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

Would you like to show yourself or others how to be  algebra 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

Numbers and quantities are known, given, measured or estimated with varying precision. For instance, the cost of a hot dog could be 2.25 dollars. This cost is given exactly. In contrast, the height of a man might be between 5[1/2] and 6 feet and the weight of a truck could be between one and ten tons. In these two cases, the quantities in question are sandwiched or bracketed between two extreme values: the least and greatest possible. (The term sandwiched is preferred. It is more graphic.) The distance between the bracketing values measures the uncertainty in our knowledge.

⁷NOTE FOR ADVANCED STUDENTS: More precisely, if x is a number whose value is known to be between two positive number a and b with a £ b, then the mean value c = [(a+b)/2] gives an approximation to x. The absolute error in this approximation is £ [1/2]|b-a|. The percentage error in this approximation is £ 100·[1/2][(|b-a|)/(a)]%. The relative error in this approximation is £ [1/2][(|b-a|)/(a)]. To say that the percentage error is at most 1% indicates a better approximation than a percentage error of at most 5% or even 100%. In the above examples, note for instance the following: The height of the man is known within 100[(0.25ft)/(5.5ft)] = 4.55% £ 5%, a small (?) uncertainty. The weight of the truck is known within 100[(4.5tons)/(1ton)] = 450%. The uncertainty in the latter is large.The symbol £ is shorthand for the expression less than or equal to.

Knowledge of numbers and quantities may be exact or approximate. But we can still speak about them. We can also use approximate values in calculations and then hope the resulting error is not too large. Estimating errors in calculations is a useful topic which cannot be fully explored here. Error estimation is limited by the observation that perfect knowledge of the error in a computation would provide a means for removing the error. So error estimates must remain imperfect.

⁸Significant Digits etc: When you say that the height of a building is 10.47 meters (approximately) without giving any further information, the uncertainty in the last digit 7 should be £ [1/2]. When a single decimal is used to approximate a number or quantity, the digits in it are said to be significant when and only when the uncertainty in the last digit written is £ [1/2] of a unit. Digits which are uncertain by more than [1/2] should not be written when we report the result of a measurement or calculation.

Exception: When a single quantity x is bracketed between two others, say a and b, their mean value c = [(a+b)/2] provides an approximation to x with an error of at most d = [(|b-a|)/2]. In this case we may write x = c±d and keep some digits in the decimal expansion of c with an uncertainty in them of more than one half unit. Writing x = (10.472±0.003) meters for example provides more information about x than the single estimate x = 10.47 meters.

In some situations, the location of the last digit with an uncertainty of less than [1/2] of a unit may be unknown and this convention may be difficult to follow. Errors in long calculations may be minimized if rounding-off is postponed as long as possible, for instance done at the end of all calculations and not for intermediate results.

 [ Back ] [ Up ] [ Next ] [Top of this Page]  
  www.whyslopes.com

Road Safety Message  Do not walk on a road with your back to the traffic - rule of thumb
Please report by email,  errors in mathematics or grammar or terminology to site author
If a mathematics topic you need is not covered in site pages,  report that as well. Topics in most demand
will be covered first in site growth.  

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.