Introduction

Pick any number larger than one — say 16. Take its square root: you get 4. Take the square root again: roughly 2. Keep going. 1.41, 1.19, 1.09, 1.04 … The number shrinks towards ONE, closer and closer, but it never actually gets there. No matter how many times you press the square root button, no matter how powerful your computer, the result will always be slightly above ONE.

Now try the same thing with a number between zero and one — say 0.0625. Take its square root: 0.25. Again: 0.5. Again: 0.707 … The number climbs towards ONE from below, but again it never quite arrives.

ONE acts as an infinite boundary that no root calculation can cross.

This is not a quirk of arithmetic. It is a fundamental structural property of the number line — one that turns out to have deep consequences for how we understand powers, roots, geometry, and even the imaginary number i.

Why Does This Matter?

Most people think of ONE as ordinary — the starting point for counting, nothing special. But ONE and ZERO together divide the entire number line into four distinct regions, and both numbers act as infinite boundaries that arithmetic cannot breach. Understanding this changes how we think about the relationship between whole numbers and fractions, and it raises important questions about whether our current number system — including the complex numbers built around i — is truly complete.

Root Boundaries

The number ONE divides what mathematicians call reciprocal space (numbers between ZERO and ONE) from whole number space (numbers greater than ONE). ZERO does something similar: you can divide by ever-smaller numbers and approach ZERO, but you can never actually reach it through division alone.

ONE mirrors this behaviour for roots. A root calculation (square root, cube root, and so on) always pulls the result towards ONE — but the boundary holds in both directions:

  • Starting above ONE: repeated roots bring you closer and closer to ONE from above, never below.
  • Starting below ONE (but above ZERO): repeated roots bring you closer and closer to ONE from below, never above.
Number line divided into four sections by ZERO and ONE: negative whole numbers, negative reciprocals, positive reciprocals (0 to 1), and positive whole numbers (greater than 1)
ZERO and ONE divide the number line into four distinct regions. Roots pull any value towards ONE; division pulls any value towards ZERO — these are not the same operation.

An important distinction follows from this: roots and divisions are not the same thing. Division of a reciprocal (a number between ZERO and ONE) moves the result away from ONE, towards ZERO. Rooting that same reciprocal moves the result towards ONE. They are opposite in effect, even though both are often loosely grouped as "making numbers smaller."

This observation points to something deeper. ZERO and ONE are the only two numbers on the number line that create these kinds of boundary under simple arithmetic. Every other number — including all fractions — lacks this property. That raises a genuine question: what is the origin and nature of all numbers other than ONE and ZERO?

The Roots of Infinity

To understand why ONE behaves this way, it helps to understand what a root actually is.

When you square a number, you multiply it by itself: 3 × 3 = 9. When you cube a number, you multiply it by itself three times: 3 × 3 × 3 = 27. The count of how many equal values are multiplied together is the power of the equation, written mathematically as x². The reverse operation is a root: the square root asks "what number, multiplied by itself, gives this result?" The cube root asks the same for three equal factors.

Crucially, there is no reason to stop at squares and cubes. You can raise a number to the fourth power, the hundredth power, the millionth power — or to the power of infinity itself (x^∞). The same holds for roots. Every power has a corresponding root, and there are infinitely many of both.

Tree diagram showing an infinite branching structure of roots and powers for any number
There are an infinite number of roots and powers for every whole and fractional number. Squares and square roots are just the most familiar pair.

Finding the Infinity of ONE

You can verify the boundary of ONE directly on any calculator with a square root button.

Starting above ONE: Enter any number greater than ONE — for example, 100. Press the square root button. The result is 10. Press it again: 3.162. Again: 1.778. Again: 1.334. Keep pressing. The number shrinks steadily towards ONE but never dips below it.

Starting below ONE: Enter any number between ZERO and ONE — for example, 0.01. Press the square root button. The result is 0.1. Again: 0.316. Again: 0.562. Again: 0.749. Keep pressing. The number grows steadily towards ONE but never rises above it.

Graph showing two curves — one descending from above ONE and one rising from below — both approaching ONE asymptotically through repeated root calculations
Repeated root calculations from above and below ONE both converge towards ONE without ever reaching it. This infinite approach is the "infinity of ONE."

Note: if your calculator eventually displays exactly ONE, it has run out of decimal places and is rounding. Mathematically, the value only ever approaches ONE — it never arrives. True infinity lies beyond what any computer can calculate.

Geometric Powers and Roots

The reason ONE acts as this boundary becomes clearer when we look at powers and roots geometrically rather than purely algebraically.

Rotation Around ZERO

Think about squaring a number geometrically. Squaring takes a one-dimensional length and produces a two-dimensional area — a square. Geometrically, this is equivalent to rotating the number line by 90 degrees around the zero point. A three-dimensional cube adds a third axis, rotated again in space. Each increase in power adds another dimension.

This means that numbers raised to any power cannot be accurately described on a one-dimensional number line — they inherently occupy higher-dimensional space.

Diagram showing a number line rotated 90 degrees to form a two-dimensional square number plane around the origin
Squaring a number is geometrically equivalent to rotating the number line 90 degrees. Note that in the traditional view shown here, the negative portion of the line does not rotate — a limitation addressed in our geometric mathematics framework.

Implications for the Number i

The imaginary number i (pronounced "eye") is defined as the square root of −1. Because no real number multiplied by itself produces a negative result, √−1 does not appear anywhere on the standard number line. Mathematicians introduced i as a new kind of number to fill this gap, and from it constructed the complex number plane — a two-dimensional space that combines ordinary (real) numbers with imaginary ones.

The complex number plane is powerful and has countless applications in physics and engineering. However, i only addresses one specific case: the square root of −1. If powers and roots are truly infinite in number, then i represents just one out of an infinite family of such quantities. We have no agreed-upon definition for the cube root of −1, the fourth root of −1, or any higher-dimensional equivalent.

Diagram of the complex number plane with the real axis horizontal and the imaginary axis vertical, showing how i is defined as a 90-degree rotation
The complex number plane introduces the imaginary axis to accommodate √−1. While this extends mathematics considerably, it leaves the cube root of −1, the fourth root of −1, and all higher equivalents undefined.

This is not a minor technicality. If the complex plane were truly complete — capturing all possible number types — then it would need to account for all powers and roots, not just the second. The claim that complex numbers complete our number theory has not been formally proved in a way that addresses this infinite hierarchy. This also has implications for the Continuum Hypothesis, which concerns the nature of infinity in the number line and which we consider to be an open question rather than a settled one.

Conclusion

The number ONE is far from ordinary. Like ZERO, it acts as an infinite boundary — but where ZERO's boundary is revealed by division, ONE's boundary is revealed by roots. No root calculation, applied however many times, can push a result across the threshold of ONE.

Together, ZERO and ONE divide the number line into four distinct regions: positive whole numbers, negative whole numbers, and their reciprocal counterparts between −1 and 0 and between 0 and 1. These two numbers stand apart from every other number in the way they anchor arithmetic structure.

Extending this insight to geometry shows that powers and roots are not purely numerical operations — they are dimensional shifts. Squaring moves from a line to a plane; cubing moves to three-dimensional space. The imaginary number i captures one instance of this dimensional extension (rotation by 90 degrees for the square root of −1), but an infinite hierarchy of higher-dimensional roots remains mathematically uncharted.

Understanding the infinity of ONE opens a window onto a more complete geometric picture of numbers — one that treats the number line not as the whole story, but as the first dimension of a much richer structure.

FAQ

What about negative numbers — don't they form negative squares?

In our geometric theory of mathematics, they do. Traditional mathematics cannot define a negative square number due to the way it has been set up. See the article on negative square numbers for details.

Is a cube root the diagonal of a cube?

No — a cube root tells you the side length of a cube that has a specific volume. The diagonal of a square is found through the square root of 2, and the diagonal of a cube through the square root of 3. You multiply the side length by either of these factors to find the respective diagonal.

Why does a calculator sometimes show exactly ONE when you keep pressing the square root button?

The calculator has run out of decimal places and rounds the result to ONE. Mathematically, the value only ever approaches ONE — it never actually reaches it. Computers cannot represent true infinity.

Does the infinity of ONE apply to cube roots and higher roots too?

Yes. The boundary effect holds for all powers and roots, not just squares. Whether you take the cube root, the fourth root, or any root of a number greater than ONE, the result always stays above ONE. The same applies in reciprocal space: any root of a number between ZERO and ONE stays below ONE.