|
Tutors - All Subjects YOU are better than YOU think. Show yourself how:
|
-/[]\- 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;
|
-/[]\- 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. |
More About Real NumbersDivision of Real NumbersWhen a and b are nonzero unsigned numbers, decimal arithmetic tells us how to compute the quotient a/b. Decimal arithmetic also says how to compute the reciprocal 1/b, another unsigned number. Here 1/b denotes the result of 1 divided by b. The latter reciprocal has the property b (1/b) = (1/b) b = 1. Division by zero is not defined. In decimal arithmetic, a/b = (a)(1/b) As the proof of this property belongs to a rigorous discussion of arithmetic, we assume it instead. Assumption A8: If a and b are unsigned numbers > 0, then division of a by b gives the same result as multiplying a by the reciprocal (1/b) of the divisor b. a/b = (a)(1/b) So division by an nonzero b > 0 can be replaced by multiplication by its reciprocal 1/b. This allows multiplication rules or properties of arithmetic to be applied in calculations where there is division. The latter can be replaced by multiplication by a reciprocal. Assumption A5: If a and b are unsigned numbers, the decimal computation of the reciprocal of the unsigned number (a/b) yields the same result as The reciprocal of a quotient.
The foregoing indented material implies (a/b) and (b/a) are reciprocals of each other in the sense the 1 divided by the quotient a/b gives the same result as b/a. This conclusion should agree with what you have learnt previously about fractions. Assumption A9: If a and b are nonzero unsigned numbers > 0 then [1 divided by (a/b)] = b/a Exercise: Use the commutative law for multiplication to show (1/b)(1/c)(bc) = 1 when b and c are nonzero unsigned numbers. Then compute (1/bc) - (1/b)(1/c). Division of Real Numbers:The foregoing handles the division with unsigned numbers. When a and b are both positive, the quotient a/b = a (1/b). Here the quotient a/b and the reciprocal1/b may be computed using decimal arithmetic. When a and b are not both positive, and both are nonzero, we may assume a = sign(a)p and b = sign(b) p where p and q are positive. Here the quotient p/q and the reciprocal1/q can be computed with decimal arithmetic. When a and b are positive, we perversely observe a/b = p/q = { (1)/(1) }(p/q) Now when a and b are not both positive, put a/b = (sign(a)/sign(b)) (p/q) with the convention that (1)/(-1) = -1 = (-1)/(1) and (1)/(1) = 1= (-1)/(-1) This computation rule defines multiplication of real numbers with signs. The formula a/b = (sign(a)/sign(b)) (p/q) also hold when a and b are unsigned numbers > 0. Exercise: Show 1/sign(b) = sign(b) Assumption A10 (Computation Rule): if a = sign(a)p and b = sign(b) p with p > 0 then a/b = (sign(a)/sign(b)) (p/q) With this computation rule, put 1/b = (1/sign(b)) (1/q). This agrees with the previous definition when sign(b) = 1. Division by zero remains undefined. Try the following exercises with the foregoing computation rule for quotients a/b
The expression of division in terms of multiplication allows rules for arithmetic (the equality of calculations) to avoid the discussion or mention of division. Addition and Subtraction of Unsigned NumbersWhen a and b are both unsigned number, or both positive, decimal arithmetic says how to compute a + b. Decimal arithmetic also says how to compute a - b when a > b. Comparison of decimal expansions determines when a > b, a = b and a < b. On a coordinate line, a + b may be represented by a steps of one unit to the right of the origin followed by b further steps of one unit. If a and b are represented by vectors of length a and b respectively, with tails at the origin, you may visualize this addition in terms of the parallelogram law (slightly squashed). In the case a > b, we may compute c = a - b by decimal arithmetic. The result c > 0 has the property a = b + c. Example 1. In the case where a and b are whole numbers with a > b, think of two situations.
In the first case 1, the inequality a > b allows you to take or subtract b pennies from the a in the bottle with a remainder of c = a - b pennies. In the second case 2, you may take all the b pennies in the bottle for your spending plans, but there will be a shortage of c = a - b pennies for those plans. Further subtraction of pennies from the bottle is impossible. Examples like case 2 this led people to say that negative numbers were imaginary. Example 2. In the case where a and b are unsigned numbers with a > b > 0, think of two situations.
In the first case 1, a > b and c = a - b implies a = c + b. Therefore subtraction of b units from a units leaves you c = a - b units to the right of the origin. Here subtraction of b units means b movement to the left. In the second case, a > b and c = a - b implies a = b + c as before. To move a = b + c units to the left of b on the coordinate line, first b movements to the left and then move a further c units. This leaves you c units to the left of the origin at a point with coordinates - c. This point represents the result of "taking" a = b +c units from b units. Exercise: Express the calculation in examples 1 and 2 with vectors drawn on or parallel to a coordinate line. Assumption A11 (A computational Rule for subtraction of unsigned numbers). If a > b, then c = a - b may be computed by decimal arithmetic, and we may put b - a = -c = (-1) c Addition and Subtraction of Signed Numbers.Successive movement along a coordinate line may be described in terms of vectors or distance & direction traveled. This viewpoint gives a vectorial framework for describing and explaining the addition and subtraction of real numbers. Suppose a = sign(a)p and b = sign(b) p with p > q then a is farther from the origin than b. We consider four cases
Exercise: Draw all four cases and illustrate them with vector addition.(or come for more details to appear when I have the time and patience) With the above definition, if a = sign(a)p and b = sign(b) p, then we put -b = (-1) b and we put a - b = a + (-b). The expression of subtraction in terms of addition allows rules for arithmetic (the equality of calculations) to avoid the discussion or mention of subtraction. |
|
|
|