YOU are better than YOU think. Show yourself  how:  

      |      
//  _   _ \\
/\             /\
  <|  (o)   (o)   |> 
 \     | |      / 

 -/[]\- 
||
   / \_ 
 ||||||||||||||||||||||||||||

 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

      |      
//  _   _ \\
/\             /\
<|   (o)   (o)  |> 
|      | |     |
   \             /   
\    =   /

 -/[]\- 
||
  _ / \     
 ||||||||||||||||||||||||||||

What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Calculations with Simple and Compound Units

Mastery of formal operation with expression involving units is needed for proportionality computations or relationships between quantities or between numbers and quantities. We  do operations on (formal) expressions without requiring those expression have any meaning or physical interpretation.  That provide on paper exercises with marks or expression on paper.  In a following section on proportionality, multiples of simple and compound units will serve as proportionality constants and rates  in exercises that may some physical or social meaning.

Short Explanation

Examples show how to do formal operations on expressions  involving units. If you can see reproduce the patterns in these calculations, the longer version will be become optional or easier to follow.

Instructors:  (1) These operations with monomials involving units and their quotients  are similar to operations on monomials in variables x, y, z etc  and their quotients. The latter too (ouch) may represent formal operations on expressions that have no meaning for students other being marks on paper, albeit operations on monomials in  variables x, y, z etc could represent operations on potential calculations - the calculations that would result by replacing the variables by numbers or quantities. (2)  An operational command of calculations with units could be sufficient for further use in the representation of proportionality constants and for further use in calculation in chemistry, physics and money matters where units appear.  The short version above by itself or with further examples may provide that operational command. (3) Formal operations with monomials involving units, their products and  quotients takes on utility if not  meaning  in the subsequent appearance as rates and proportionality constants.  There-in lies a replacement or additions for examples of exponent addition and subtraction with monomials in one to several variables x, y, z, ... and their products or quotients.

Examples of Multiplication by a whole number or fraction: 

and

Example of Factoring coefficients: 

and

Addition Example

Terms in which units of each are identical can be combined by adding coefficients.

and

Multiplication Example

terms need not be identical: any pair of terms can be multiplied.

Reciprocal Calculation Example: 

Division Example

Division by a "fraction" is given by multiplication by the reciprocal of the fraction.

Pure Mathematics  (1) In the modern set-theoretic codification and axiomization of mathematics as seen in school and colleges, the codification is limited to pure numbers alone, in pairs, triplets and longer sequences, finite or infinite. Units are not discussed.  But as a service to the physical and social sciences, money matters included, the algebraic role of units in their computations can be codified or modeled using the theory of polynomials or formal power series and Laurent Expansions with real coefficients, with the formal variables being given by the units of a system of measurement.  (2) Development in one to four dimensions of the Lattice Point Codification of Product and Division Operations with Units could be an exercise with formal power series  for advanced undergraduate students of mathematitcs, those able to read  chapters I (all) and III (section 4) in Henri Cartan's work Elementary theory of analytic functions of one or several complex variables, translation of THEORIE ELEMENTAIRE DES FONCTIONS ANALYTIC D;UNE OU PLUSIEURS VARIABLES COMPLEX, to provide a codification of operation with units in which addition is limited perhaps to those monomials having the corresponding units to identical powers.  (3) Units may provide the basis of an algebra over the real (or complex numbers). Units of the same type are real (or complex) number multiples of each other.  Here changing units or changing the basis of an algebra corresponds to a  similarity transformation  on monomials with applications to similarity considerations in physical situations where computations should be and are required to be independent of the choice of units of measurement. 

[Top] [ Back ] [ Area Map & Intro ] [ Next ] [Site Exit] 
site entrance

Favourite Sites:  BBC News  and  mathematics portion of  English National Curriculum  
// Solving Linear equations //Fractions, Ratios, Rates, Proportionality //Logic & Algebra Revisited//
// Parent Center // Mathematics Curriculum Notes//Number Theory//
www.whyslopes.com

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, 1983 McGill Ph. D. in maths,  All Rights Reserved.