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Chapter 18
Arithmetic Rules and Patterns (algebraically described)

Rule-based reasoning is used in the changing of formulas and equations. Somewhat flexible rules say how or what is permitted. The flexible rules in algebra can be applied one at a time or one after another to arrive at formulas and equations or to draw conclusions on or from them (the formulas and equations, that is). But understanding the rules requires the algebraic way of writing and thinking to be well understood beforehand, else they, the rules, will not make sense.  This chapter aims to  make the algebraic description of the properties of real numbers understandable and useable.  For many students, the algebraic shorthand description of numerical properties is gibberish - ouch. Explicit and deliberate rationalization or explanation is needed. 

Chapter Sections: [Order of Operations] [ 18 Changing Formulas ] [ 18. Proper Use of Equal Sign ] [ 18. Replacement & Substitution ] [ 18 Real Numbers & Quantities ] [ 18 Rules for Algebra ] [ 18 Sums as Factors I ] [ 18 Sums as Factors II ] [ 18 Addition Properties ] [ 18 Sum Associative Property ] [ 18 Sums and Number 0 ] [ 18 Replacing Subtraction by Addition ] [ 18 Times Properties ] [ 18 Sum Grouping and Ordering ] [ 18 Product Associative Property ] [ 18 Products with the Number 1 ] [ 18 Product Grouping and Ordering ] [ 18 Power Rules ] [ 18 To Divide, Multiply ] [ PS: Rules for Fractions and Division ] [ 18 Inconsistent Notation ]

1  Order of Operations

Parentheses are (often) used to show the order in which arithmetic (+, -, ¸ and ×) is done in a calculation. The order can sometimes be changed without changing the result. Rules or properties of arithmetic say when. These rules and properties say how to move the parentheses about, and how to omit them, without changing the result obtained. Here the arithmetic may change, but the result does not.

In this section, we talk about the use of parentheses in the description of calculations that are or could be done. The rules of arithmetic for shifting or omitting parentheses state when two would-be calculations should give the same result are described in the following sections.

In arithmetic, the order in which the arithmetic is done may change the result. So some caution is required. In describing calculations we also need to give the order in which the additions, subtractions, multiplications and divisions can be correctly done. The order is based on the following conventions:

• expressions within a pair of parentheses (¼) are to be computed before those outside. So the stuff ¼, whatever it is, within a pair of innermost parenthesis are done before those outside.
• without parentheses to show what calculation is to be done, multiplications and divisions are to be done before additions and subtractions. Multiplication and division are said to have a higher priority.

1. The expression 17-(10-3) gives 17-7 = 10 but (17-10)-3 gives 7-3 = 4.
2. The expression ([4/5] ¸[5/16])×[2/3] gives

   æ ç è 4 5 · 16 5 ö ÷ ø × 2 3 = 4×16 5×5 · 2 3 = 64 25 · 2 3 = 128 75

   This is different from

   4 5 ¸ æ ç è 5 16 · 2 3 ö ÷ ø = 4 5 ¸ æ ç è 5 24 ö ÷ ø = 4 5 · 24 5 = 96 25

3. The expression (5¸6)¸2 = ([5/6])¸2 = [5/12] but 5 ¸(6¸2) = 5¸3 = [5/3]. The parentheses cannot be omitted.
4. (8-5)-2 = 3-2 = 1 while 8-(5-2) = 8-3 = 5. So the parentheses are important.
5. But (5·4) ·3 and 5 ·(4 ·3) both give the same result.

In teaching, I had respect for the student who would identify in his or her arithmetic (or reasoning) what was uncertain. That was a sign of careful thinking. I tried not to reward students who tried to hide their guesses. In learning, once a student has identified the limits and uncertainties in his or her knowledge, that student is ready and able to learn more.

Chapter Sections: [ 18 Changing Formulas ] [ 18. Proper Use of Equal Sign ] [ 18. Replacement & Substitution ] [ 18 Real Numbers & Quantities ] [ 18 Rules for Algebra ] [ 18 Sums as Factors I ] [ 18 Sums as Factors II ] [ 18 Addition Properties ] [ 18 Sum Associative Property ] [ 18 Sums and Number 0 ] [ 18 Replacing Subtraction by Addition ] [ 18 Times Properties ] [ 18 Sum Grouping and Ordering ] [ 18 Product Associative Property ] [ 18 Products with the Number 1 ] [ 18 Product Grouping and Ordering ] [ 18 Power Rules ] [ 18 To Divide, Multiply ] [ PS: Rules for Fractions and Division ] [ 18 Inconsistent Notation ]

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