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.

Footnotes:

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.

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.

⁵expressions which give real numbers when computed

⁶Empirical Observation: If you use decimal arithmetic to add two positive numbers together, the result will be positive. If you use decimal arithmetic to multiply two positive numbers together, the result will again be positive. This implies the Nonzero Product Rule in the setting of decimal arithmetic.

⁷A physical analogy for this is as follows: imagine umpteen bags of marbles, all of which are to be placed in a larger container. The total number of marbles put the larger container does not depend on the order in which the smaller bags are put in, and it does not depend on how the smallers bags are grouped together before they are put in. Discussing about this physical analogy departs from the pure development of mathematical concepts from the long chains of reasoning starting with rules or assumptions that involve no physics.

More Chapter Sections: [ Up ] [ 18 Changing Formulas ] [ 18 Rules for Algebra ] [ 18. Field Properties ] [ 18 Sums as Factors I ] [ 18 Sums as Factors II ] [ 18 Addition Properties ] [ 18 Times Properties ] [ 18 Power Rules ] [ 18 To Divide, Multiply ] [ 18 Inconsistent Nttn ]