INTERMEDIATE ALGEBRA

SUMMER TERM 2001

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** –

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.

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.

* *

4** . The set L of irrational
numbers** –

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

- rational numbers which contains a pattern of digits to the right of the decimal point that repeats itself over and over again,
- irrational numbers are non-repeating decimals; i.e. no pattern is ever established. Two well known examples are -

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** -

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.