Abstract Thinking and Numbers

When abstract thinking is applied to numbers, whole new sets of numbers begin to emrege. A creation story of numbers could be envisioned as follows:

Numbers are directly related to objects or events that exist in the tangible world. eg. 2 means 2 apples or 2 days.
Numbers are abstracted from the object or event that they describe in the sense that a meaning from the number may be applied to the objects or events that the number describes. Symbolic thinking is emergent here in that there may be a special meaning of twoness. Symbolism and Number
Natural numbers - The numbers 1 thru infinity (or the maximum expressible by the numbering system) are the set of natural numbers. These numbers have a direct link to the real world in that they count things or events. You can add and multiply these numbers and get more natural numbers.
Integers - The mathematical relation of substraction produces some numbers that do not directly relate to objects in the world. eg. 5-3 = -2. There is no -2 things in the world. There is a relationship that is implied by -2 to the people using the number in the sense that two things may be owed to another. In a sense, negative numbers and zero are created from the natural numbers by substraction.
Rational numbers - The mathematical relation of division can produce fractional parts such as 1/2, 1/3, 3/5, etc. These numbers can subdivide an integer into very small parts. This allows for very great precision in specifiying a value.
Irrational numbers - The mathematical relation of a square root can produce numbers that never seem to resolve to a specific value. eg. the fraction 1/2 can be expressed in decimal notation as .5 or 5 tenths. The fraction 1/3 can be expressed in decimal notation as .333333 ->continuing as many digits as necesarry. The square root of 2, however produces a number 1.4142135 -> continuing as many digits as necessary, however, the numbers never repeat a sequence. In the 1/3 example the 3 was repeated over and over as is the case for all Rational fractions. Irrational numbers do not have a recurring pattern of numbers.
Real numbers - The set of all of the above numbers.
Real valued functions - A function is a relationship between two real numbers in the sense that a function takes a real number and produces another real number. For example a function that squares a number could be written as f(x) = x*x. This relationship may be graphed on a Cartesian coordinate system as y=f(x). So if x = 1 then y = 1*1 or 1. If x = 2 then y = 2*2 or 4. This function works with any value of x and produces a value of y.

Back to Number Base

Link to Poseidon Software and Invention Home Page

Last update on 01/26/98