Three Skills
For
Algebra
Volume 2
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Chapters and Appendices
Book Entrance
9 Numbers & Quantities
9 Everyday Words
9 Words Math Usage
9 Precision or Not
9 Numbers & Quantities
9 Changing Units
9 Further Readings
Foreword
1. Introduction
2. Implication Rules [4]
3. Chains of Reason [3]
4. Induction Mathematical
4. Romeo and Juliet
6 Old Language
5 Knowledge Islands [2]
7 Arith Skill Check [4 X 2]
Arith Webvideos
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
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
PS I. Distributive Law
PS II. Polynomials
18 Rules of Algebra [20]
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]
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
Pathways for Learning
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power users?
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
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Chapter 9
Talking about Numbers or Quantities
Chapter Sections: [ Up ] [ 9 Numbers & Quantities ] [ 9 Everyday Words ] [ 9 Words Math Usage ] [ 9 Precision or Not ] [ 9 Numbers & Quantities ] [ 9 Changing Units ] [ 9 Further Readings ]
3 Mathematical Usage of Words
The above examples show how everyday words are used to describe numbers and
quantities. Our next task is to say further or more precisely how the words
variable, constant, known and unknown are used to both describe and refer to
numbers and quantities.
See the postscript: What is a
Variable
A. Variables Versus Constants
To say that a number or quantity is variable means that the number or
quantity may vary or change. To say that a number or quantity is constant means
that its value remains unchanged. For example:
- The letter p »
3.14159... stands for or denotes a constant - a value or number which will
never change.
- The time of day is always changing. So time is varying. It is an example
of a number or quantity which is always increasing and therefore variable.
When you ask what time it is, you will get an approximate answer.
To complicate matters further, numbers and quantities may change in one
period and not in another. The height of house increases slowly as it built,
remains constant while it used, and decreases rapidly if it torn down. So this
height may be variable in some situations and constant in others. In everyday
life and in mathematics, when a number or quantity is called a constant, we
expect its value not to change in the situation at hand. Similarly, when a
number or quantity is called a variable, we should expect or suspect that its
value may change.
More examples: Your height is a variable or it was a variable while
you were growing. The speed of a car or a bicycle is an example of a variable (a
variable number or quantity that is). The speed of a car can be almost constant.
The zero speed of a stationary car or a parked car is constant - in one
reference system at least. Note that a number or quantity can be variable in one
situation, and constant in another. We can further talk about a previously
constant or a previously variable number or quantity.
In summary, the terms constant and variable can be used to talk about and
describe numbers and quantities. A constant is a number or quantity whose value
is expected not to change - whose value should not change. A variable is a
number or quantity whose value does or might change. The use of these terms is
flexible and context dependent. What is constant in one situation may be varying
or changing in another.5
5In
some algebra texts and in some dictionaries, the term variable means or refers
to the letters that appear in formulas. That use of the term variable departs
from the use and meaning given above. In my view, the mathematical usage of
everyday words should be in the first instance linked and extracted from their
ordinary usage. Where the mathematical usage has departed from the everyday
usage, we need to ask if that departure is necessary, and whether or not the
departure should be corrected. Documenting reasons or possibly causes
for such departures could be material for a thesis in linguistics.
B. Known Versus Unknown
Numbers and quantities can be known or unknown. You may know your own height,
age and weight, but I don't know your personal measurements. To you these
quantities are known. To me they are unknown. Whether they are known or not
depends on the company you keep - that is to whom you speak. When you see the
instruction find the unknown, you should ask the question: unknown to
whom? Note further in solving an equation, the solution of the equation goes
from being unknown to being known.
This is a note mainly for people who know how to solve
equations. See
the following chapter or chapters to learn how.
There is only one number x solving the equation
2x = 10. Before you solve this equation, its solution, the number
x is unknown to you. The solution is x = 5. When or as you
solve the equation (or see the solution), the number x becomes
known.
When you are only speaking about the solution x
of the equation 2x = 10, the solution is given by a constant. The
letter x stands for the constant, non-changing number 5.
Now in two different problems in which you solve for x,
their solutions x are often given by different numbers
(constants). Thus the value of the solution x may change as you
go from one problem to another. From this perspective, the solution x
can be also called a variable.
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For the sake of variety in our speech, numbers and quantities are also called
parameters. A parameter is another name for a number or a quantity. When we say
a number or quantity is a parameter, we have no immediate expectation that the
number or quantity in question will be constant nor that it will be variable.
The term parameter gives a vague expectation somewhere between constant and
variable. We can talk about numbers and quantities in precise and imprecise
ways.
Chapter Sections: [ Up ] [ 9 Numbers & Quantities ] [ 9 Everyday Words ] [ 9 Words Math Usage ] [ 9 Precision or Not ] [ 9 Numbers & Quantities ] [ 9 Changing Units ] [ 9 Further Readings ]
Next Section: 9-Talking about Numbers or
Quantities, Approximate Knowledge, Precision or not.
Next Topic: What
is a Variable:
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Online Volumes (orders)
1, Elements of Reason.
1996
1A. Pattern Based
Reason 1995
1B. Math
Curriculum Notes 1996
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3 .Why.Slopes.&.More.Math.1995
Skill &
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2 Fractions
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