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YOU are better than YOU think. Show yourself how:
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-/[]\- Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite. Logic
Mastery 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;
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-/[]\- 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. |
Real Numbers AxiomsIn your studies may meet axioms (assumptions) for real numbers after your mastery of arithmetic. This page provides a manual for computations with real numbers: how to compute if you must. This page also aims to provide a better understanding of assumptions made, explicitly or not, in mathematics from elementary school to college. There is a gap in that elementary mathematics teaches decimals but high school and college courses offer a thought-based description of mathematics in which decimals are not mentioned. The last pure mathematics books on arithmetic were written in the 1920's or 30's. Assumption A1. There is a set R = (-oo, + oo) of "real numbers", each of which has a finite decimal expansion preceded by a sign (+ or -). This set can be represented by a coordinate system on a straight line where the unit of distance is taken to be 1. The scale of the representation depends on the length of this unit distance on the line. (In the next lesson on complex numbers, we will take this line to be the horizontal axis of a rectangular coordinate system for the plane.)
Assumption A2. If a is real then a times 0 = 0. Multiplication of Unsigned NumbersFrom elementary arithmetic, we can compute or approximate products ab from their decimal expansions when a and b are nonnegative. Here we assume or observe the following assumptions. Assumption A3 (Commutative law for Multiplication): if a and b are unsigned real numbers then ab = ba Assumption A4 (Associative Law of Multiplication): if a, b and c are unsigned real numbers then (ab)c = a(bc) Multiplication of Signed Numberssign multiplicationPut sign(a) = 1 when a > 0, put sign(a) = 0 when a = 0 and put sign(a) = -1 when a < 0. We assume the law of signs (1)(1) = 1 (1)(-1) = -1 (-1)(1) = -1 (-1)(-1) = 1 with 0 = (0)(0) = (0)(-1) = (0)(1) = (-1)(0) = (1)(0). The law of signs implies sign(a) * sign(a) = 1 except when a = 0. Assumption A6: if a is a real number then a = sign(a) p where p is zero or an unsigned decimal expansion. Now if a = sign(a) p and b = sign(b) q then we may compute sign(a) * sign(b) from the law of signs and we may compute or approximate pq using decimal arithmetic. When both a and b are positive, a = q and b = p and ab = pq. But when when a or b is non-positive, we need to say how to compute ab. The computation rule is a follows. We put ab = (sign(a) * sign(b)) pq and take sign(ab) = sign(a) * sign(b). Observe this formula first given for the case when one factor a or b is non-positive yields pq, the same result as before, when a and b are positive. So ab = (sign(a) * sign(b)) pq for all real numbers. Assumption A7 (computational rule): If a = sign(a) p and b = sign(b) q are real numbers then ab = (sign(a) * sign(b)) pq This describes the multiplication of real numbers in terms of signs and decimal expansions. This definition implies no real squared gives a negative result.
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