Through examining the reciprocal number space between  zero and one, we find that the infinite set of natural numbers each exhibits a particular point within this number space. As the whole numbers extend, so the reciprocal diminishes towards infinity. The implication is that infinity and zero are a mirror opposite of the entire infinite set.


We can choose any whole number and identify its reciprocal value. This means that every whole number can be represented within the unit space between zero and one. As the whole numbers increase towards infinity, so its equivalent reciprocal value diminishes toward zero.

Therefore, we can perceive that at the concept of infinity, that appears at the outermost limit of the whole numbers, holds the same quality as zero within reciprocal space.

This concept can be demonstrated by drawing a line of two equal unit length. We label the first unit ‘reciprocal’ number space, and the second unit, whole number space. When the line is folded, the points that form the reciprocal values of all whole numbers can be ‘mirrored’ into whole number space. Once folded we find that zero and infinity occupy the same space. 

Just as any number will never reach infinity, so all reciprocal numbers will never equal zero. Zero therefore holds the same quality as infinity, which is clearly demonstrated by the ‘folding’ of number space, where the infinity at the end of the number line fall in exactly the same place as zero.

KEy Points

  • All whole numbers can be represented as a reciprocal value between zero and one
  • As numbers increace to infinity, so thier reciprocal value decreaces towards zero
  • This demonstrates that infinity and zero are similar in concept, forming the boundary that contains the infinite set of numbers and thier reciprocals.



How to count infinity

At first, the idea that zero should start numbers seems fairly logical. But the idea that you should count 1,2,3,4… and so on, until we reach…ZERO? Yes, the last number in the series of infinite whole numbers will actually be zero. You might think that that sounds rather odd. However, once you begin to look into the world of infinity, you will find out that there can be no other truth. Let us begin by taking the concept of numbers.

When we count a number, what we’re really doing is, finding and identifying a unit within infinite number space. What this means is that we are taking a number, (i.e. one, two), in between which, each exhibits an infinite number space. The space between 1 and 2 can be divided into as many infinite small decimal fractions as the number space proceeding it.

Also, there are an infinite number of numbers that can be created just through whole fractions. This means that each whole number unit contains an infinite number of potential numbers. So amongst all this infinity, how can we say the zero is the last number? Does that even make sense?

When we’re counting numbers from the perspective of infinity, we do not view numbers as units added to an ever-growing number line. Instead, we need to subtract a single infinite set from the total set of infinite numbers for each step counted. In this way we can note how many units of infinite sets are missing from the whole infinite set of real numbers. This is one way we can compare the density of infinity. By identifying, which numbers are missing.

A complete set will be an infinity of infinities. If we count the number one we can exclude it from the set, so 1 becomes ∞-1. If we move into the next number two, we represent that as ∞-2, number three, ∞-3, and so on. Each number unit contains an infinite number space. As we carry on 1,2,3,4,5…. Into infinity, we are subtracting an infinity as each step, until we reach infinity.

subtracting infinities

Key point

From the perspective of the infinite We don't count, we subtract infinite sets for each unit.

The infinity between infinities

There are as many numbers between zero and one as there are between one and infinity. If we divide a line, we are shortening the length section. As we divide the line into equal parts in reciprocal space, so we are also creating countable units in whole numbers space.

These are in a 1:1 ratio that are equal infinities. As the division process continues to infinity, the reciprocal values approach zero. We call this the ZERO boundary. At no point through division will the result ever reach zero, and zero is the last number in the series. On the inverse, we see that while numbers increase towards infinity, they decrease in size when located in reciprocal space, exactly the opposite direction. At no point will the number above ONE reach infinity, similarly, no reciprocal representation of a whole number will never reach zero. Just as infinity is at the end of the set of infinite numbers, so zero is at the end of their reciprocal counterpart.

If you think about the infinity symbol, it looks like two zeroes that are joined at a single point on the ‘circles’, circumference. Infinity is a type of zero that limits ‘whole’ number space, whereas zero limits ‘reciprocal’ number space. When we look at the difference between whole and reciprocal numbers, we find that both are fractions defined by the number one. Like numbers pairs that appear on opposite sides of number space. So it all depends on which side of the mirror reflection the numbers appears.

zero at the end of infinity

The limit of infinity

If we consider the nature of counting to infinity as the subtraction of a set of infinite fractions from the whole number set at each step, we can begin understand the relationship of zero and infinity. By continuing this subtractive process up to its infinite limit, we find that the calculation will be ∞=0. As the result of infinity minus infinity equals zero, we can be assured that the number zero is the final number that exists at the end of the number line.

zero at the end of infinity e

Infinity Squared

In the image above we see the number line represents at the top counting numbers into infinity. However, from the perspective of infinite density, we are removing and infinite set at each step. If we continue to subtract infinite sets then when the line reaches infinity we need to subtract infinity, which leaves the result, zero, at the end of the number line.

Can you spot the logical problem?

Actually, this is only true of the whole numbers. Each unit, however, contains an infinite variety of decimal fractions. So really at the end of the number line is ∞² – ∞² = 0. This is true of all the real numbers that include decimal fractions. Therefore, the more exact expression subtracts a single ∞ set at each step. You can find out more about this identity by exploring the infinity equation, and our solution to the nature of infinity.

Aleph 0.5

Begin Learning about our solution to the worlds number 1 mathematical challenge of the 20th century. They thought it could not be done, but in fact our solution is so simple, anyone can understand it



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What does this tell us about the relationship of zero and infinity?

As infinity terminates the set of whole natural numbers, so zero terminates the infinite set of reciprocals. Each sits at the opposite side of the infinite set. We can represent these two sets by defining a line length of TWO units, one to represent reciprocal numbers, into which the whole numbers are ‘reflected.

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What is important about this realisation

In this post, we begin to outline the concept of folding the number line. This is important as it begins to answer the Russell Paradox. By folding the number line, we are performing a rotation of numerical space. This is one of the key concepts of Geometric Maths, that is able to put a bound on the infinite, which in turn allows us greater insight into the nature of the infinite that lies at the foundation of modern number theory.


Find out more about 4th dimensional mathematics by reading these interesting articles.





Got a Question? Then leave a comment below.

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john hammond
JOHN - Engineer

I did not quite understand how the whole number units should be contained within two units of space.


In fact, in our theory of Universal Maths we suggest that the unit numbers are restricted to 0,1, and 2 in 4D space. These form the numerical spaces from which all other numbers originate. Outside of this, geometric constants operate outside the structure of number.


Why is infinity squared in the final section of the article?


If each unit has an infinite set of fractional numbers, then the entire number line is actually constructed from an infinities worth of infinity. When we count to ONE, we are subtracting a whole ∞ set of decimal fractions that begin with 0.n. In the next step we subtract a second ∞ set, all fractions beginning with 1.n. and so on into infinity. So 1× infinity, then 2× infinity, and so on. When we reach infinity itself, we have ∞×∞ or put another way ∞².

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