Category Archives: Mathematics

Numbers & Consciousness

July 10, 2010 – 2:53 PM

Numbers, in a crude way, seem to represent consciousness. In ancient times, the shepherd counted his sheep and, thus, accounted for them. That represented his consciousness of the number of sheep, whether all were there or if any were missing.

And so we have counting numbers that start with 1. The next number is obtained by adding 1 to a number. A person counts by calling out 1 for the first item, 2 for the second item, 3 for the third item, and so on. These numbers can be very large. For example, there are so many stars in the sky that it is difficult to count all of them.

Besides, a number may be dreamed up that is considered the largest. But, anyone may add 1 to that number to get a still larger number. Thus, there is no limit as to how large a counting number can be.

These counting numbers are also called natural numbers because they follow from the natural process of counting.

NATURAL NUMBERS: 1, 2, 3, 4, 5, 6, 7, 8, 9, …

SMALLEST NUMBER = 1

NEXT NUMBER = Number + 1

LARGEST NUMBER = Undetermined

The natural numbers provide us with a sequence that may be plotted as follows.

Here each number represents a whole unit, and it is set apart equally from the previous number. These numbers go on for ever.

As commerce developed, need arose for numbers that could represent parts of a unit, such as half of a unit, or quarter of a unit.  Half of a unit was written as 1/2 (one of two equal parts of a unit). Quarter of a unit was written as 1/4 (one of four equal parts of a unit). Three quaters was written as 3/4 (three of four equal parts of a unit).

Thus, fractions came to be written as ratio of two natural numbers. It was found that a natural number could be written as a ratio of itself and one. This new expanded set of numbers came to be known as rational numbers.

The fractions could be plotted on the number line before 1. A number and a fraction could be plotted between two natural numbers. For example, “one and a half” could be plotted between 1 and 2. In fact, it appeared that a rational number could be found for each and every point on the number line.

The consciousness of numbers had thus deepened and expanded greatly. It seemed at that point in time that rational numbers represented all possible numbers that could ever exist.  A unit could be broken into smaller and smaller units making it possible to account for smaller and smaller quantities. This system seemed to provide an adequate abstraction for the unfathomable and intricate quantitative universe.

The rational numbers came to be regarded mystical by those at the forefront of research, such as the Pythagorean Brotherhood.

What happened next is quite mystical. Please see  Going Beyond Counting.

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By vinaire | Comments (2)

The Abacus and Numbers

June 25, 2010 – 11:22 PM

The system of how counting numbers are written today can be demonstrated visually on a device called Abacus. You count by moving 1 bead at a time on the first wire on the top.

The digit 0 corresponds to no bead moved.
The digit 1 corresponds to one bead moved.
The digit 2 corresponds to two beads moved.
The digit 3 corresponds to three beads moved...

And so on.

When all the ten beads are moved to the right on the wire they are regrouped as one bead to the right on the next wire. Thus, TEN is expressed as a combination of one bead on the second wire and no bead on the first.

In terms of digits, TEN is expressed as “10”.

Please note that the place of a digit in a number corresponds to the order of the wire from the top.

As you continue to count, the beads are regrouped not only on the first wire but also on subsequent wires per the following rule:

WHEN ALL BEADS ON A WIRE ARE TO THE RIGHT, THEY ARE REGROUPED AS ONE BEAD TO THE RIGHT ON THE NEXT WIRE.

For further details, please see, Numbers and Place Values.

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Number Base

We are used to dealing with numbers in the decimal system, where we use a base of 10, counting up from 0 to 9 and then resetting our number to 0 and carrying 1 into another column. This is probably a result of having ten fingers.

Suppose we had only eight fingers, then we would most probably work in base 8, counting from 0 up to 7 and then resetting to 0 and carrying 1. So the number 10 in this system would mean 8 in the decimal system. Base 8 is called the Octal system.

OCTAL NUMBERS MAY BE REPRESENTED BY A SPECIAL VERSION OF ABACUS WITH EIGHT BEADS ON EACH WIRE INSTEAD OF TEN.

If an intelligent race had only two fingers, then it would most probably work in base 2, counting from 0 up to 1 and then resetting to 0 and carrying 1. So the number 10 in this system would mean 2 in the decimal system. Base 2 is called the Binary system.

BINARY NUMBERS MAY BE REPRESENTED BY A SPECIAL VERSION OF ABACUS WITH ONLY TWO BEADS ON EACH WIRE INSTEAD OF TEN.

Using this “binary abacus” it can be seen that the decimal numbers 1, 2, 3, 4, 5, 6, 7, 8, etc., will appear as 0, 1, 10, 11, 100, 101, 110, 111, 1000, etc., respectively.

The binary numbers may be represented very accurately with a series of switches, each of which can be set to either OFF or ON corresponding to 0 and 1. This is the secret underlying the powerful computers.

The elegance of abacus is that young children can use this device to learn different number systems quite easily.

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By vinaire | Comments (0)