Definition:Negative/Number

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Definition

The concept of negative can be applied to the following sets of numbers:

$(1): \quad$ The integers $\Z$
$(2): \quad$ The rational numbers $\Q$
$(3): \quad$ The real numbers $\R$

The Complex Numbers cannot be Ordered Compatibly with Ring Structure, so there is no such concept as a negative complex number.

Integers

The negative integers comprise the set:

$\set {0, -1, -2, -3, \ldots}$

As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same.

Thus negative can be formally defined on $\Z$ as a relation induced on those equivalence classes as specified in the definition of integers.

That is, the integers being defined as all the difference congruence classes, negative can be defined directly as the relation specified as follows:

The integer $z \in \Z: z = \eqclass {\tuple {a, b} } \boxminus$ is negative if and only if $b > a$.

The set of negative integers is denoted $\Z_{\le 0}$.

An element of $\Z$ can be specifically indicated as being negative by prepending a $-$ sign:

$-x \in \Z_{\le 0} \iff x \in \Z_{\ge 0}$

Rational Numbers

The negative rational numbers are the set defined as:

$\Q_{\le 0} := \set {x \in \Q: x \le 0}$

That is, all the rational numbers that are less than or equal to zero.

Real Numbers

The negative real numbers are the set defined as:

$\R_{\le 0} := \set {x \in \R: x \le 0}$

That is, all the real numbers that are less than or equal to zero.

Complex Numbers

As the Complex Numbers cannot be Ordered Compatibly with Ring Structure, the concept of a negative complex number, relative to a specified zero, is not defined.

However, the negative of a complex number is defined as follows:

Let $z = a + i b$ be a complex number.

Then the negative of $z$ is defined as:

$-z = -a - i b$

Also known as

The notations $R_-$ and $R^-$ are also frequently seen for $\set {x \in R: x \le 0_R}$.

However, these notations are also used to denote $\set {x \in R: x < 0_R}$, that is $R_{< 0_R}$, and hence are ambiguous.

Some treatments of this subject use the term define non-positive to define $x \in R$ where $0_R \le x$, reserving the term negative for what is defined on this website as strictly negative.

With the conveniently unambiguous notation that has been adopted on this site, the distinction between the terms loses its importance, as the symbology removes the confusion.

Historical Note

The idea of a negative number was the cause of considerable philosophical difficulty.

Negative numbers made no sense to the ancient Greeks. A number expressed a magnitude and that was all.

Diophantus of Alexandria recognised that certain equations yielded both a positive root and a negative root, but rejected the negative root as nonsensical. If an equation had no positive root, for example $x + 10 = 5$, then it was not a proper equation.

The early Hindu mathematicians recognised the existence of negative roots, but were still wary of them. As Bhaskara II Acharya put it:

The second value is in this case not to be taken, for it is inadequate; people do not approve of negative roots.

The Chinese were using negative numbers by the $12$th century as a matter of course. However, they still did not recognise negative roots.

The general acceptance of negative numbers in calculations seems in fact to have started with merchants and accountants. The symbols $+$ and $-$ originated in $15$th century German warehouses for indicating whether a container was over or underweight.

Michael Stifel referred to negative numbers as absurd or fictitious.

It was Gerolamo Cardano who was one of the first to accept negative numbers, and he even went as far as considering their square roots in his Ars Magna.

However, as late as the end of the $18$th century, William Frend, together with Francis Maseres, published between them a number of works, most notably The Principles of Algebra in $1796$, which rejected the concept of negative numbers as invalid.