Three Notions of a Variable

What is a Variable, Sections:  [ Up ] [ Variation between Examples ] [ Variation of Letters ] [ When does a letter denote a variable ] [ Cases of Double Variation ] [ Three Notions of a Variable ] [ Constants ] [ Talking about numbers ] [ Dependent or Independent Variables ]

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.

Chapter subsections: [ Up ] [ Variation between Examples ] [ Variation of Letters ] [ When does a letter denote a variable ] [ Cases of Double Variation ] [ Three Notions of a Variable ] [ Constants ] [ Talking about numbers ] [ Dependent or Independent Variables ]

Next: Constants and Parameters

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  Three Skills for Algebra 
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Foreword, Chapters 
& Appendices 

Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Romeo and Juliet
4. Induction Mathematical
5 Knowledge Islands
6  Old Language
7  Arith Skill Check
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable
9. Algebra Talk
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
PS I.  Distributive Law
PS II. Polynomials
16 Painless Proofs
17 Pythagoras
18 Rules of Algebra
19  Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums
23 Summation Notation
24 Your Money
25 Induction & Recursion
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Advice For Learning

Words Before Symbols: 
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent or Independent
Variable, a Matter of Choice

Complex number: starter lesson  

Solving Linear Equations:

A. Letters and Lengths
B. & C. Solving Linear Eq'ns
with stick diagrams.

(i) x + 20 = 29
(ii) 2x + 5 = 20
(iii) 3x + 10 = 32
(iv) 5a + 16 = 3a+ 24
(v)  (½)x + 8 = 24½
(vI)  (¾)a + 16 = (¼)a+ 24
(vii) (¾)q + 17 = 32
(viii) 13 =[2/3]x +7 twice
(x) Animated Examples
(i) Integral Coefficients (A)
(ii) Integral Coefficients (B)
(iii) Fractional Coefficients
(iv) With Parameters

Problem Solving with Linear
Equations in one or many
unknowns, and in essentially 
one unknown - Symbols before
words. 

C. Solving Linear Eq'ns 
without
Stick Diagrams
D. Problems in 
essentially one unknown
E: 2D Systems - Sub Methods.
F. Larger Systems

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Wordy Site Tour:  Not too Bad ,  Vol 2. Foreword,  1A Foreword,  2  Logic Puzzles,  
What is a Variable  Calculus Starter Guide  Vol 1 Foreword,    
 
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