Arithmetic Rules and Patterns

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What They Do. The rules of arithmetic say when the order of operations can be changed in a first calculation, so that we obtain a second calculation which gives the same result as the first. These rules apply to arithmetic involving real numbers and/or real quantities.³

³ High school mathematics (circa 1990) talks only about real numbers, and leaves talk about quantities to physic courses and commerce courses. But calculations involve both real numbers and units of measurements. The convention in algebra textbooks is to emphasize the connection with real numbers but not real quantities. But in dealing with quantities in physical and monetary calculations, students need some guidance. Since the rules of algebra apply to calculations involving units, an algebraic tradition involving the manipulation of units and their powers needs to be presented and sanctioned in high school mathematics courses.

The order of arithmetic operations, suggested by parentheses, matters in some calculations, but there is some flexibility. In some but not all, we can change the order in which arithmetic is done without changing the arithmetic result. The properties of arithmetic (rules) given below say how this can be done.

Explaining Some Rules.

Next, you may meet more words than you ever wanted on the rules of arithmetic. Read on and look for the ideas new to you. Some, just a few, could be worth repeating to others.

To describe the properties rules for changing calculations without changing their results, we introduce four shorthand letters a, b, c and d to stand-in for real numbers (or real quantities). The use of these letters is a tradition. Other letters could be used. Sometimes it is convenient to describe or rewrite these rules or properties using other letters.⁴ You could pick four different letters if you wish.

⁴You should imagine these rules written with other letters of your choice, when in the calculations you meet, at least one letter a, b, c and d, that has been previously assigned a different role or meaning. In any plot, each actor should have only one role.

The following table describes properties of addition and multiplication which you can use in doing arithmetic or describing arithmetic that could be done. In these laws and properties, the expressions on either side of the equal sign, always give the same result.

Properties of Addition and Multiplication

First expression = Second expression name of the property (or rule)

(a+b)+c = a+(b+c)
associative law for addition

(ab)c = a(bc)
associative law for multiplication

(a+b)c = ac+bc
(right) distributive law

c(a+b) = ca+cb
(left) distributive law

a+b = b+a
commutative law of addition

ab = ba
commutative law for multiplication

a+0 = a
additive identity: the effect of adding zero

a·1 = a
multiplicative identity: the effect of multiplying by one.

In each row of the above table, the first expression always gives the same result as the second expression, no matter what real numbers or quantities the letters a, b and c represent. In describing a calculation, either expression can be replaced by the other, or a symbol (pronoun) representing the result of either calculation.

The above rules only involve addition and multiplication. We will talk next about the above properties and rules and about how they are used, next. How to apply these rules to expressions involving subtraction or division will also be described later.

Reminder. The product a×b is also written as a·b or as ab. Which notation is used to signal multiplication is a matter of taste and convenience. When the times symbol × might be confused with the letter x, remember to use the dot · instead, write a·b or ab.

Remark. The above properties are assumed and used in doing arithmetic and in changing and manipulating formulas. They are often called the laws of algebra. This author prefers to call them laws or properties for arithmetic.

Chapter Sections: [ Up ] [ 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 Nttn ]

Chapter 18 Rules for Algebra

Arithmetic Rules and Patterns

Chapter 18
Rules for Algebra