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YOU are better than YOU think. Show yourself how: |
-/[]\- Logic
Mastery Do not leave here without it - 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 linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6; 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. |
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Rename The Greater Than Sign
Shifty Mathematics: Primary school students are taught to use the phrases greater than and less than in comparing the size or magnitude of decimals, unsigned numbers. High school students in the discussion of real numbers have this meaning taken away and replaced by another. This shift in meaning while comparing signed and unsigned numbers if not noticed, could be source of confusion. Shifty mathematics terminology is not recommended.
The mathematical usage of terms or words sometimes drifts away from the common usage. Here the drift is from the first usage in mathematics and the common usage. Both coincide.
Why Say "More Positive Than" Instead of "Greater Than"
The concept of greater than or more than is understood by students when dealing with counts or unsigned whole numbers. Before the introduction of signs, that is negative and positive numbers, finite decimal expansions extend this idea of greater than or more than. A finite decimal expansion in particular counts the number of units, tenths, thousandths and so on that the number it represents can be divided into. Beyond this, students may be shown or pointed to the comparison of (unsigned) numbers with infinite decimal expansions.
With the introduction of positive and negative numbers and zero on say the real number line, the technical ideas of greater than differs from the common usage, or the introductory idea of comparison of by size or magnitude (apart from any signs that may be present). Because of this students are tempted to say that a real number a is greater than another real number b if the magnitude of a is greater than the magnitude of b. The latter means real number a is greater in magnitude than another real number b.
The task is to remove the temptation or conflict. The symbol > traditional has been called the greater than sign. Technically, given two real numbers a and b we write a>b if and only if there is positive number c such that a = b + c. So a is c units more positive than b.
To avoid confusion, and to align mathematical terminology with the common usage, the symbol > should be named or renamed the more positive than symbol on first usage in mathematics. This new name corresponds precisely to the technical meaning. With this new convention, the phrase a greater than b can revert to the common usage and mean |a| > |b|, a comparison of magnitude. Similarly, a < b can be read not as a is less than b but as a is more negative than b.
This new terminology means there is a positive number c such that a = b - c or equivalently such that a + c = b. The signs <= and >= now may be read as more negative or equal to and more positive or equal to.
Linear and Nonlinear Orderings (optional)
A number b is said to between two other numbers a and c if and only if there is a positive number q < 1 such that b=qa + (1-q)c.
Ordering of the real line by the relationship more positive than provides a linear ordering of the real line: for any three points a, b and c on the real line the relationships a < b <c imply that b is between a and c.
Ordering by magnitude provides a linear ordering of the positive numbers. For any three unsigned (positive) numbers a, b and c, the relationship a < b < c implies that b is between a and c. But for any three points a, b and c on the real line the relationship |a| < |b| < |c| does not imply that b is between a and c. So ordering by magnitude (or absolute value) of points on the whole real line is nonlinear.
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In Volume 2, Three Skills for Algebra, Chapters 8 to 14 and postscript What is a Variable point to a greater & clear use of words in algebra. Chapter 14 introduces a 4th skill for algebra, an elaboration of the third: - The direct and indirect use of formulas, numerically and algebraically, is unifying theme that should be mentioned aloud, with words, in each and every use of formula.