An Introduction to Transreal Numbers

../_images/number_line.png

The line, shown above is the real number-line, which contains all of the finite numbers. The big dots ⬤ show the non-finite numbers. The extended-real number-line is the real number-line with infinity (∞) and minus infinity (-∞). Every number to the right of zero, on the extended-real number-line, is positive; every number to the left of zero, on the extended-real number-line, is negative. Zero is neither positive nor negative. The sign of zero is zero. This definition, which is widely accepted in the world, contradicts the teaching of sign in French speaking primary and secondary schools, where it is taught that zero is a positive number.

The transreal number-line is the extended-real numberline with nullity. Nullity (𝜱) lies off the extended-real number-line [1]. Nullity is neither positive nor negative. The sign of nullity is nullity. Infinity (∞) is the most positive number and minus infinity (-∞) is the most negative number. Infinity is bigger, meaning further to the right on the extended real number-line, than any number apart from itself and nullity. Minus infinity is smaller, meaning further to the left on the extended-real number-line, than any number apart from itself and nullity. Nullity is equal to itself but is not bigger than or smaller than any number! This is because it is not on the extended-real number line.

This paints a mental picture of how the transreal numbers relate to each other and gives a vocabulary for talking about these relationships. Both aspects are extended in more advanced study.

The standard form of certain numbers is as follows. Transreal one (1) is real one divided by real one: \(\frac{1}{1}\). Transreal minus-one (-1) is real minus-one divided by real one: \(\frac{-1}{1}\). Transreal zero (0) is real zero divided by real one: \(\frac{0}{1}\). Transreal infinity (∞) is real one divided by real zero: \(\frac{1}{0}\). Transreal minus-infinity (-∞) is real minus-one divided by real zero: \(\frac{-1}{0}\). Transreal nullity (𝜱) is real zero divided by real zero: \(\frac{0}{0}\). Any irrational number \(x\) is \(x\) divided by real one: \(\frac{x}{1}\).

Transreal numbers can be expressed as transreal fractions \(\frac{n}{d}\) of a real numerator \(n\) and a non-negative, real denominator \(d\). Transreal numbers with a non-finite numerator or denominator simplify to this form. An improper fraction can be written with a negative denominator, but it must be converted to a proper fraction, by carrying the sign up to the numerator, before applying any transreal, arithmetical operation. This can be done by multiplying both the numerator and denominator by minus one; it can be done by negating both the numerator and the denominator, using subtraction; and it can be done, lexically, by moving the minus sign from the denominator to the numerator.

(1)\[\frac{n}{-d} = \frac{-1 \times n}{-1 \times (-d)} = \frac{-n}{-(-d)} = \frac{-n}{d}\]

Transreal infinity is equal to any positive number divided by zero. Transreal minus-infinity is equal to any negative number divided by zero. Zero is equal to zero divided by any positive or negative number. That is, with \(k > 0\) we have:

(2)\[ \begin{align}\begin{aligned}\infty = \frac{1}{0} = \frac{k}{0}\\-\infty = \frac{-1}{0} = \frac{-k}{0}\\0 = \frac{0}{1} = \frac{0}{k} = \frac{0}{-k}\end{aligned}\end{align} \]

The ordinary rules for multiplication and division apply universally to proper transreal-fractions. That is, they apply without side conditions. In particular, division by zero is allowed.

(3)\[ \begin{align}\begin{aligned}\frac{a}{b} \times \frac{c}{d} = \frac{a \times c }{b \times d}\\\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\end{aligned}\end{align} \]

Addition is more difficult than multiplication and division because it breaks into two cases: the addition of two signed infinities and the general case.

The general rule of addition looks like this.

(4)\[\frac{a}{b} + \frac{c}{d} = \frac{a \times d + b \times c}{b \times d}\]

However, two infinities are added using the rule for adding fractions with a common denominator [2]. The sign of each infinity, \(\pm \infty\), may be chosen independently but the chosen sign is then carried into the corresponding term \(\pm 1\) so that \(+ \infty\) has \(+1\) and \(- \infty\) has \(-1\):

(5)\[\pm \infty + \pm \infty = \frac{\pm 1}{0} + \frac{\pm 1}{0} = \frac{(\pm 1) + (\pm 1)}{0}\]

Subtraction is the addition of a negated number:

(6)\[\frac{a}{b} - \frac{c}{d} = \frac{a}{b} + \frac{-c}{d}\]

Transreal arithmetic is totally associative and totally commutative but it is only partially distributive at infinity. The axiom of transreal distributivity [P8] can be broken down into a number of cases. As usual, a number \(a\) distributes over \(b + c\) only when:

(7)\[a \times ( b + c ) = a \times b + a \times c\]

If \(a\) is finite or nullity then \(a\) distributes over any \(b + c\). If \(a\) is infinity or minus infinity then \(a\) distributes if \(b + c = \Phi\) or \(b + c = 0\) or \(b\) and \(c\) have the same sign [3].


[1]At distance and angle nullity.
[2]If infinities were added by the general rule we would have \(\infty + \infty = \Phi\), but this is inconsistent with various arithmetics ([A1], [A2], [A3]) of the infinite that all have \(\infty + \infty = \infty\).
[3]Two numbers have the same sign if they are both positive, both negative, both zero or both nullity.