Study Notes and Discussion
The basic properties listed in the table on page 4 of the text all have geometric interpretations in terms of lengths, areas or volumes. For example commutativity of addition corresponds to the fact that if we place two sticks end to end the combined length is the same no matter which of the two is on the left or right. What about commutativity of multiplication? Think for yourself first, then check our answer.
Note: Questions like the one above will be included from time to time to help you to check your understanding. Try to resist the temptation to just read the answers without thinking.
It would also be a good idea to think of similar reasons for the basic rules of signed arithmetic, and for the properties of quotients listed on page 8 of the text.
For the product of two negatives giving a positive, you might think for example of the effect on your bank balance of having debts cancelled, or geometrically, of the fact that reversing direction twice has you end up facing in the original direction; and to motivate the rules of fractional arithmetic you might recall the “pizza sharing” examples that you probably remember from elementary school.
If you take the trouble to think again about these things now, you may save yourself from making silly mistakes later on; and a rule for which you have a reason is much easier to remember than one which seems arbitrary.
Real mathematics is more about why things are true and why methods work than what is true and applying the methods. This is more true now than ever, as any method that can be taught can also be programmed into a computer or calculator. So no-one is likely to pay you for just applying a method. The skills that are valuable are analysis of what method is appropriate to a given problem, and recognizing the mathematical nature of an applied problem. This doesn’t mean that computational exercises are a waste of time; they can help you build understanding, and speed of calculation will help you to follow arguments in future courses. But the most important thing is to focus on understanding.
So you should try to insist on having an explanation for anything that you are expected to believe.
One of the first things mentioned, both in the text and in our discussion, is that the real number system includes more than just the rational numbers.
Where does the text explain why cannot be a rational number?(Answer)
Before leaving this section, you should be sure that you correctly understand all of the terms used in the text section, and review any that are unfamiliar.
Note in particular that the mathematical terms greater and less are not equivalent to the common language ‘bigger’ and ‘smaller’. In terms of the number line, Less means on the Left, and gReater means on the Right. So a ‘big’ negative number is less than a small one.
Check your understanding, and practice for speed, by working through some of the Exercises on pages 16-18. Do enough of the odd numbered questions of each type to convince yourself that you can get the right answers. (Note that, as usual, the answers are in the back of the text and complete worked solutions are in the student study guide - but try to avoid looking at answers or solutions until you have made your own best effort.)
As a bare minimum you should do ##1,5,11,15,21,25,31,35,41,45,47,51and 55, and when done, compare your solutions with those in the student solutions guide