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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 Totally Ordered, so there is no such concept as a negative complex number.


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 $z \in \Z: z = \left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$ is negative iff $b > a$.

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:

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

Rational Numbers

The negative rational numbers are the set defined as:

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

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:

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

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

Complex Numbers

As the Complex Numbers cannot be Totally Ordered, 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:

$-z = -a - i b$

Historical Note

The idea of a negative number 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 Bhāskara II Āchārya 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.