The Real Number Line

Peter Oye Sagay

The number line is a line containing numbers. It is a fundamental concept in mathematics. Numbers are specific symbols that have been uniquely mapped onto quantities. They originated from primitive counting.

A line is a commonly known shape. The great Greeek mathematician Euclid, attempted to define a line about 300 BC in his famous geometry book called the Elements. In the Elements, Euclid stated that two points uniquely determine a straight line. What then is a point? We often assume that the description of a point is self-evident. Nonetheless, let us use a physical quantity to describe a point. Since dust is a universal quantity, let us assume that a particle of dust as minute as the eye can see, is a fair description of a point. Now suppose we are able to arrange in a row many similar minute particles of dust such that there is no space between consecutive particles. We can describe the resulting shape as a straight line. Another way to visualize a straight line is to tie one end of a thread to the forefinger of the left palm and the other end to the forefinger of the right palm, then to hold the palms as far apart as possible in order to have a straight stretch of thread. The stretch of thread between the forefingers is a straight line. Conceptually, the number line is basically a straight line made up of numerous points. Each of these points have been assigned a number with a unique identity and meaning.

Now given that the number system is the positional number system with the numerals 0,1,2,3,4,5,6,7,8,9 (the decimal number system), let us examine some of the basic properties of the number line.
(1) The end points of the number line reside in infinity. In other words, numbers at both ends of the number line are exceedingly large
(2) The number 0 resides at the mid-point of the number line.
(3) If the number line is horizontal and the direction above 0 is north, the eastward direction of the number line is called the positive direction, and the numbers along this segment of the number line are called positive numbers. The westward direction is called the negative direction, and the numbers along this segment of the number line are called negative numbers.
(4) If the number line is vertical, the northward direction (the direction above 0) is the positive direction and the southward direction is the negative direction.

The procedure used to position the numbers on the number line is simple. The single digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are consecutively positioned such that there is a unit distance between adjacent digits. After the digit 9 has been positioned, the next number to be positioned a unit distance from 9 is the two digit number, 10. Each of the digits of the number 10, has a specific meaning. The leftmost digit which is 1, indicates that we have marked ten unit distances on the number line, one time. The rightmost digit which is 0, indicates that we are beginning another round of marking ten unit distances on the number line and so the next ten unit distances will be 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and at the end of the round, the number positioned on the number line is 20, that is, ten unit distances have been marked on the number line, two times. This procedure is used to position the numbers 30, 40, 50, 60, 70, 80, 90 and at the end of the next ten unit distances, the number positioned on the number line is 100, a three digit number. The interpretation of this number is that ten ten-unit distances have been marked on the number line. As numbers that represent larger quantities are positioned on the number line using this procedure, the relationship between the number of ten unit distances marked on the number line and the number of digits contained in a number, becomes apparent.

Numbers are single digits until the end of the first ten unit distances. The number 10 (ten) is the first two digit number. Numbers contain two digits until ten ten-unit distances have been marked on the number line. The number 100 (one hundred) is the first three digit number. Numbers contain three digits until one-hundred ten-unit distances have been marked on the number line. The number 1,000 (one thousand) is the first four digit number. From this pattern, we can make the following generalization:
The number of ten-unit distances between 0 and a given number, is equal to the number remaining, after the rightmost digit has been removed from the given number. For example, consider the number 1734. If we remove the rightmost digit which is 4, we have the number 173. So, the number of ten-unit distances between 0 and 1734 is 173. This means that 1734 = (173)10 + 4. The continued application of this principle eventually results in the following:
1734 = [ ((1x10)) + 7)10 + 3]10 +4, that is,
1734 = 1x103 + 7x102 + 3x101 + 4x100
. So, each digit of a number has a positional value on the number line which is equal to d x Np, where d is the digit, N is the number of unique digits that make up the number system and p is the position number of the digit in the number. For a whole number, the position number p, ranges from 0 (the position number of the rightmost digit) to n-1 (the position number of the leftmost digit), where n is the number of digits in the number. For a fractional number, the position number ranges from -1 (the position number of the leftmost digit, that is, the digit immediately after the decimal point) to -n (the position number of the rightmost digit), where n is the number of digits in the fractional number. Consequently, the value of a number on a number line is equal to the sum of the positional values of the digits in the numbers.