All numbers beyond ONE have a reciprocal representation between the numbers ZERO and ONE. It is this mirror-like nature of reciprocal space that allows us to conclude that the number ZERO terminates the series of infinite whole numbers.


Within the ‘Reciprocal Space’ between Zero and One, find all numbers above one can be created by dividing the space into a specified number of equal parts. As both the boundaries that define reciprocal space (zero and one), are also defined through a process of infinite division, we can begin to see that division is the prime mover for the creation of all other numbers.

Geometrically there is no difference between dividing a line, or adding numbers units to a given length. The difference comes when we apply a specific scale. This is related to the nature of our very consciousness to distinguish ‘things’ in our environment.

All whole numbers are just a reflection of the divisions of reciprocal space, each whole unit ‘contains’ the same potential for infinite division as reciprocal space.

By comprehending this we are able to conclude that zero terminates the set of infinite numbers, as the greater the value in whole number space, so its reciprocal moves closer to zero.


KEy Points

  • As all numbers above one are found in reciprocal space, any number outside of this can be createed from its reciprocal value without the need for addition or subtaction.
  • As division is the mathematical process that defines the Zero Boundary, and Infinity of ONE, so it also manifest all numbers greater than one
  • when we view reciprocal space as the reflection of whole number space, we can determine that the number Zero sits at the end of the infintie series ofwhole numbers numbers



Between Zero and One

Did you know that there are as many numbers between 0 and 1 as there are between 0 and 10, and between 0 and 100? This might sound confusing at first, so let’s use an example to help make it clearer:

Draw a line, it doesn’t matter how long, mark one end 0 and the other end 1. Divide the line roughly into two with a mark. By dividing the line of a random length into two, you create two spaces that are similar to the first, but half as small (from 1 to 0.5). Go on to divide these two new half-lines again into two, and you have divided 1 into four (from 1 to 0.25).

You can carry on indefinitely, dividing the new line lengths into two, getting a new number. The same will happen with the number three. This is the nature of division. We can divide a line into an infinite number of equal parts, a fact that is the lies at the foundations of all fractal geometry.


A Line can be divided into an infinite number of smaller parts. what does this tell us about the nature of numbers and infinity?

This means that all numbers, including fractions, can be found by dividing reciprocal space. This is an important realisation when considering our new mathematics of infinity. The fact that ZERO and ONE are excluded from this set is significant, as it demonstrates these numbers have a special quality. Through the process of division, they form the boundaries of reciprocal at their periphery. For more information, you can view our posts on the nature of infinity around ZERO and ONE.

Key point

Between Zero and One, all numbers are manifest

Division requires a pre-defined space into which is it enacted, as opposed to addition and subtraction that increase or decrease the value of a calculation. Just like a string on a Guitar. We can divide the string in half to create the musical octave, but the whole string still exists. It is this subtle nature of division that makes it unique amongst the four mathematical operators. And which leads to a very interesting conclusion as to the underlying fabric of current mathematical axioms.

One with everything

When we consider the way that our consciousness is able to perceive reality, we find it can fall into two broad camps. Seeing things as separated, or seeing ourselves as part of a larger structure. We do this quite naturally every day, yet rarely do we stop to consider what is actually happening.

Let us take a minute to review these two perspectives and the relationship to the construct of numbers

Individuated - Units

The physical world is a place of manifest things. This is how most of us are first introduced to the concept of numbers. We learn to count objects, add and subtract, as we become accustomed to our environment.

This is the perception of whole numbers, that is most common to us. Individual things that can be bought, sold; units and quantities.

Collection - UNIFIED

Next is the unified perspective. This is the recognition of the part that operates to make the whole.

When we work for a company, often we begin to regard ourselves as a part of that organisation. In conversation, we might catch someone referring to a company they work for as ‘we’. Even though you know they are not actually the whole company, they identify themselves as a unified part of the whole.

Shifting the boundary

These two types of perspective we use every day in combination to make sense of the world.

When we get into a car, we are normally only aware of the object as a whole. Only when it breaks down, and we open the bonnet (hood), do we see the small individuated parts that make up the whole.

The same can be said of almost all phenomena. Governments, companies, and other institutions, made of up people, forests arise from trees, and birthday cake is for someone’s celebration, until they share it around, and then we all take part.

It is our capacity to shift the boundary of our awareness that is probably the most defining factor of human consciousness.

car made of parts
A car is made of thousands of smaller parts, that work so well together we normally only consider the car as a single whole.
Find out more about GEOMETRIC THINKING

Division or addition?

The nature of division is unique, as through infinite iteration it defines the zero boundary. Also, we find that a square root, that form the Infinity of ONE, has its mathematical foundations in the nature of division. This sets the function of division in a different category from the other mathematical operators. 

Yet, appearances can be deceiving.

Number Perception

As with our examples of conscious perception above, the same applies to mathematics. We can draw a line and divided it into two equal parts.

If we place the zero at the start and a one at the end, then we say the halfway point is the number 0.5. However, we can also place the number two at the end, in which case the midpoint can be labelled 1. Which is correct?

From either perspective both appear to be logically true, however, as we drew the line first and then ‘divided’ it, the correct perception is that of division. If we had drawn a line and the added another unit to the end, then the process is that of addition. The final result would double in size, whereas the size of the divided line will never change.

It is division and square roots that form the infinite boundaries around ZERO and ONE. Only after this mathematical nature is established can all other whole numbers arise. The establishes what Aristotle called ‘the Unit measure’, from which the mathematical functions of addition and subtraction can be enacted. Division is the precursor to all other mathematical operators.

Key point

Division is the first mathematical function to be enacted.

[1] In the beginning, God created the heaven and the earth.
[2] And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.
[3] And God said, Let there be light: and there was light.
[4] And God saw the light, that it was good: and God divided the light from the darkness.

It's all within

From this realisation about the nature of division, and the boundaries on the number line that are manifest through it, we begin to produce an interesting description of numbers. As a line becomes divided into smaller and smaller equal parts, so a uniform scale is created that represent the complete set of infinite whole numbers. Each will have a specific division which creates a unit of specific size in relationship to the whole. This means that whole numbers can be generated purely from the space between zero and one.

Learning to count starts from one and leads us on a numerical pathway, all the way to infinity. Viewed like this, it would appear that infinity has no end. However, as we now know it is contained within reciprocal space, was can see this cannot be the case. So what is actually going on?

The number observer

The answer is that we are creating the number, through observation. When we ‘count’ a number, we are actually forming a mathematical boundary as we process the information.

Similarly, quantum physics suggests that reality does not exist until it is observed, so the same can be said of numbers.

We are decoding the infinite (internal) space between the zero and one, and applying it to the physical worlds as a unit number, a quantity, or a value.

You only know the number, once the card is pulled

Think of it like this.

Within the reciprocal space between zero and one, there exists more numbers than all the whole numbers in existence. Similarly, the mind is able to imagine a wide range of phenomena and ideas. Many of these are not able to manifest in reality, as they are outside the realm of possibility. Reality is limited in scope by the Law of Nature, science. Similarly, whole numbers are an expression of all that can possible manifest into reality. Units of things that work collectively to produce an outcome. Mind is therefore the reciprocal space, from which all whole numbers are generated. The implication is that thoughts come before matter (reality).

Did you ever try to count all the stars at night?

If you did, then you know that it is an impossible task. However, by observing the patterns of the Universe as an integrated whole, we can begin to make sense of it. It’s all about open our perception towards noticing the patterns of the infinite.

Interested in the spiritual aspect of numbers?

Check out our post on the Numbers of Creation

Infinite Reflections

These observations expose to an interesting thought, that suggests whole numbers are just a reflection of reciprocal space. Each whole number is represented on a number line as a unit measure, inside which we can write the same number of infinite fractions as the space between one and zero. Viewed like this, reciprocal space can be seen to ‘reflect’ into the other number units. Just like a mirror that represents all the feature of the actual object standing in front of it.

At the end of Infinity

The question of ‘where do numbers end’, is therefore answered by the observations of reciprocal space. We observe that root numbers will diminish towards one. Therefore squaring, or any other power, will move the reciprocal result towards zero.

If zero is at the end of the sequence of infinite numbers found in reciprocal space, which is then reflected into whole number space, then we can clearly see an answer.


But how can zero be at the end of an infinite set of numbers that appears to be moving in exactly the opposite direction? 

Zero sits at the centre of the positive and negative numbers. Whenever a positive number exists, so does its counterpart in the negative. Regardless of the two values, the number zero will always be directly in between. 

This concept is explored in more detail in our post, the zero line.

zeroline e




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what did we learn about Reciprocal Space?

All numbers are contained within the numerical space between ZERO and ONE. This fact, combined with the comprehension of the Zero Boundary and infinity of ONE, suggests that all numbers above ONE are just reflections generated from reciprocal space

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What did we learn about Mathematics?

Whilst for most of us, the first mathematical process we are taught is that of addition and subtraction, inverse to this, the mathematics of infinity suggest that division is the first mathematical process that gives rise to all other mathematical functions.

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What did we learn Numbers?

The recognition of whole number space, being a ‘mirror’ reflection of reciprocal space, allows us to determine that the ZERO (boundary) must sit at the end of the infinite sequence of whole numbers. This is an important step in grasping the nature of infinity, which is explored in greater detail in our new theory of geometric maths.


Redefining our concept of numbers through geometry

Carry On Learning

This article is part of our new theory, ‘Maths of Infinity
Read the main article or browse more interesting post from the list below


Got a Question? Then leave a comment below.

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Abbey Brown Big
ABBY - Manager

Wow, some of that blew my mind. But I got a bit lost which the idea that zero is at the end of the infinite number line, seems to be a bit illogical?


At first, many things in the world of the infinite seem not to make sense. However, when we view reciprocal space, we can see that the zero boundary is the ‘reflection’ of the end of the whole number series, represented in reciprocal space. That is logical. Now we need to reconcile that fact that whole number units seem to move away from the number zero. The solution to this is found in the geometric transformation of the number line into a circle that is placed on a ‘Number Cross’. This is described in great detail in our new theory of Geometric Maths.

john hammond
JOHN - Engineer

But aren’t there more than just two ways to view something, apart from either unified or separate? For example, I can be a part of a company and still recognise it as not ‘being’ me.


In fact, the process of recognising you are both a part of something and at the same time separate is what we call ‘bi-location’. This is a particular state of awareness that is cultivated through the practice of 4D thinking. As an example, we can look at ourselves on the earth as an individual, yet at the same time recognise we are all sharing one planet, which helps us to develop an awareness of our actions at the larger scale. The same can be said of our bodies, that operate as a single unit, yet, at the same time, are composed of millions of cells. More on this can be found on our new theory of Harmonic Philosophy.

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