Traditional mathematics sees numbers and mathematical operations from the perspective of a 1 dimensional number line. However, from a geometric perspective, numbers also occupy higher dimensional spaces. This idea when applied to the axioms of mathematics reveal a whole new dimension to numbers, and the possibility of producing calculations that have no form of expression in conventional theory.


The 4 mathematical operators, addition, subtraction, multiplication and divide, that lie at the heart of all mathematical calculations are often taken for granted. Yet upon closer examination we can identify particular numerical conditions that define 8 distinct types of infinity on the number line.

These exist around the boundaries of ZERO and ONE, and extend into two opposite infinite extremities. By recognising this limitation, we are able to grasp a deeper understanding of the nature of infinity, which begins to challenge many mathematical conventions of our present day.



The Nature of Zero

Mathematics is founded upon the notion of 4 types of numerical operators; addition, subtraction, multiplication, and division. Each exhibits unique qualities that can help us comprehend the nature of infinity.

A whole number can be divided into infinitely smaller parts. The process can be reiterated, diminishing towards zero, yet the result will never cross the zero point. The same can be said for numbers in the negative.

This demonstrates that the number zero exhibits an infinite boundary on both its positive and negative side.

division creates the zero boundary

Zero is not Nothing

As the two boundaries are distinguished from each other, then the zero that sits at the centre cannot be ‘nothing’.


Did you ever question?

Why are there only 4 mathematical operators?

Can cross the Zero point

Can NOT cross the Zero point

Equal Values

When values of the same quantities are operated upon we notice the situation changes. Addition now doubles the quantity, whereas subtraction dissolves it back to zero. When we multiply or divide equal quantities we call it the square or root of the number.

If the squaring or rooting function is reiterated to an infinite degree, and, the initial value resides between zero and one, then, the result will tend move either toward zero, in the case of squaring, or, toward the number one, when the root value is found.

sections of a number line e


Yet no matter how many times the calculation is reiterated, the result will never be able to cross the boundary of ONE.


Try It for Yourself!

1: Take a simple calculator and press the number 2.
2: Find the square root button (√), and press it: the result is 1.414... or √2.
3: Keep Pressing the same square root button over and over again.
Notice that the answer never goes below ZERO. NOTE: If your calculator does go to zero, that is because it has run out of memory. Infinitity is beyond the ability for a machine to calculate

Reciprocal Number Space

The reciprocal number space between ZERO and ONE contains all of the possible whole numbers. With the exception of ONE and ZERO all numbers, even primes, can be divided by one to find its reciprocal fraction. We call this space between zero and one ‘Reciprocal Number Space’

This means ONE and ZERO are a particular class of number, separate from the rest.


8 Infinities on the Number Line

By identifying the limitations of the number zero and one in reciprocal space, we are able make an examination infinity as found on the number line

By performing squaring and root calculations for any whole number, we find that multiplication expands into infinity. Whereas a root function will diminish towards ONE. Here we see that, similar to the number ZERO, the number ONE exhibits two boundaries, one on the whole number side, and one on the reciprocal.

Generally we can see that negative numbers should perform in the same way as positive integers. However, do to the establish axioms of current mathematics, two negative numbers will make a positive square. However, in Geometric Maths, this issue is resolved by a cleaner definition of the concept of powers. and negative powers.

Once we resolve the ‘negative square/root number issue, then we are able to place and additional boundary set around the number -ONE. This is a key step that has been missing from modern mathematics.

Is it just a coincidence that the number 8 and the infinity sign should share the same symbol, rotated at 90° to each other?

Once we include the two infinite series of positive and negative whole numbers, which expand into infinity, we have correctly identified the 8 types of infinity that can be found on the number line.


Reciprocal Prime Numbers

As Reciprocal Space contains all prime numbers, we can conclude that the number 0 and 1 are particular precursors to all other numbers, including primes. In the consideration of reciprocal prime numbers, find that numerical space is divided into smaller parts as we progress through the infinite set. If a number create a division that falls over the same place as one that proceeds it then the number is not a prime. Based on this logic, we can see that we should be able to take the closest point of division and be able to predict the series of prime numbers. Unfortunately this in not the case.
Space Blue Stars

Why are prime numbers so unpredictable?

Due to the number powers, and in particular the nature of the number 37, all prime number predictions begin to fail. However, what this highlights is that the number line is not an accurate depiction of numerical space.


Beyond the number ¡

The traditional expansion of numerical space is termed the complex plain. The number ‘i’ was developed to compensate for the inability for the traditional number line to accommodate the square root of -1. However, what our new mathematics observations begin to allude to is the fact that both square and root numbers, whether positive or negative express a particular quality in relation to the number ONE and Infinity.

As the number ‘i’ only accounts for the square root of -1 it is not adequate enough to express this infinite relationship for other root or square numbers. The picture that begins to emerge is that the number ‘i’ is only one of a whole set of root and square numbers, that have yet to be accounted for.

Space Blue Stars

There's more to maths than meets the ¡

The significance of this is that it provides a new mathematical solutions to the ‘Continuum Hypothesis', that lies right at the heart of mathematical debate around the topic of infinity.




The GEOMETRIC MATHS examines numerical space primarily in terms of a MULTI  dimension number line.

In this introduction what we have shown is that:

Divider Menu
  1. Zero is not ‘nothing’.
  2. Division creates the boundary around the number ZERO.
  3. Square and Root function that operate on reciprocal number space can never exceed the boundaries of ZERO and ONE.
  4. There are 8 types of infinity that are found on the number line.
  5. By examining prime number reciprocal space, we can ascertain that the number line and number i are missing expressions such as √2, √3, etc. despite appearances.


This is just an introduction to our new theory of THE ‘MATHS OF INFINITY’ Explore MORE BY READING some of the ascociated articles below.


In2infinity's new solution
to the Continuum Hypothesis.

Listed as the most important mathematical challenge of the 1800's, In2infinity sheds new light on the answer, disproving the present solution, and opening the doors for a whole new maths of infinity.

Explore our theory

THE ‘MATHS OF INFINITY’ is JUST A SMALL part of our new theory
FIND OUT MORE BY READING THE main articles below

4D Maths

4D Maths is a fundamental step into a new dimension of mathematics. Founded upon the idea of space, time and dimension, and the infinity equation, 0 ±1= ±∞, the 4D calculator generates new insight into the infinite nature of numbers, and introduces the ‘Theory of Effort’. Overview How numbers are created 4D Maths suggest a new type of categorisation of

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