Nombor negatif dan nombor bukan negatif

Sesuatu nombor negatif merupakan suatu nombor yang lebih kurang daripada 0, seperti −3. Sesuatu nombor positif merupakan suatu nombor yang lebih besar daripada 0, seperti 3. 0 sendiri bukan negatif mahupun positif. Nombor-nombor bukan negatif merupakan nombor nyata yang bukan negatif (sama ada positif atau kosong). Nombor-nombor bukan positif merupakan nombor yang bukan positif (sama ada negatif atau kosong).

Dalam konteks nombor kompleks, positif adalah nyata, tetapi demi kejelasan seseorang juga boleh sebut "nombor nyata positif".

Fungsi signum

Adalah mungkin untuk menentukan sesuatu fungsi sgn(x) pada nombor nyata yang adalah 1 bagi nombor positif, −1 bagi nombor negatif, dan 0 bagi kosong (kadangkala digelar fungsi signum):

${\displaystyle \operatorname {sgn}(x)=\left\{{\begin{matrix}-1&:x<0\\\;0&:x=0\\\;1&:x>0\end{matrix}}\right.}$

Dengan ini kita ada (kecuali bagi x=0):

${\displaystyle \operatorname {sgn}(x)={\frac {x}{|x|}}={\frac {|x|}{x}}={\frac {d{|x|}}{d{x}}}=2H(x)-1.}$

Di mana |x| merupakan nilai mutlak untuk x dan H(x) merupakan Heaviside step function. Lihat juga terbitan (matematik).

Fungsi Signum kompleks

It is possible to define a function csgn(x) on the complex numbers which is 1 for positive numbers, −1 for negative numbers and 0 for zero (sometimes called the complex signum function):

${\displaystyle \operatorname {csgn} (x)=\left\{{\begin{matrix}-1&:x<0\\\;0&:x=0\\\;1&:x>0\end{matrix}}\right.}$

Where the complex inequality should be interpreted as follows

${\displaystyle {\begin{cases}x>0\iff \operatorname {Re} (x)>0\vee (\operatorname {Re} (x)=0\land \operatorname {Im} (x)>0)\\x<0\iff \operatorname {Re} (x)<0\vee (\operatorname {Re} (x)=0\land \operatorname {Im} (x)<0)\\\end{cases}}}$

We then have (except for x=0):

${\displaystyle \operatorname {csgn} (x)={\frac {x}{\sqrt {x^{2}}}}={\frac {\sqrt {x^{2}}}{x}}={\frac {d{\sqrt {x^{2}}}}{d{x}}}=2H(x)-1.}$

Lihat juga

• Nombor hipernyata
• Integer
• Rational number
• Nombor nyata
• Surreal number
• −0
• Sejarah matematik

Catatan dan rujukan

• Templat:Ent Maseres, Francis, 1731–1824. A dissertation on the use of the negative sign in algebra: containing a demonstration of the rules usually given concerning it; and shewing how quadratic and cubic equations may be explained, without the consideration of negative roots. To which is added, as an appendix, Mr. Machin's Quadrature of the Circle, 1758. Quoting from Maseres' work, "If any single quantity is marked either with the sign + or the sign − without affecting some other quantity, the mark will have no meaning or significance, thus if it be said that the square of −5, or the product of −5 into −5, is equal to +25, such an assertion must either signify no more than 5 times 5 is equal to 25 without any regard for the signs, or it must be mere nonsense or unintelligible jargon."
• Templat:Ent Colva Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews, stated this on the BBC Radio 4 "In Our Time", on Negative Numbers, 9 March 2006.
• Templat:Ent Knowledge Transfer and Perceptions of the Passage of Time, ICEE-2002 Keynote Address by Colin Adamson-Macedo. "Referring again to Brahmagupta's great work, all the necessary rules for algebra, including the 'rule of signs', were stipulated, but in a form which used the language and imagery of commerce and the market place. Thus 'dhana' (= fortunes) is used to represent positive numbers, whereas 'rina' (= debts) were negative". [1]
• Templat:Ent Alberto A. Martinez, Negative Math: How Mathematical Rules Can Be Positively Bent, Princeton University Press, 2006; a history of controversies on negative numbers, mainly from the 1600s until the early 1900s.

Pautan luar

• BBC Radio 4 series "In Our Time", on Negative Numbers, March 9, 2006
• Endless Examples & Exercises: Operations With Signed Integers

• VERY intersting and insightful concept of negative numbers