Six Rhetoric Quadratic:
the Six Types of Quadratic Equations Presented by Abū Ja’far Muḥammad ibn Mūsā
al-Khwārizmī in al-Kitāb al-Mukhtaṣor fī Ḥisāb al-Jabr wa al-Muqōbala
Abstract
Abū Ja’far Muḥammad ibn Mūsā al-Khwārizmī
(Arabic: أبو جعفر محمد بن موسی الخوارزمی) presented the six types of quadratic equations in al-Kitāb
al-Mukhtaṣor fī Ḥisāb al-Jabr wa al-Muqōbalah (Arabic: الكتاب المختصر في حساب الجبر والمقابلة). In this mathematical treatise
written approximately 820 CE, equations are verbally described in terms of
“squares” (Arabic: مربع الجذر;
what would today be “x2”),
“roots” (Arabic: الجذور;
what would today be “x”) and
“numbers” (Arabic: الأعداد;
“constants”: ordinary spelled out numbers, like ‘twenty-six’). The six types
equations are:
[1] <المربعات تساوي الجذور> or squares equal
roots or with current notations ax2=bx;
[2] <المربعات تساوي الأعداد> or squares equal
number or ax2=c;
[3] <الجذور تساوي الأعداد> or roots equal
number or bx=c;
[4] <المربعات والجذور تساوي الأعداد> or squares and
roots equal number or ax2+bx=c;
[5] <المربعات والأعداد تساوي الجذور> or squares and
number equal roots or ax2+c=bx; and
[6] <الجذور والأعداد تساوي المربعات> or roots and
number equal squares or bx+c=ax2.
Al-Kitāb al-Mukhtaṣor fī Ḥisāb al-Jabr wa
al-Muqōbala thoroughly rhetorical, with the
syncopation that is the numbers were written out in words rather than symbols.
However, in author’s day, most of this notation had not yet
been invented, so he had to use ordinary text to present problems and their
solutions. The types of problems which the book discusses reveals middle east
mathematicians didn’t deal with negative numbers at all. Hence an equation like
bx+c=0
doesn’t
appear in the classification, because it has no positive solutions if all the
coefficients are positive. Similarly equation types 4, 5 and 6, which look
equivalent to our current view, were distinguished because the coefficients
must all be positive.
Keywords:
Abū
Ja’far Muḥammad ibn Mūsā al-Khwārizmī; algebra;
al-Kitāb al-Mukhtaṣor fī Ḥisāb al-Jabr wa al-Muqōbala;
mathematical treatise;
To
cite this work:
Setiawan,
Adib Rifqi. (2018, 2 May). Six Rhetoric Quadratic: the Six Types of Quadratic
Equations Presented by Abū Ja’far Muḥammad ibn Mūsā al-Khwārizmī in al-Kitāb
al-Mukhtaṣor fī Ḥisāb al-Jabr wa al-Muqōbala. ΛLΟBΑΤИIƆ. URL: https://alobatnic.blogspot.com/2018/05/six-rhetoric-quadratic.html