Module 1 – Introduction and Review
Section 1.1 – The Real Number System
The Real Numbers include positive and negative counting numbers as well as all their possible ratios (fractions), but that’s not all. In order to describe physical measurements we also need to consider “irrational numbers”.
In this section you will be reminded why these are necessary and how such numbers all correspond to points on a “Number Line”. You will also review the basic arithmetic operations and some of their algebraic properties.
The Real Number System
Most of the mathematics studied in high school and college has to do with the Real Numbers. These were invented to give us a way of describing and calculating with physical measurements such as length, time, temperature, etc., and so they are essential for most practical applications of mathematics.
An important point is that the real numbers correspond to all possible measurements along a linear scale and so include not just whole number positions but also others in between (so they can be represented graphically on a “Number Line“).
For calculus, you will need both a good facility with basic algebraic operations and also an understanding of how to deal with the concepts of ordering and distance in terms of inequalities and absolute value.
There are a number of other web sites that offer further explanation and practice with these basic concepts.Most introductory calculus courses assume an intuitive understanding of the number system and do not go into a rigorous analysis or justification of its properties. But an important aspect of mathematics is the fact that its results are logically provable from more “elementary” assumptions. Where not adressed in first year calculus, these issues are often introduced in an Introductory Analysis course at the second year level. There are several other sites with on-line analysis course materials)
What are the ‘Real’ Numbers?
The so-called ‘Real Number System’ is an attempt to extend the ideas of number and arithmetic (which originate in counting) to describe more general measurements such as length, weight and temperature.
In fact, the way we measure anything is based on counting. When we use a ruler or tape to measure a length, we do so by counting the number of marked intervals of some specific length or “unit” (eg cm,m, or inch). But this is often not exact – and we divide each unit interval into smaller parts (or fractions) to get a more accurate (but still not perfect) measure of the length.
It might be expected that if we take all possible fractional divisions of the basic unit then we’d be able to give an exact description for any concievable length and so that every length can be measured exactly by a “rational” number of units.
But this expectation is FALSE! (Do you know why?)
In fact, we can get arbitrarily close to any real length with rationals, but we can’t always match it exactly. (This idea of arbitrarily close approximation is the source of the mathematical concept of a limit which underlies many of the concepts of Calculus.)
On the other hand, if that was the bad news, the good news is that the operations of arithmetic (defined for whole numbers in terms of combinations of sets) can be extended in a natural way to fractions and also to those “limits of fractions” that we need to account for all possible measurements.
One way to define the Real Number System is to “construct” it by identifying real numbers with sequences of approximating rationals and showing that the arithmetic operations do extend consistently. Another is to specify the basic properties that the resulting system should satisfy and to take these as “Axioms” from which less obvious properties can be proved (or disproved as the case may be).
- Construction of Real Number System
- Axioms for the Real Number System
- The Number Line
- Irrational Numbers
- Transcendental Numbers
Module 1 Introduction and Review
Section 1.1 – The Real Number System
(see Text section 1.1)
Almost all physical measurements involve the use of a linear scale (i.e. a set of marked positions on a line – like those on a ruler of thermometer for example). The Real Number System is intended to include all possible positions on such a scale (or “Number Line”) and so is exactly what we need for most applications of mathematics. In this section your objectives are to review some of the basic features that system.
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)
What to do
- Follow the link above to review why we need to include more than just ratios of whole numbers if our number system is to be used for physical measurements.
- Read through section 1.1 of the text. As you proceed, use the questions in this ‘Reading Guide’ to confirm your understanding, keep in mind your Learning Objectives for this section, and when you have finished be sure to follow the suggestions regarding Further Practice.
- When you feel that you are ready, go on to the next section.