Quadratic equations have been considered and solved
since Old Babylonian times (c. 1800 BC), but the
quadratic formula students memorize today is an
18th century AD development. What did people do
in the meantime?
The difficulty with the general quadratic equation
( as we write it today) is that, unlike a linear equation,
it cannot be solved by
arithmetic manipulation of the terms themselves:
a creative intervention, the addition and subtraction of a
new term, is required to change the form of the
equation into one which is arithmetically solvable. We call this
maneuver completing the square.
In this column I will explore some episodes in this history,
starting with the Old Babylonians and ending with an occurence
of square-completion in
an important calculation in 20th century physics.
In 20th century physics
In Old Babylonian Mathematics

The Yale Babylonian Collection's tablet YBC 6967,
as transcribed in in Neugebauer and Sachs,
Mathematical Cuneiform Texts,
American Oriental Society, New Haven, 1986. Size 4.5 6.5cm.
The Old Babylonian tablet YBC 6967 (about 1900 BC)
contains a problem and its solution.
Here is a line-by-line literal translation
from Jöran Friberg, A Remarkable Collection of Babylonian Mathematical Texts, Springer, New York, 2007.
- The igi.bi over the igi 7 is beyond.
The igi and the igi.bi are what?
You:
7 the the igi.bi over the igi is beyond
to two break, then 3 30.
3 30 with 3 30 let them eat each other, then 12 15
To 12 15 that came up for you
1, the field, add, then 1 12 15.
The equalside of 1 12 15 is what? 8 30.
8 30 and 8 30, its equal, lay down, then
3 30, the holder,
from one tear out,
to one add.
one is 12, the second 5.
12 is the igi.bi, 5 the igi.
The Old Babylonians used a base-60 floating-point notation for numbers, so
that the symbol corresponding to 1 can represent for example 60 or 1 or 1/60.
In the context of YBC 6967,
the reciprocal numbers, the igi and the igi.bi,
have product 1 0. Their difference
is given as 7.
Diagram of solution of the YBC 6967 problem, adapted
from Eleanor Robson's
"Words and Pictures: New Light on Plimpton 322" (MAA Monthly, February 2002, 105-120).
Robson uses a semi-colon
to separate the whole and the fractional part of a number,
but this is a modern insertion for our convenience.
The two unknown reciprocals are conceptualized as the
sides of a rectangle of area (yellow) 1 0 [or 60 in decimal notation]. A
rectangle with one side 3;30 [] is moved from the side
of the figure to the top, creating an L-shaped figure of area 1 0 which can
be completed to a square by adding a small square of area
3;30 3;30 = 12;15 []. The area of the large square
is 1 0 + 12;15 = 1 12;15 [] with square root 8;30 []. It follows that our unknown reciprocals must be 8;30 + 3;30 = 12 and 8;30 − 3;30 = 5 respectively.
In modern notation, the YBC 6967 problem would be ,
or . In this case the term to be added
in completing the square is ,
corresponding exactly to the area of the small square in the diagram.
This tablet, and the several related ones from the same period that
exist in various collections (they are cataloged in Friberg's book
mentioned above), are significant because they hold a piece of
real mathematics:
a calculation that goes well beyond tallying to a creative attack on
a problem. It should also be noted that none of these tablets contains
a figure, even though Old Babylonian tablets often have diagrams. It is
as if those mathematicians thought of "breaking," "laying down" and
"tearing out" as purely abstract operations on quantities, despite
the geometrical/physical language and the clear (to us) geometrical
conceptualization.
(The connection to Plimpton 322)
The Old Babylonian tablet Plimpton 322 (approx.
cm) is in the
Columbia University Library. Image courtesy of Bill Casselman
(larger image). For
scholarly discussions of its content see Friberg's book
or (available online)
Eleanor Robson's article referenced above.
There have
been many attempts at understanding why Plimpton 322 was made
and also how that particular set of numbers was generated. It has
been described as a list of Pythagorean triples and as
an exact sexagesimal trigonometry table. An interpretation mentioned
by R. Creighton
Buck and attributed to D. L. Voils (in a paper that was never
published) is "that the Plimpton tablet has nothing to do with
Pythagorean triplets or trigonometry but, instead, is a pedagogical tool intended to help a
mathematics teacher of the period make up a large number of
igi-igibi quadratic equation
exercises having known solutions and intermediate solution steps that are easily checked" (igi, igi.bi problems involve a number and
its reciprocal; the one on YBC 6769 is exactly of this type).
In the solution of the problem of YBC 6769, we squared 3;30 to
get 12;15, added 1 to get and took the square root
to get 8;30. Then the solutions were 8;30 + 3;30 and 8;30 3;30.
So to set up an
igi, igi.bi problem which will work out evenly we need a
square like 12;15 which, when a 1 is placed in the
next base-60
place to the left, becomes another square.
Now the first column of Plimpton 322 contains exactly numbers
of this type. For example, in row 5, the first undamaged row,
the column I entry is [=10562500] with
square root [=3250]. Adding 1 gives
[=23522500]
with square root [=4850].
The corresponding igi, igi.bi problem would ask
for two reciprocals differing by two times , i.e. ;
the answer would be and
.
Unfortunately, neither the problem on YBC 6967 nor any of the
five igi, igi.bi problems recorded by Friberg from tablet
MS 3971 correspond to parameters on Plimpton 322. It is possible
that lines on the bottom and reverse of the tablet mean
that it was supposed to be extended to additional 20 or so rows,
where those missing parameters would appear. In fact, none
of the proposed explanations is completely satisfactory.
As Robson remarked, "The Mystery of the Cuneiform Tablet has not
yet been fully solved."
In Islamic Mathematics
Solving quadratic equations by completing the square was treated
by Diophantus (c.200-c.285 AD) in his Arithmetica, but the
explanations are in the six lost books of that work.
Here we'll look at the topic as covered by Muhammad ibn Musa, born
in Khworaezm (Khiva in present-day Uzbekistan) and known as al-Khwarizmi,
on his Compendium on Calculation by Completion and Reduction
dating to c. 820 AD. (I'm using the
translation published by Frederic Rosen in 1831).
Negative numbers were still unavailable, so
al-Khwarizmi, to solve a general quadratic, has to consider three
cases. In each case he supposes a preliminary division has been
done so the coefficient
of the squares is equal to one ("Whenever you meet with a multiple or
sub-multiple of a square, reduce it to the entire square").
- "roots and squares are equal to numbers" []
- "squares and numbers are equal to roots" []
- "roots and numbers are equal to squares" []
Case 1. al-Khwarizmi works out a specific numerical
example, which can serve as a template for any other equation
of this form: "what must be the square which, when increased by ten
of its roots, amounts to thirty-nine."
- "The solution is this: you halve the number of the roots,
which in the present case equals five. This you multiply by
itself; the product is twenty-five. Add this to thirty-nine,
the sum is sixty-four. Now take the root of this, which is eight,
and subtract from it half the number of the roots, which is five;
the remainder is three. This is the root of the square which you
sought for; the square itself is nine."
Note that this is exactly the Old Babylonian recipe, updated
from to , and that the figure Eleanor Robson
uses for her explanation is essentially identical to the
one al-Khwarizmi gives for his second demonstration, reproduced here:
"We proceed from the quadrate AB, which represents
the square. It is our next business to add to it the ten roots of the
same. We halve for this purpose the ten, so it becomes five, and construct
two quadrangles on two sides of the quadrate AB, namely, G and D, the
length of each of them being five, as the moiety of the ten roots, whilst
the breadth of each is equal to a side of the quadrate AB. Then a quadrate
remains opposite the corner of the quadrate AB. This is equal to five
multiplied by five: this five being half of the number of roots which we
have added to each side of the first quadrate. Thus we know that the
first quadrate, which is the square, and the two quadrangles on its sides,
which are the ten roots, make together thirty-nine. In order to complete
the great quadrate, there wants only a square of five multiplied by five,
or twenty-five. This we add to the thirty-nine, in order to complete
the great square SH. The sum is sixty-four. We extract its root, eight,
which is one of the sides of the great quadrangle. By subtracting from this
the same quantity which we have before added, namely five, we obtain three
as the remainder. This is the side of the quadrangle AB, which represents
the square; it is the root of this square, and the square itself is nine."
Case 2. "for instance, 'a square and twenty-one in numbers
are equal to ten roots of the same square."
- Solution: Halve the number of the roots; the moiety is five.
Multiply this by itself; the product is twenty-five. Subtract from
this the twenty-one which are connected with the square; the remainder
is four. Extract its root; it is two. Subtract this from the
moiety of the roots, which is five; the remainder is three. This
is the root of the square you required, and the square is nine.
Here is summary of al-Khwarizmi's demonstration. The last of the
four figures appears (minus the modern embellishments) in Rosen, p. 18.
The problem set up geometrically. I have labeled
the unknown root for modern convenience. The square ABCD has area ,
the rectangle CHND has area , the rectangle AHNB has area 21,
so .
The side CH is divided in half at G, so the segment
AG measures . The segment TG parallel to DC is extended by
GK with length also . Al-Khwarizmi says this is done "in order
to complete the square."
The segment TK then measures 5, so the figure KMNT,
obtained by drawing KM parallel to GH and adding MH, is a square with area 25.
Measuring off KL equal to KG, and drawing LR parallel
to KG leads to a square KLRG. Since HR has length the rectangles
LMHR and AGTB have the same area, so the area of the region formed by adding
LMHR to GHNT is the same as that of the rectangle formed by adding
AGTB to GHNT, i.e. 21. And since that region together with the square KLRG
makes up the square KMNT of area 25, it follows that the area of
KLRG is , and that its side-length is equal to 2. Hence
, and the sought-for square is 9.
Al-Khwarizmi remarks that if you add that 2
to the length of CG then "the sum is seven, represented by the line CR,
which is the root to a larger square," and that this square is also
a solution to the problem.
Case 3. Example: "Three roots and four simple numbers are equal
to a square."
- Solution: "Halve the roots; the moiety is one and a half. Multiply
this by itself; the product is two and a quarter. Add this to the four;
the sum is six and a quarter. Extract its root; it is two and
a half. Add this to the moiety of the roots, which was one and a
half; the sum is four. This is the root of the square, and
the square is sixteen."
As above, we summarize al-Khwarizmi's demonstration. The last figure
minus decoration appears on Rosen, p. 20.
We represent the unknown square as ABDC,
with side-length . We cut off the rectangle HRDC with side-lengths
3 and . Since the remaining rectangle ABRH has
area 4.
Halve the side HC at the point G, and construct the
square HKTG. Since HG has length , the square HKTG has
area .
Extend CT by a segment TL equal to AH. Then
the segments GL and AG have the same length, so drawing LM parallel
to AG gives a square AGLM. Now TL = AH = MN and NL = HG = GC = BM,
so the rectangles MBRN and KNLT have equal area, and so the region
formed by AMNH and KNLT has the same area as ABRH, namely 4. It
follows that the square AMLG has area
and consequently side-length AG = . Since GC = ,
it follows that .
Aryabhata and Brahmagupta
The study of quadratic equations in India dates back to Aryabhata (476-550)
and Brahmagupta (598-c.665). Aryabhata's work on the topic
was referenced at the time but
is now lost; Brahmagupta's has been preserved. He gives a solution
algorithm in words (in verse!) which
turns out to be equivalent to part of the quadratic formula —it
only gives the root involving the radical. Here's
Brahmagupta's rule with a translation, from Brahmagupta as an Algebraist
(a chapter of Brahmasphutasiddhanta, Vol. 1):
-
"The quadratic: the absolute quantities multiplied by four times the
coefficient of the square of the unknown are increased by the square
of the coefficient of the middle (i.e. unknown); the square-root of the
result being diminished by the coefficient of the middle and divided by
twice the coefficient of the square of the unknown, is (the value of)
the middle."
There is only the rule, and no indication of how it was derived.
In Fibonacci's Liber Abaci
The third section of chapter 15 of the Liber Abaci, written by Fibonacci (Leonardo of Pisa,
c. 1170-c. 1245) in 1202, concerns "Problems of Algebra and Almuchabala,"
referring directly to the Arabic words for "Completion and Reduction"
in the title of al-Khwarizmi's compendium. I am using the translation
of Liber Abaci by L. E. Sigler, Springer, 2002. In that section, a short introduction
presenting the
methods is followed by a collection of over a hundred problems.
Fibonacci follows al-Khwarizmi
in distinguishing three "modes" of compound problems involving a square
("census") and its roots (his second mode corresponds to al_Khwarizmi's
case 3 and vice-versa).
- "the first mode is when census plus roots are equal to a number."
The example Fibonacci gives, "the census plus ten roots is equal
to 39" is inherited directly from al Khwarizmi. But the demonstration
he gives is different: he starts with a square of side length
greater than 5, marks off lengths of 5 in two directions from
one corner and thus partitions into a square of area 25,
two rectangles and another square, which he identifies with the
unknown census. Then the two rectangles add up to 10 times the root,
but since "the census plus 10 roots is equal to 39 denari" the
area of must be , so its side length is 8, our
root is and the census is 9. The figure and the calculation
are essentially
the same as al-Khwarizmi's, but
the "completion" narrative has disappeared.
- "let the census be equal to 10 roots plus
39 denari." Here the example is new.
(Figure from Sigler, p. 557).
Fibonacci starts by representing the census as
a square of side length
larger than 10. He draws a segment parallel to so that
the segments and each have length 10 and assigns a midpoint
to . So the ten roots will be equal to the area of
, and the area of the rectangle , which is
also , is 39.
But , therefore . "If to this is added the
square of the line segment , then 64 results for the square of the
line segment ." [
]. So and is the root
of the sought census, and the census is 169.
- "when it will occur that the census plus a number will equal a number
of roots, then you know that you can operate whenever the number is equal
to or less than the square of half the number of roots." Here
Fibonacci goes beyond al-Khwarizmi in remarking that (in modern
notation) has no solution if . The example
he gives is "let the census plus 40 equal 14 roots."
(Figures from Sigler, pp. 557, 558). Fibonacci
gives two construction for the two positive roots. In each case he
starts with a segment of length 14, with center at . He
chooses another point dividing into two unequal parts.
For the
first root he constructs a square over --this will
be the census--
and extends to matching . The rectangle now
represents 14 roots, so that measures (in modern notation)
. Now so and
. It follows that and the root is
.
For the second root the square is over ;
the segment is extended to matching , and measures
. Now so again
and . This time the root is .
In Simon Stevin's L'Arithmétique
Stevin's L'Arithmétique was published in 1594
(excerpts here are from the 1625 edition,
supervised by Albert Girard).
Among other things it treated quadratic equations.
Stevin, following Bombelli, used a notation for powers
that turns out to be intermediate
between cossic notation (which used different
symbols for the unknown, its square, its cube etc.)
and the exponents that started with Descartes.
For Stevin, the
unknown was represented by a 1 in a circle, its square by a 2 in
a circle, etc., and the numerical unit by a 0 in a circle. Here let us write
for 1 in a circle, etc. He also had an
idiosyncratic way of expressing the solution to an equation in one
unknown, using
the "fourth proportional." For example, his setting of the
problem the problem of finding
such that could be schematized as
(he would actually write: given the three terms for the problem -- the first ,
the second , the third , find the
fourth proportional). As Girard explains,
the "proportion" is equality. So the problem should be read as:
"if , then what?"
One can read "[Stevin's Arithmetic] brought to the western world for the first time a general solution of the quadratic equation ..." but in fact there is only
this step towards modern notation, his use of minus signs in his
equations and
his admission of irrational numbers as coefficients to separate him from Fibonacci. Notably he
also considers separately
three types of quadratic equations [his examples, in post-Descartes notation]
- "second term " []
- "second term " []
- "second term " [].
and does not entertain negative solutions, so for the first
equation he gives only and not , for the second he gives
and not ; for the third he gives the two (positive) solutions
and .
Stevin gives a geometric justification for his solution of each type
of equation. For example for the first type his solution is:
Half of the 4 (from the ) is | 2 |
Its square | 4 |
To the same added the given , which is | 12 |
Gives the sum | 16 |
Its square root | 4 |
To the same added the 2 from the first line | 6 |
I say that 6 is the sought-for fourth proportional term.
Here is his geometrical proof:
Figure from Stevin, L'Arithmétique p. 267.
With modern notation:
Stevin starts with a square ABCD representing ,
so its side BC has length . He draws EF parallel to AD, with AE . So
the rectangle ADFE measures and then the rectangle EFCB has area
. Pick G the midpoint of AE, and draw the square GLKB.
The rest of the argument as it appears in L'Arithmétique:
Half of AE which is GE | 2 |
Its square GEHI | 4 |
Add to the same the given , i.e. EFCB
| 12 |
Gives sum for the gnomon HIGBCF | |
or for the square GBKL of same area | 16 |
Its root BK | 4 |
Add to the same GE or instead KC = GE, makes for BC | 6 |
Q.E.D. [My translations -TP] | |
Notice that the argument and even the diagram are inherited directly from
al-Khwarizmi.
Descartes' La Géometrie
René Descartes (1596-1650) published
La Géometrie
(1885 edition in modern French)
in 1637. One of the first topics he covers is the solution of quadratic
equations. Besides the actual geometry in his book, two notational and
conceptual features started the modern era in mathematics. The first
was his use of
exponents for powers. This started as a typographical convenience:
he still usually wrote instead of . It turned out to be a short series of steps (but one he
could not have imagined) from his to Euler's . The second
was his use of and as coordinates in the plane. The first
would eventually allow the general quadratic equation to be written
as , and the second would allow the solution to be
viewed as the intersection of a parabola with the -axis. But
Descartes' avoidance of negative numbers (the original cartesian plane
was a quadrant) kept him from the full exploitation of his own
inventions.
In particular, he still had to follow al-Khwarizmi
and Stevin in distinguishing three forms of quadratic equations. In his notation:
His solutions, however, are completely different from the earlier methods. He
shows in all three cases how a ruler-and-compass construction can
lead from the lengths and to a segment whose length is
the solution.
Cases 1 and 2. "I construct the right
triangle NLM with one leg equal to , and the other
is , half of the other known quantity which was multiplied
by , which I suppose to be the unknown length; then extending , the
hypothenuse of this triangle, to the point , such that is equal
to , the entirety is the sought-for length; and it can be
expressed in this way:
Whereas if I have , with the unknown quantity, I construct
the same right triangle , and from the hypothenuse I subtract
equal to , and the remainder is , the sought-for root.
So that I have
Case 3. "Finally, if I have
I draw equal to and
equal to , as before; then, instead of joining the
points , I draw parallel to , and having drawn
a circle with center which cuts it
at the points and , the sought-for length is
, or ; since in this case it can be expressed two
ways, to wit,
and
And if the circle, which has its center at and passes
through neither cuts nor touches the straight line
, the equation has no root, so one can state
that the construction of the proposed problem is
impossible." [My translation. Checking Descartes'
constructions involves some standard Euclidean geometry.-TP]
In Maria Gaetana Agnesi's Instituzioni analitiche ad uso della
gioventù italiana
Agnesi's textbook was published in 1748. Agnesi does algebraically
complete the square for one case of the quadratic equation:
-
"Consider the equation ; add to one and
to the other sides the square of half the coefficient of the second term,
i.e. , and it becomes and, extracting the
root, ."
Note that she uses the notation. She explains: "So the ambiguity of
the sign affected to the square root implies two values for the unknown,
which can be both positive, both negative, one positive and the other
negative, or even both imaginary, depending on the quantities from which they
have been computed."
But when Agnesi comes to a general treatment she follows Descartes,
using identical geometric constructions, with one significant
improvement, as implied above: negative roots are calculated and given the same status as
positive ones:
- "Negative values, which are still called false, are
no less real than the positive, and have the only difference, that if
in the solution to the problem the positives are taken from the fixed
starting point of the unknown toward one side, the negatives are taken
from the same point in the opposite direction."
Specifically (with the letters Descartes used) she takes as a positive
quantity (necessary for the geometric construction) and gives
- MP and MO as roots of
- MO and MP as roots of
- MQ and MR as roots of
- MQ and MR as roots of .
Imaginary numbers do not yet have a geometric
meaning. "Whenever the equation, to which the particulars of the
problems have led us, produces only imaginary values, this means that the
problem has no solution, and that it is impossible." [My translations -TP]
In Leonhard Euler's
Vollständige Anleitung zur Algebra
Euler's text was published in St. Petersburg in 1770;
John Hewitt's
English
translation, Elements of Algebra, appeared in 1840.
Chapter VI covers the
general quadratic equation: Euler writes it as ,
and then remarks that it can always be put in the form
, where and can be positive or negative.
He explains how the left-hand side can be made into
the square of by
adding to both sides, leading to
and,
"as every square root may be taken either affirmitively
or negatively,"
In deriving this solution, completely equivalent to the
quadratic formula, Euler has completed the square in a
purely algebraic manner. The gnomons and Euclidean diagrams,
that for some 2500 years had seemed necessary to justify
the maneuver,
have evaporated.
In Elements of Algebra Euler is
receptive to imaginary numbers
- §145. "But notwithstanding [their being impossible]
these numbers present
themselves to the mind; they exist in our imagination,
and we still have a sufficient idea of them; since we know
that by is meant a number which, multiplied by
itself, produces ; for this reason also, nothing prevents
us from making use of these imaginary numbers,
and employing them in calculation."
but does not consider them "possible" as roots of
quadratic equations. For example:
- §700. "A very remarkable case sometimes occurs, in which
both values of become imaginary, or impossible; and it is
then wholly impossible to assign any value for , that would
satisfy the terms of the equation."
A Gaussian function corresponds to
a familiar "bell-shaped curve." In multivariable calculus we
learn that ;
this also holds for , with
any finite number.
The
width of the Gaussian ,
defined as the distance between the two
points where , can be calculated to be . In this image with , , the width is 3.33. Note that the width does not depend on the factor .
The Fourier transform of (taking )
is the function
This integral can be computed by completing the square: write
as . Then
This
means that the Fourier transform of is again
a Gaussian; the parameter has become , so
the product
of the widths of and its Fourier transform is
constant. The wider is, the narrower must be,
and vice-versa. This phenomenon is the mathematical form
of the uncertainty principle.