ELH (Elhan numbers) Designation of large and very large numbers in calculations in programming

ELH ( elhan-numbers ) is a special notation in programming for all numbers (an abstract set of numbers) that:

• are outside the range acceptable for the given data type; (cannot be calculated using the given data type (e.g. int32, float64))

• cannot be calculated using standard operations (are outside the range available for standard operations.)

• cannot be physically displayed (for example, on a screen or in the device's memory).

An example of the appearance of such numbers:

tetration 7^^5 online

History of origin

• The term ELH is derived from the name Elhan, which is a combination of the last names of two programmers, El and Han .

• Their research led to the idea of ​​introducing a special notation for numbers that go beyond the capabilities of standard data types.

• This is how the concept of elhan numbers emerged—a kind of marker for “impossible” values.

In other words, ELH is the "zone beyond the horizon" of calculations, where all values ​​that exceed the technical capabilities of the system fall.

Basic properties of ELH

• Origin: The term is derived from the surnames of two programmers, El and Han.

• Meaning: Marker for an "impossible value" that occurs during overflow or when computing hyperoperations (tetration, pentation, and higher).

• Behavior:

  • like NaN, ELH is not equal to itself (ELH != ELH);

  • differs from Infinity, as it does not mean infinity, but rather an “uncomputable number.”

• Usage: Used to signal that a result is out of bounds in computable space.

Comparison with other programming symbols

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Designation	Meaning	Origin	Meaning	Example of occurrence	Property
NaN	Calculation error (0/0, sqrt(-1))	IEEE-754	"Not a Number" — result of incorrect operations (eg 0/0)	Division 0/0	NaN != NaN
INF	Infinity on float overflow	IEEE-754	Infinity on overflow floating-point numbers	1e308 * 1e308	∞ == ∞
ELH	Uncomputable in computer science or aorithm finite a number outside the bounds of the data type (variable)	El + Han	"Elhan-numbers" - all meanings, which are impossible calculate or display	10↑↑5 or int32 overflow	ELH != ELH

Thus, ELH is closer to NaN, but it reflects overflow and impossibility of representation, rather than a mathematical error.

Practical significance

• In algorithms: allows you to explicitly record the moment of overflow.

• In hyperoperations: serves as a marker for numbers that cannot be represented even theoretically within the framework of a computing system.

• In complexity theory: helps separate the realm of the computable from the realm of the "impossible."

# Python: Working with Large Numbers
x = 10**100 # number with 100 zeros
y = 1e308 # scientific notation
print(x)
print(y)

x = MAX_INT32
y = x + 1

if y == y:
    print("The value is correct")
else:
    print("Overflow: ELH")

Elhan numbers (ELH) and hyperoperations

As everyone remembers, there are different types of hyperoperations

• Raising to a power: a^b.

• Tetration: a↑↑b — repeated raising to a power. Example: 2↑↑4= =65,536

• Pentation: a↑↑↑b .

• And so on - each operation grows faster than the previous one.

Even for small values ​​of arguments, the results of computing tetration and higher hyperoperations become astronomical.

Where does ELH occur?

• When calculating the titration and above, the values ​​go beyond the limits:

  • standard types (int32, float64);

  • even extended libraries (BigInteger).

• For example:

  • 10↑↑5 = 10^{10^{10^{10^{10}}}}. This number cannot be stored or displayed.

• In such cases, the result can be interpreted as an ELH number—a marker of the impossibility of calculation, but the number is finite.

Practical significance of ELH

• Overflow signal: ELH is returned instead of infinity or error, which clearly indicates that the computation is beyond its limits.

• Marker in algorithms: when modeling hyperoperations, ELH helps to distinguish:

  • computable values ​​(up to a certain level);

  • "the zone of the impossible" (above the threshold).

• In mathematical models, ELH can be interpreted as an abstract container for all “supernumbers”.

function tetration(a, b):
    result = a
    for i in 1..b:
        result = pow(a, result)
        if result is out of type:
            return ELH
    return result

print(tetration(2, 4)) # 65536
print(tetration(10, 5)) # ELH

function add(a, b):
    result = a + b
    if result is out of type:
        return ELH
    else:
        return result

x = MAX_INT32
y = x + 1000

if y == y:
    print("Correct value")
else:
    print("Overflow: ELH")

Here, on overflow, ELH is returned, which is not equal to itself, making it easy to catch such situations.

5. Practical significance

• ELH can be seen as a conceptual tool for future programming languages.

• It helps to clearly separate "impossible" values ​​from normal calculation errors.

• Unlike NaN, ELH signals precisely the boundaries of computability.