I have been trying to really comprehend how we (as a species) got to the quadratic formula. I know what it is, and why it’s useful; what I’m interested in is the story of its development.
I understand that an Indian mathematician by the name of Brahmagupta wrote out an algebraic version of a geometric process called “completing the square”, and his version is (more or less) the quadratic formula we use today.
Okay, cool. But here’s the problem: the Interwebs is great at explaining how to “complete the square” and even how to derive the quadratic formula from the process, but it’s not so strong on explaining what it was that people were originally doing with this pre-algebraic, “completing the square” math -- why they were doing it. Most of what I could find seemed pretty breezy -- variations on “people did it for <waving hands> engineering and parceling land and stuff like that.”
What I really needed was an actual, practical examples of a problem that a person -- pre-algebra -- would be figuring out by completing the square. I couldn’t find anything specific, so I kludged one up for myself. And this is what I eventually came up with: the story of an architect and her apprentice.
SCENE: Babylonia, sometime in or around 1750 BCE...
[ Warning: what follows, under the cut, is long, and contains gruesome mathematics.]
Merchant: <enters an architect’s office> I would like to commission you to expand my warehouse. My business is growing and I need more storage space.
Architect: By how much would you like to expand it?
Merchant: I would like to double my storage space.
Architect: What is the current size and shape of your warehouse?
Merchant: It is rectangular — one side is 12 rods in length and the other is 5 rods.
Architect: Are there any constraints on the available land?
Merchant: Not really.
Architect: Very good. We should have an estimate ready for you after noon. Will you stay and take tea while we work?
Merchant: I appreciate the offer, but I have other business in the city. I will return after noon. <leaves>
Architect: <turns to apprentice> Now, young apprentice, take you some clay and show me how you will work out the problem. Do you see what we need to do?
Apprentice: Yes, Mom — I mean, Boss. I think the easiest thing is to extend each wall by some amount, so that the area of the new rectangle is now twice as before.
[ Image description: sketch of a rectangle. Within the rectangle and aligned to its lower left-hand corner is another rectangle, labeled “old warehouse (60)”, with sides annotated with “12″ and “5″. The edges of the smaller rectangle are carried out across the larger rectangle by dotted lines, showing that the larger rectangle has a side of “5″ and “?” and another side of “12″ and “?”. The “L-shaped” extra area of the larger rectangle is hatched out, and a legend indicates that this is “new warehouse: +60″. End ID. ]
Apprentice: the old warehouse: a rectangle of 60 square rods. The new warehouse: a rectangle of 120 square rods.
Architect: Very good. We know the area that currently exists — this is the old warehouse. What is the area we need to build?
Apprentice: The same again: 60 square rods, in this new area.
Architect: If we only concentrate on the new area, what shapes do you see?
Apprentice: I see a square, and a rectangle, and another rectangle, whose area when all added together needs to be the same as 60. But if I rearrange those rectangles in this way…
[ Image description: a square, its sides labeled “?”, a plus sign, a rectangle, its sides labeled “12″ and “?”, another plus sign, another rectangle, its sides labeled “5″ and “?”, an equal sign, and then “60″ enclosed within a dotted square frame. End ID. ]
Apprentice: …I have a square and since I can add those other two rectangles together, that would be a square and a single rectangle…hey! Wait a minute — this looks like an igi/igi.bi problem! The value of igi is what we need to add to the walls!
[ Image description: a rectangle that has been divided into a square, its sides labelled “igi”, and a smaller rectangle, one side of which is marked “17″. There is a division within the smaller rectangle to show that it is “12″ and “5″. There is a long line along the bottom of the entire larger rectangular figure labeled “igi.bi”, and to the right of it, there is an equal sign, and “60″ enclosed within a dotted square frame. End ID. ]
[ Author’s note: an igi/igi.bi problem is basically a problem in which you work out the sides of a rectangle knowing only its area and the difference between the long side (igi.bi) and the short side (igi). It is essentially the process of “completing the square” condensed into a series of mathematical operations. ]
[ In the “completing the square” geometric (that is, drawn) solution, the “hack” is to split off the rectangle from the unknown square, divide it in half along the known side (that is, the side that has an actual value), then rearrange those two smaller rectangles around the unknown square so that you can make (that is, “complete”) an even larger square. The value that you would use to do this, you then add to the other side of the equation. Then you can continue to solve for x using algebra, as follows… ]
[ Image description: A series of drawings showing the steps of “completing the square”: the original rectangle, divided into a square, its side now labelled “x” and its area “x^2″ and a smaller rectangle with sides “17″ and “x”, equal to “60″ with a dashed line around it. A dashed line splits the 17x rectangle into two rectangles of 8 1/2x each. Lines show how these rectangles can be moved and attached to the x^2 square to form a larger square, its sides now “x + 8 1/2″. To complete this square, the empty area (which is conveniently square) is labeled “72 1/4″ (because its sides are 8 1/2 each). Text reads: “Adding this square completes the bigger, total square. So you have to add it to the other side of the equation too”. The “60″ in its dashed line is now followed by a plus sign and a square with an area of 71 1/4. Next, the whole square is redrawn to show the larger square, its sides “x + 8 1/2″, an equal sign, and 132 1/4. Text indicates that this resolves to “x + 8 1/2″ times “x + 8 1/2″ equal 132 1/4, and then writes out the rest of the algebraic solution: (x + 8 1/2)^2 = 132 1/4 becomes x + 8 1/2 = (square root of) 132 1/4 which itself is plus or minus 11 1/2; so x either equals -8 1/2 + 11 1/2 or =8 1/2 - 11 1/2. Therefore: x equal 3 or x equals -20. End ID. ]
[ …but let’s do it in the igi/igi.bi style, as per Yale Babylonian Collection tablet #6967. The following text is modified from Jøran Friberg’s translation of YBC 6967 in A Remarkable Collection of Babylonian Mathematical Texts (Springer, New York, 2007):]
Architect: <smiling> That’s my girl. So: the igi.bi over the igi 17 is beyond. The igi and igi.bi are what?
Apprentice:
17 the igi.bi over the igi is beyond, [ y - x = 17 (the remainder) ]
to two break, then 8 ½. [ 17 / 2 = 8 1/2 ]
8 ½ with 8 ½ let them eat each other, then 72 ¼. [ 8 1/2^2 = 72 1/4 ]
To the 72 ¼ that came up for you, 60, the field, add, then 132 ¼
[ 72 1/4 + 60 (the total area) = 132 1/4 ]
The equalside of 132 ¼ is what? 11 ½. [ the square root of 132 1/4 is 11 1/2 ]
11 ½ and 11 ½, its equal, lay down [ write down 11 1/2 twice ]
8 ½, the holder, from one tear out, [ subtract the half remainder (8 1/2) from one]
to one add [ and add it to the other ]
one is 20, the second 3.
20 is the igi.bi, 3 the igi.
Apprentice: The igi is 3. So we lengthen the walls by 3 rods each, then build the other two walls to finish the warehouse. Or we could push out the current walls by 1 ½ rods each in every direction.
[ Image description: sketch of the two solutions: in the first, the walls of the original warehouse have each been extended by 3 for a new size of 15 by 8, with text “8 x 15 = 120″. In the second, the walls of the new warehouse are centered around those of the old; arrows indicated that the space between them is now 1 1/2 on the outer sides of the old walls. Again, the total side lengths are “15″ and “8″ respectively, and text reads “9 x 15 = 120″. End ID. ]
Architect: Excellent. Let’s work out the estimate, now.
<later>
Merchant: Well met — is the estimate ready?
Architect: <pushes a clay-filled frame over to the merchant with calculated information> Here it is.
Merchant: <looks it over> This seems reasonable. Ah! I see you are proposing to reuse the bricks from the inside once the roof is supported. Very frugal.
Architect: It will save the cost of using all-new bricks, and also save us some time. A win-win, yes?
Merchant: Very good. I accept your proposal. Please begin immediately.
Architect: How will you pay?
Merchant: I primarily trade in copper —the finest kind. I believe that 18 talents’ worth of my most excellent copper ingots should be enough to cover the cost.
Architect: This is very generous. I accept.
<merchant leaves>
Apprentice: That seemed like a really good deal, Mom — I mean, Boss.
Architect: Yes, apparently he recognizes and rewards quality work.
Apprentice: Um. I mean, it was a really, really good deal. Too good, maybe.
Architect: I’m sure it will be fine.
<sometime later>
[ Image description: a photograph of the complaint tablet to Ea-Nasir in the British Museum, a rectangular baked-clay tablet covered in cuneiform writing, courtesy Wikimedia Commons user Zukir (CC BY-SA 4.0). This is just a personal observation, but it seems to me that the writing at the bottom is more deeply and sloppily impressed, which possibly says something about the original author’s state of mind. End ID. ]
