Division by Zero

DIVISION BY ZERO

"Zero, a dangerous Idea" (Charles Seife)

"Nullity is the missing link that turns real arithmetic into total arithmetic" (James Anderson).

The Division by Zero Problem
A historical problem

Since the invention of zero by the Hindus, more than a thousand years ago, people have been perplexed, astonished, doubtful when confronted with dividing a number by zero. Faced with this situation there are several opinions about the result:

It is an infinite number (∞) because the denominator is the limit of a positive real number tending to zero.
It is −∞ because the denominator is the limit of a negative real number tending to zero.
It is both +∞ and −∞.
It is the number itself, because by dividing it by nothing, that is, by not dividing it, it remains invariant.
It is an imaginary number.
It is an indeterminate number.
It is not a number.
The operation is impossible.

Of all these interpretations, the most widely accepted is that the result is infinite.

Another problem, related to the previous one, is the expression 0/0. In this case, there are also several opinions about the result:

It is 1, since numerator and denominator are equal.
It is 0, because by dividing a number by nothing, that is, not dividing it, it remains unchanged.
It is an "indeterminate" value, not a concrete number. It was based on the fact that if 0÷0 = c, then, by the inverse operation, c×0 = 0, and this is satisfied by any c.

In this case, the most accepted interpretation is the latter.

Two conclusions are drawn from these two problems:

The arithmetic is not complete, since the operations 1/0 and 0/0 are not defined.
The two interpretations of 1/0 and 0/0 imply transcending arithmetic itself, since they are abstractions superior to the abstraction of number. Moreover, they can give rise to mathematical paradoxes. One of them is: 0×1 = 0×2 = 0. Dividing by zero, (0/0)×1 = (0/0)×2. Eliminating the common factor, 1 = 2.

The IEEE 754 standard

In computing, an attempt to divide by zero produces an exception, an error message, and the result is "Not a Number" (NaN). This acronym is part of the IEEE 754 standard, a term used in computer floating-point arithmetic.

To prevent division by zero, programs should always ask if the divisor is zero, and if so, branch to a routine that deals with this circumstance. It should also be checked if the divisor is less than a certain value, as the result may cause overflow. It should also be checked that it is not too large so that it does not produce underflow.

According to IEEE 754, the result NaN can be represented in floating point format with ones in the exponent and non-zero numbers in the mantissa.

Arithmetic operations involving a NaN produce a NaN, whereupon this value is propagated through the calculations. In the standard there are two types of NaN: quiet NaN, which generate no additional exceptions in propagation), and signalling NaN, where it propagates.

The proposed revised standard (IEEE 754r) also establishes maximum (maxnum) and minimum (minnum) values, so that a NaN can occur under various circumstances: division by zero, when trying to calculate the square root of a negative number, by underflow (generating a value less than minnum, by overflow (generating a value greater than maxnum), and so on.

The IEEE 754 standard contemplates, in addition to NaN, signed zero (+0 and −0) and signed infinity (+∞ and −∞). It contemplates a series of axioms involving these three elements (they are detailed in the following section). The justification of the sign is based on knowing the sign of the result, in the case of underflow.

A case of severe divide-by-zero problem occurred in the Remote Data Base Manager aboard the USS Yorktown (CG-48), causing the ship's propulsion system to fail.

James Anderson's solution

James Arthur Dean Dean Anderson (known as James Anderson), a professor of computer science at the University of Reading (UK), gained public prominence in 2006 when he claimed to have "solved a 1200-year-old problem: division by zero." It was eventually concluded that it was actually a variant of the NaN concept.

Anderson invented transreal arithmetic, later called transarithmetic or transmathematics. It included all real numbers and three others: +∞, −∞ and Φ (nullity), where Φ is the representation of 0÷0, a number that falls outside the real line. Φ symbolizes a zero divided by itself.

The axioms of transreal arithmetic are identical to those of IEEE standard floating arithmetic, except for the last one. The comparative table is :

Transreal arithmetic	IEEE
0 ÷ 0 = Φ	0 ÷ 0 = NaN
+1 ÷ 0 = +∞	1 ÷ +0 = −1 ÷ −0 = +∞
−1 ÷ 0 = −∞	1 ÷ −0 = −1 ÷ 0 = −∞
∞ × 0 = Φ	∞ × 0 = NaN
∞ − ∞ = Φ	∞ − ∞ = NaN
Φ + a = Φ	NaN + a = NaN
Φ × a = Φ	NaN × a = NaN
−Φ = Φ	−NaN = NaN
PΦ = Φ? = T	PΦ = Φ? = F

In transreal arithmetic x − x is not always 0, because Φ − Φ = Φ. And neither x÷x = 1 because Φ÷Φ = Φ.

Anderson calls his arithmetic "total" in the sense that division is applicable to all numbers, as in the operations of addition, subtraction, and multiplication.

Jesper Carlström's solution

Carlström (Stockholm University) invented the concept of the "wheel" (wheel), the wheel theory (wheel theory), where the real numbers are extended to a wheel. The term "wheel" is inspired by the image of a circle projected onto the real line, which includes the point 0/0.

It is a type of algebra where division is always defined, and where division by zero makes sense. In this algebra division is not a binary (two-argument) operation, but division is considered a unary (one-argument) operation. The operation /x indicates something similar (but not identical) to x⁻¹, where.

A wheel is an algebraic structure (commutative ring) (W, 0, 1, +, ·, /) with the binary operations of addition and product, the unary operator "/" and the constants 0 and 1, with the following axioms:

In this algebra it is fulfilled:

The Solution in MENTAL
Division by zero is a problem, but it is also an opportunity. Zero symbolizes consciousness. And division by zero is a moment of maximum awareness. It is not a curiosity. It is of great interest, as is the case with paradoxes, with everything that defies the rational and logic, that forces us to seek a solution that transcends arithmetic.

In MENTAL we can capture the event of a division by zero by means of the generic expression

⟨( x÷0 → process) ⟩

In this way, programs need not always ask at each operation whether the divisor is zero, since this generic expression covers all cases.

It is clear that 0/0 is an indeterminate value because if we consider that result is r, the product of r by 0 is zero. Then any r satisfies it.

In MENTAL we already have an expression that represents any (non-zero) expression, numeric or not: α, the existential expression. Then we can infer and define that
0÷0 = α, and it is fulfilled:

α*0 = α(1−1) = α& −α = 0 Analogously we can define 1÷0 = β.

The calculation with α and β

When a calculation process finds a value of 0÷0, by inverse evaluation, it is replaced by α. And, similarly, when it finds 1÷0, it is replaced by β.

The tables of arithmetic operations are:

+	0	1	α	β
0	0	0	α	β
1	1	2	α	β
α	α	α	α	α
β	β	β	α	α
r	r	r+1	α	β

−	0	1	α	β
0	0	−1	α	−β
1	β	1	α	0
α	α	α	α	α
β	β	β	α	α
r	r	r−1	α	−β

×	0	1	α	β
0	0	0	α	α
1	0	1	α	β
α	α	α	α	α
β	α	β	α	β
r	0	r	α	rβ

÷	0	1	α	β
0	α	0	α	0
1	β	1	α	0
α	α	α	α	α
β	β	β	α	α
r	rβ	r	α	0

With the elements α and β, and with these operations defined, computer programs produce no exceptions and can arrive at a final result, which may include α, β or both.

Notes:

α and β are imaginary expressions with which direct operations are performed, but the properties of traditional arithmetic are not satisfied.

If x×y = z, then x = z/ y, and if x/y = z, then x = z×y

For example, from α×1 = α it does not follow that α/α = 1, since α/α = α.
The expression 0⁰ is also α. Indeed: 0⁰ = 0ⁿ/0ⁿ = 0/0. Therefore, (0^0 = α).
The number r can be negative. For example, (−1)×α = α.
α is invariant with respect to all arithmetic operations:

r+α = α, r−α = α, r·α = α, r/α = α

The transcendent

The expressions α and β are higher order abstractions, and can be considered two imaginary numbers. They represent two dual and transcendent elements, where α is more transcendent (or superior) to β, because α is of qualitative type: it is an invariant with respect to all arithmetic operations, including itself and β (α×β = α). And β is a quantitative element: β+β = 2β.

Advances in mathematics have been linked to the discovery of unknown elements in operations and expressions. This is how they were discovered:

Negative numbers, for example, in 5+x = 2. There was a time when negative numbers were considered imaginary.
Rational numbers, for example in 5x = 2.
The irrational numbers, as in x² = 2.
Imaginary numbers, with x² = −1.
The infinitesimal ε, an imaginary number defined as ε² = 0. This is a substitution (because if it were an equation it would be ε = 0). This definition is much simpler than the traditional definition of limit.

Now it is necessary to consider the expressions α and β as the foundation of an extended and complete arithmetic based on division by zero. A new paradigm.

Transreal arithmetic vs. MENTAL

Both systems are similar, but with some differences:

In transreal arithmetic axioms are needed to define extended arithmetic, namely 32, 13 more than in ordinary arithmetic. The extra axioms are the axioms involving strictly transreal numbers (−∞, +∞, Φ).

In MENTAL, only two axioms are needed, which are the definitions of α and β. The result is obtained by applying the rules of arithmetic, namely these four:
For example,
At all times semantics are maintained.
In transreal arithmetic, nullity is an unknown, fixed number. In MENTAL, its equivalent is α, which is a nonzero expression. Since α is an expression, and not just a number, arithmetic connects to something higher, namely logic, since θ and α represent the null and non-null expressions (or existential meta-expressions), respectively, which replace the values F (false) and T (true) of binary logic. The expression α is not something unknown or fixed, but something on a higher level, beyond arithmetic and definite semantics.
In transreal arithmetic, 1/0 is considered to be infinite (∞), a quality, not a quantity. In MENTAL, 1/0 is simply called β, with no associated semantics other than the expression itself.
It makes no sense to talk about distinguishing between +0 and −0, since in the number zero all opposite numbers converge. And zero is the opposite of itself.

Comparison between expressions:

Transreal arithmetic	MENTAL
0 ÷ 0 = Φ	0 ÷ 0 = α
+1 ÷ 0 = +∞	1 ÷ 0 = β
−1 ÷ 0 = −∞	−β = −1 ÷ 0
a + ∞ = ∞	a + β = β
a − ∞ = −∞	a − β = −β
a× ∞ = ∞	a − β = β
∞ × 0 = Φ	β × 0 = α
∞ − ∞ = Φ	β − β = α
1 ÷ ∞ = 0	1 ÷ β = 0
Φ + a = Φ	α + a = α
Φ × a = Φ	α × a = α
−Φ = Φ	−α = α
1 ÷ Φ = Φ	1 ÷ α = α
Φ = Φ? = T	α = α? = T

Addenda
The diffusion of transreal arithmetic

Anderson has been trying to market his ideas of transreal arithmetic and Perspex machines to investors. He believes his work can produce computers that run several orders of magnitude faster than today's computers. And that this can help enrich our understanding of the universe and mathematics, improve our lives, as well as solve problems such as quantum gravity, the mind-body connection, consciousness and free will.

Anderson introduced his transreal arithmetic, along with the concept of Nullity, to the public on the BBC in December 2006. He was presented as the "discoverer" of the solution to the division-by-zero problem, rather than an attempt to formalize it. After the presentation, there were many, many people who were interested in his work.

Anderson has tried to generalize his transreal arithmetic to complex numbers (transcomplex numbers) and differential calculus (transdifferential numbers?).

A short history of the division by zero problem

The number zero was known in the 7th century in India. Brahmagupta (eln his text Brahmasphutasiddhanta) treated zero as a number in its own right (and not as a mere placeholder symbol) and described various rules for operating with it, and its properties relating to addition, subtraction and product, stating that "zero divided by zero is zero".

In the 9th century, in India, Mahavira, updated Brahmagupta's ideas and stated that "a number remains unchanged when divided by zero".

In the 12th century, Bhaskara wrote on the subject of division by zero. Although he did not state it clearly, he is usually interpreted as stating that division by zero is infinity.

In the 18th century, Newton and Leibniz invented differential calculus, an extraordinarily useful calculus based on infinitesimal numbers, close to zero, but not equal to zero.

In 1985 the IEEE 754 standard appeared, which was born to detect division by zero in the execution of computer programs, and where it adopted the concept NaN (Not a Number).

In 2004, Jesper Carlström invented algebraic structures called "wheels" that allow division by zero. It also uses axioms, but not very intuitive.

In 2006, James Anderson invented transreal arithmetic using a series of axioms.

Finally, MENTAL appears, a universal formal language, which solves the problem in the simplest possible way by defining two imaginary expressions (α and β) that make arithmetic complete, and without the need for axioms.

The conclusion is that in zero (in nothingness), paradoxically, resides everything, resides an enormous power, precisely because zero symbolizes consciousness (because it is the union of opposites) and the soul (because in it there is neither space nor time).

Bibliography

Anderson, James. Representing Geometrical Knowledge. Internet, 1997.
Anderson, James. Transmathematics. Internet, 2009.
Anderson, James. Transarithmetics. Internet, 2007.
Anderson, James. Perspex Machine IX: Transreal Analysis. Internet, 2006.
Anderson, James. Transreal Computing. Research and Portfolio − Company Showcase. Internet, 2006.
Anderson, James. Evolutionary and revolutionary effects of transcomputation. In 2nd IMA Conference on Mathematics in Defence. Institute of Mathematics and its Applications, Oct. 2011.
Cajori, Florian. Absurdities due to división by zero: An historical note. The Mathematics Teacher, pp. 366-8.
Carlström, Jesper. Wheels: On the división by zero. Mathematical Structures in Computer Science, vol. 14, 2004, pp. 143-184. Disponible online.
Grupo del IEEE sobre la norma 754: http://grouper.ieee.org/groups/754/
Kaplan, Robert. The Nothing that is: A Natural History of Zero. Internet, 2000.
Reis, Tiago S. dos; Anderson, James. Construction of the Transcomplex Numbers From the Complex Numbers. Internet, 2004.
Seife, Charles. Cero: la biografía de una idea peligrosa. Ellago, 2006.
Setzer, Anton. Wheels. Internet, 2007.