MATH 4
INTERMEDIATE ALGEBRA
SUMMER TERM  2001

REAL NUMBER SYSTEM

Naturals, Integers, Fractions, Decimals, Rationals, Square Roots, Irrationals, Formulas, Scientific Notation, Graphing

 

Real Number System - The real number system includes all the numbers we have studied in the previous math assignments. It can be put into the form of a hierarchy going from the simplest type to the most complicated one.

1. The set N of natural numbers -

N = {1, 2, 3, 4, ...}

These are numbers for counting things, and early man hundreds of thousands of years ago probably used them. It wasn't until about 200 BC., however, that Greek mathematicians made the jump from finite numbers to infinite numbers. This jump is suggested by the three dots placed after the 4.

 

2. The set I of integers -

I = {…-3, -2, -1, 0, 1, 2, 3, ...}

The set of integers consists of the union of the natural numbers, their negatives, and the number zero. Oddly enough, the incorporation of negative numbers was a long time coming. Certainly the concept was understood with the use of black and red entries in the ledgers of traders, but negative numbers were not fully incorporated into mathematics until the Italian mathematician, Girolamo Cardano, used them in 1545 AD. The zero was introduced much earlier, some say around 700 to 800 AD, by Hindu mathematicians.

 

3. The set Q of rational numbers

Fig11_1.gif (1306 bytes)

This set includes fractions as well as integers. Fractional numbers are quite ancient. They appear in the earliest mathematical writings, and were discussed at some length as early as 1550 BC. in the Rhind Papyrus of Egypt. All rational numbers are characterized by having a repeating decimal form.

 

The Number Line - Before continuing our study of the real number system, let us consider a geometric interpretation of numbers called the number line. It allows us to set up a one-to-one correspondence between numbers and points on the line.

 

Fig11_2.gif (1502 bytes)

 

At this point it appears that we can completely fill up our number line with rational numbers. In fact, Greek mathematicians thought that this was the case, until in the 6th century BC, the mathematical school of Pythagoreans encountered a number that could not be written as a ratio of two integers: i.e. the number was not rational. The number was the square root of two , the length of the diagonal of a square whose sides are one unit long.

Since the Greeks originally thought that all numbers were rational, this discovery was tantamount to finding that the diagonal of a square did not have a mathematical length. Most Greek mathematicians resolved this paradox by thinking of numbers as lengths; i.e. in geometric terms, which inhibited the development of algebra which might have rivaled or surpassed their work in geometry. It took centuries of development and sophistication in mathematics to resolve this problem. What was done was to fill in the gaps of the number line, which are not filled by rational numbers, with irrational numbers.

 

4The set L of irrational numbers

Fig11_3.gif (1014 bytes)

Note that the prefix 'ir' means not. Also, unlike;

Fig11_4.gif (1368 bytes)

We are now able to define a number in which addition, subtraction, multiplication, division, and roots of positive numbers are closed operations.

 

 

5. The set R of real numbers -

Fig11_5.gif (1179 bytes)

This set includes all rational and irrational numbers. The set of real numbers is closed for the operations of addition, subtraction, multiplication, division, and roots of positive numbers. Odd roots of negative numbers exist, however, even roots of negative numbers do not.

As you can see, we started with the most basic type of number, and through a series of abstractions, reached the level of the real numbers. We can represent this schematically with the following diagram.

 

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