the identity function  I(x) = x. Here is an echo of the role of sets in understanding and explaining  modern mathematics - providing a codification.

E. The mathematical study of logic may also include a further view - let me know.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science