Algebra: From solving for an unknown to studying symmetry and structure

Here is a puzzle from a book written in Baghdad about twelve hundred years ago. In Frederic Rosen's 1831 English translation, it reads: "one square, and ten roots of the same, amount to thirty-nine dirhems" (al-Khwarizmi 1831, 8). A dirhem was a coin, used here as a unit of counting. A "root" is the number you're looking for, and the "square" is that number multiplied by itself. So the puzzle asks: what number, squared and then added to ten times itself, makes thirty-nine?

Try it before you read on. Patient guessing works. Try 2: its square is 4, ten roots make 20, and the total is 24, too small. Try 4: 16 plus 40 is 56, too big. The answer lies somewhere between.

In modern notation, the letter x stands for the unknown number, and the puzzle becomes

Here x^2, read "x squared," is the square, and 10x is ten roots. You may remember a formula from school for equations like this. The book's author, Muhammad ibn Musa al-Khwarizmi, had no such formula and no symbols at all. His mathematics was written entirely in words (O'Connor and Robertson 1999).

What he had was a rule, stated in words. Halve the ten roots to get five. Multiply five by itself to get twenty-five. Add that to thirty-nine to get sixty-four. Take its root, which is eight, and subtract the five. The answer is three (al-Khwarizmi 1831, 8). You can check it: 9 plus 30 is 39.

Then he did something that mattered more than finding 3. He justified the rule with a picture: a square with rectangles attached to its sides, and a missing corner filled in to make a larger square (al-Khwarizmi 1831, 9–11). The picture shows why the rule works. It also works for every problem of the same shape, not only this one.

That step opens a story that runs in three stages:

• One problem: find the number that makes this equation true.
• One method for a type: a rule that solves every equation of the form "a square and roots equal a number."
• A question about every type: does each kind of equation have a rule like this?

The third question took about a thousand years to answer, and for equations that involve an unknown raised to the fifth power, the answer turned out to be no. There is no general rule for them built from arithmetic and roots.

Proving that took a new way of thinking. Mathematicians stopped asking only "what is the answer?" and started asking "what structure does this equation have?" The study of structure that grew out of that question, the theory of groups, now describes the symmetry of crystals, the codes that repair corrupted data, and the moves of a puzzle cube.

References

• al-Khwarizmi, Muhammad ibn Musa. 1831. The Algebra of Mohammed ben Musa. Edited and translated by Frederic Rosen. London: Oriental Translation Fund.
  • Free: Internet Archive
• O'Connor, J. J., and E. F. Robertson. 1999. "Abu Ja'far Muhammad ibn Musa Al-Khwarizmi." MacTutor History of Mathematics Archive, University of St Andrews. Last updated July 1999.
  • Free: MacTutor: Al-Khwarizmi

Introduction

Ask who invented algebra and you'll often hear one name. The historical record tells a longer story: methods for finding an unknown number grew over more than two thousand years, in several cultures. This reading traces that growth from Babylonian problem texts to al-Khwarizmi's book, whose title gave algebra its name.

Babylonian Procedures with Specific Numbers

An unknown is a quantity a problem asks you to find, such as a length or a price. Scribes in Mesopotamia, around 1800 BCE, solved problems about unknown lengths and areas that we would now write as quadratic equations, which involve an unknown multiplied by itself (O'Connor and Robertson 1996c; Katz 2018, chap. 1).

They didn't write equations. An equation is a statement that two quantities are equal, such as "this number squared, plus ten times the number, equals thirty-nine." The Babylonian texts instead give step-by-step procedures, worked on specific numbers. Their method amounts to what is now called completing the square. Their answers were always positive quantities, usually lengths (O'Connor and Robertson 1996c).

Abbreviations for the Unknown

Diophantus of Alexandria wrote a long work on number problems. His dates are uncertain; he is usually placed in the 3rd century CE, though some historians place him centuries before that (Katz 2018, chap. 6). He used abbreviations for the unknown, for its powers, and for "equals." That was a step from writing mathematics in words toward writing it in symbols.

In India, Brahmagupta, in the 7th century CE, also used abbreviations for unknowns, often the first letter of a color word. His methods admitted negative quantities, which many other traditions avoided (O'Connor and Robertson 1996c).

Al-Khwarizmi's Six Standard Forms

Muhammad ibn Musa al-Khwarizmi worked at the House of Wisdom in Baghdad, a center of scholarship and translation under the caliph al-Mamun. His dates are given as about 780 to about 850. His book Hisab al-jabr w'al-muqabala was written entirely in words, with no symbols (O'Connor and Robertson 1999). Al-Khwarizmi himself says that al-Mamun encouraged him to write it (al-Khwarizmi 1831, 3).

Al-Khwarizmi sorted problems about a number and its square into six types. Each type is a standard form: a fixed arrangement of terms that has its own rule. He needed six because he worked without zero or negative numbers as coefficients, so each arrangement had to be treated separately (O'Connor and Robertson 1996c).

The last three examples are his own (O'Connor and Robertson 1999). For each form he gave a general rule and then justified it with a geometric argument, completing a square drawn on the page.

Completion and Balancing

Two operations bring a problem into one of the six forms. The first is al-jabr (completion): removing a subtracted quantity by adding it to both sides. For example, x^2 = 40x - 4x^2 becomes 5x^2 = 40x. The second is al-muqabala (balancing): removing equal quantities that appear on both sides. For example, 50 + 3x + x^2 = 29 + 10x becomes 21 + x^2 = 7x, once 29 and 3x are taken from both sides (O'Connor and Robertson 1999).

The book's title, built from these two words, gives us the word "algebra" (O'Connor and Robertson 1999). Al-Khwarizmi's own description of the work, in Rosen's translation, is a short treatise on "Calculating by (the rules of) Completion and Reduction" (al-Khwarizmi 1831, 3).

Symbols Came Later

The letters you met in school are much younger than these methods. In 1591, François Viète used vowels for unknowns and consonants for known quantities, and Descartes later introduced the convention of using letters from the end of the alphabet for unknowns (Katz 2018, chaps. 12, 14).

Historians still disagree about how to weigh these contributions. Some have argued that al-Khwarizmi has the strongest claim to having founded algebra; others call his work unoriginal and point to older sources (O'Connor and Robertson 1999). The record supports a plainer description: Babylonian, Greek, Indian, and Arabic-language mathematicians each added methods, and European writers later added the symbols.

Conclusion

Algebra's methods for finding an unknown are older than its name and much older than its symbols. Al-Khwarizmi's book stands out because it organized the problems into standard forms with general rules and geometric justifications. Its title supplied the word "algebra."

Key Terms

• Unknown: A quantity a problem asks you to find, such as a length or a number.
• Equation: A statement that two quantities are equal.
• Standard form: One of al-Khwarizmi's six fixed arrangements of squares, roots, and numbers, each with its own rule of solution.
• Al-jabr (completion): The operation of removing a subtracted quantity by adding it to both sides of an equation.
• Al-muqabala (balancing): The operation of removing equal quantities that appear on both sides of an equation.

References

• al-Khwarizmi, Muhammad ibn Musa. 1831. The Algebra of Mohammed ben Musa. Edited and translated by Frederic Rosen. London: Oriental Translation Fund.
  • Free: Internet Archive
• Katz, Victor J. 2018. A History of Mathematics: An Introduction. 3rd ed., Classic Version. New York: Pearson.
  • Free: None
  • Publisher: Pearson
  • Free companion: O'Connor and Robertson 2018
• O'Connor, J. J., and E. F. Robertson. 1996c. "Quadratic, Cubic and Quartic Equations." MacTutor History of Mathematics Archive, University of St Andrews. Last updated February 1996.
  • Free: MacTutor: Quadratic, cubic and quartic equations
• O'Connor, J. J., and E. F. Robertson. 1999. "Abu Ja'far Muhammad ibn Musa Al-Khwarizmi." MacTutor History of Mathematics Archive, University of St Andrews. Last updated July 1999.
  • Free: MacTutor: Al-Khwarizmi

Introduction

"Do the same thing to both sides" is the first rule most people learn in school algebra. Behind it sits a larger ambition: a single recipe that solves every equation of a given kind. This reading covers the moves that keep an equation true, the formulas found for equations of degree 2, 3, and 4, and the formula for degree 5 that nobody could find.

Moves That Preserve Equality

An equation is a statement that two quantities are equal. If you add the same amount to both sides, the two sides stay equal. The same holds when you subtract, multiply, or divide both sides by the same number (as long as that number isn't zero). Most steps in solving an equation are balancing moves of this kind.

Take 3x + 4 = 19, where x is an unknown number. Subtract 4 from both sides to get 3x = 15. Divide both sides by 3 to get x = 5. Each move produces a new equation with exactly the same solutions as the old one.

An older move does the same work. If an equation has a quantity subtracted on one side, you can add that quantity to both sides, which moves it across as an addition. Al-Khwarizmi's 9th-century book called this move al-jabr, "completion," and it gave algebra its name (O'Connor and Robertson 1999).

Degree and the Quadratic Formula

The degree of an equation is the highest power of the unknown that appears in it. The equation 3x + 4 = 19 has degree 1, because x appears only to the first power. A quadratic equation has degree 2: its highest term is x^2, the unknown multiplied by itself.

For quadratics there is a general formula: an expression that gives the solutions of every equation of a given degree, using only the equation's numbers, arithmetic, and roots such as square roots. Write any quadratic as ax^2 + bx + c = 0, where a, b, and c are its fixed numbers, called coefficients, and a is not zero. Then

In words: take the negative of b, then add or subtract the square root of "b squared minus four times a times c," and divide the result by twice a. The symbol \pm means "plus or minus," which is why a quadratic can have two solutions. For x^2 + 10x - 39 = 0, the formula gives (-10 \pm 16) \div 2, so x = 3 or x = -13.

Cubic and Quartic Formulas in 16th-Century Italy

A cubic equation has degree 3, with an x^3 term. A quartic equation has degree 4, with an x^4 term. Formulas for both were found in Italy in the first half of the 1500s, and their story is also a quarrel.

Around 1515, Scipione del Ferro of Bologna solved one form of the cubic and kept his method secret until shortly before his death in 1526. Niccolò Tartaglia later found a method for the cubic on his own. Gerolamo Cardano persuaded Tartaglia to share it, after promising to keep it secret until Tartaglia published it himself. Cardano then published it in his Ars Magna in 1545, along with a solution of the quartic found by his student Lodovico Ferrari (O'Connor and Robertson 1996c).

The two sides told the story differently. Cardano had learned in Bologna that del Ferro had solved the cubic first, and he argued that this freed him to publish del Ferro's formula. Tartaglia, in a book of 1546, accused him of bad faith. Ferrari answered with a public challenge, and the two men met in a contest in Milan in 1548 (O'Connor and Robertson 2005). The formula is often called the Cardano–Tartaglia formula, a name that keeps both claims in view.

The Next Step: Degree 5

A quintic is an equation of degree 5, with an x^5 term. After 1545 the next target was a formula for it, and mathematicians tried for more than two centuries. In 1770, Joseph-Louis Lagrange studied why the cubic and quartic could be solved at all, by looking at how their solutions could be rearranged (O'Connor and Robertson 1996a). His question shifted attention from finding formulas to understanding why they work.

Conclusion

Algebra's balancing moves turn an equation into simpler equations with the same solutions. Built from those moves, general formulas exist for equations of degree 2, 3, and 4, the last two found amid a priority dispute that is still told from both sides. For degree 5, centuries of effort produced no formula.

Key Terms

• Degree: The highest power of the unknown that appears in an equation.
• Quadratic: An equation of degree 2, whose highest term is the unknown squared.
• General formula: An expression that gives the solutions of every equation of a given degree, using only the equation's coefficients, arithmetic, and roots.
• Cubic: An equation of degree 3, whose highest term is the unknown cubed.
• Quartic: An equation of degree 4, whose highest term is the unknown to the fourth power.
• Quintic: An equation of degree 5, whose highest term is the unknown to the fifth power.

References

• O'Connor, J. J., and E. F. Robertson. 1996a. "The Development of Group Theory." MacTutor History of Mathematics Archive, University of St Andrews. Last updated May 1996.
  • Free: MacTutor: The development of group theory
• O'Connor, J. J., and E. F. Robertson. 1996c. "Quadratic, Cubic and Quartic Equations." MacTutor History of Mathematics Archive, University of St Andrews. Last updated February 1996.
  • Free: MacTutor: Quadratic, cubic and quartic equations
• O'Connor, J. J., and E. F. Robertson. 1999. "Abu Ja'far Muhammad ibn Musa Al-Khwarizmi." MacTutor History of Mathematics Archive, University of St Andrews. Last updated July 1999.
  • Free: MacTutor: Al-Khwarizmi
• O'Connor, J. J., and E. F. Robertson. 2005. "Niccolò Tartaglia." MacTutor History of Mathematics Archive, University of St Andrews. Last updated September 2005.
  • Free: MacTutor: Niccolò Tartaglia

Introduction

An equilateral triangle cut from paper can be picked up and set back down in its outline in several different ways. Counting those ways, and combining them, leads to one of the central objects of modern algebra. This reading covers the symmetries of a triangle, the three rules that make a collection of actions a group, a second example from clock arithmetic, and where the idea came from.

The Six Symmetries of a Triangle

A symmetry of a shape is an action that moves the shape so it lands exactly in its own outline. Label the corners of an equilateral triangle A, B, and C, and trace its outline on the table. There are six symmetries:

• Three rotations about the center: by 0°, by 120°, and by 240°. The rotation by 0° leaves the triangle where it is.
• Three flips, each across a line through one corner and the midpoint of the opposite side.

Each one puts the corner labels in a different arrangement, and no other move lands the triangle in its outline.

Combining and Undoing

You can do one symmetry and then another. Doing two actions in sequence is called composition. Rotate by 120° and then by 240°, and the triangle has turned 360°, a full turn: every corner is back where it started. Flip across one line and then across a different line, and the result is a rotation.

Two features matter here. First, the composition of any two symmetries is again one of the six; it never produces a seventh. Second, the rotation by 0° acts as a "do nothing" move. That move is called the identity: the action that leaves everything as it is. Each symmetry also has an inverse, an action that undoes it. The inverse of rotating by 120° is rotating by 240°, and each flip is its own inverse.

The Group Idea

A group is a collection of actions, with a way of combining them, that follows three rules:

1. Combining any two actions in the collection gives an action in the collection.
2. The collection contains an identity, a "do nothing" action.
3. Every action has an inverse in the collection.

The formal definition adds one technical rule, that combining is associative: when you combine three actions in a row, it doesn't matter which pair you work out first. Actions done in sequence always satisfy it.

The power of the idea is that the rules matter more than what the actions are. Clock arithmetic shows this. Take the hours 0 through 11 on a 12-hour clock, and combine them by adding and wrapping around past 12. Nine o'clock plus 5 hours is 2 o'clock. Adding two hours always gives another hour on the clock, adding 0 is the identity, and adding 7 undoes adding 5. Hours on a clock and moves of a triangle have nothing physical in common, yet both are groups, and facts proved from the rules hold for both.

Where the Idea Came From

The group concept grew out of the study of equations. A permutation is a rearrangement of a list of objects, such as the solutions of an equation. In 1770, Joseph-Louis Lagrange studied permutations of the solutions of cubic and quartic equations, to understand why those equations could be solved. He never combined his permutations, though, so he never quite worked with groups (O'Connor and Robertson 1996a).

The link between solving equations and groups of permutations was first understood by Évariste Galois in 1831. In 1854, Arthur Cayley gave an abstract definition of a group, along with a table showing how its elements combine. Cayley's papers drew little attention when they appeared (O'Connor and Robertson 1996a).

Conclusion

A group is a collection of actions that can be combined, includes a "do nothing" action, and can always be undone. The six symmetries of an equilateral triangle and the twelve hours of a clock are both groups. The idea came out of the study of equations, starting with permutations of their solutions, and became an object of study in its own right.

Key Terms

• Symmetry: An action that moves a shape so it lands exactly in its own outline.
• Composition: Doing one action and then another, treated as a single combined action.
• Identity: The "do nothing" action, which leaves everything as it is.
• Inverse: The action that undoes a given action.
• Group: A collection of actions, with a way of combining them, in which combining any two gives another action in the collection, there is an identity, and every action has an inverse.
• Permutation: A rearrangement of a list of objects, such as the solutions of an equation.

References

• O'Connor, J. J., and E. F. Robertson. 1996a. "The Development of Group Theory." MacTutor History of Mathematics Archive, University of St Andrews. Last updated May 1996.
  • Free: MacTutor: The development of group theory

Introduction

For equations of degree 2, 3, and 4, mathematicians found formulas that solve every case. For degree 5 they searched for more than 250 years and found nothing, and then someone proved there was nothing to find. This reading covers what "no formula" means, the proofs of Ruffini and Abel, the insight of Galois, and why the result changed what algebra studies.

What "No Formula" Means

A quintic is an equation of degree 5, whose highest term is the unknown to the fifth power, x^5. An equation is solvable by radicals when its solutions can be written using only its coefficients, addition, subtraction, multiplication, division, and roots (square roots, cube roots, fifth roots, and so on). The quadratic formula shows that every degree-2 equation is solvable by radicals, and the 16th-century cubic and quartic formulas do the same for degrees 3 and 4.

The question for degree 5 was whether one formula of this kind works for every quintic. The answer is no. That doesn't mean quintics have no solutions; they do. It means the solutions of a general quintic can't always be written with arithmetic and roots.

Specific quintics can still be solved. The equation x^5 = 32 has the solution x = 2, because 2^5 = 32. The equation x^5 = 2 has the solution x = \sqrt[5]{2}, the fifth root of 2. What fails is a single formula that covers them all.

Ruffini and Abel

The first person to claim that the general quintic couldn't be solved by radicals was Paolo Ruffini, who published an argument in 1799. His proof had gaps, and he published further versions later. In 1824, Niels Henrik Abel gave the first proof that mathematicians accepted (O'Connor and Robertson 1996a).

Abel's result is an impossibility proof: a proof that something can't be done, no matter how clever the attempt. Such proofs are different from failed searches. Centuries of failure suggested that no formula existed. Abel's proof showed that none could.

Galois and the Structure of Solutions

Abel showed that no general formula exists. Évariste Galois, in work of 1831 and 1832, explained which equations are solvable by radicals and why (O'Connor and Robertson 1996a).

His idea was to study how an equation's solutions can be rearranged. Some rearrangements keep every relationship among the solutions true, where a relationship is anything you can state with whole-number arithmetic.

A small case shows the idea. The equation x^2 = 2 has two solutions, \sqrt{2} and -\sqrt{2}, where \sqrt{2} is the positive number whose square is 2. Their sum is 0 and their product is -2. Swap the two solutions, and the sum is still 0 and the product still -2. Swapping them breaks no relationship of this kind, so the swap is one of the allowed rearrangements. Those rearrangements form a group: combining two gives another, there is a "do nothing" rearrangement, and each can be undone. This is the equation's Galois group. Galois showed that an equation is solvable by radicals exactly when its Galois group has a particular structure: it can be broken down, step by step, into simple layers of a specific kind. The general quintic's group can't be broken down that way, so no formula exists.

Galois died in 1832, at age 20, after a duel (O'Connor and Robertson 1996d). His papers were not widely known until Joseph Liouville published them in 1846 (O'Connor and Robertson 1996a).

Why It Is a Landmark

The result is a landmark for two reasons. First, it settled a centuries-old question with a proof of impossibility, a kind of answer that searching alone can never give. Second, it changed the question. Instead of asking "what is the formula?", Galois asked "what structure do the solutions have?" Studying that structure, through groups, became a central part of algebra.

Conclusion

No formula using arithmetic and roots solves every quintic. Ruffini argued it first, Abel proved it in 1824, and Galois explained it through the group of each equation's solutions. Individual quintics can still be solved, sometimes by radicals and always by numerical methods, but the search for a universal formula ended with a proof that it can't exist.

Key Terms

• Quintic: An equation of degree 5, whose highest term is the unknown to the fifth power.
• Solvable by radicals: Describes an equation whose solutions can be written using only its coefficients, the four arithmetic operations, and roots.
• Impossibility proof: A proof that something can't be done, however the attempt is made.
• Galois group: The group of rearrangements of an equation's solutions that keep every relationship among them true.
• Structure: The way a group is built, including whether it can be broken down step by step into simpler layers.

References

• O'Connor, J. J., and E. F. Robertson. 1996a. "The Development of Group Theory." MacTutor History of Mathematics Archive, University of St Andrews. Last updated May 1996.
  • Free: MacTutor: The development of group theory
• O'Connor, J. J., and E. F. Robertson. 1996d. "Évariste Galois." MacTutor History of Mathematics Archive, University of St Andrews. Last updated December 1996.
  • Free: MacTutor: Évariste Galois

Introduction

"Completing the square" sounds like a phrase from a textbook. In al-Khwarizmi's 9th-century algebra, it describes something you can draw: a square with a corner missing, and the piece that fills it. This worked example solves his best-known problem with his picture, step by step, and then translates each step into modern symbols, so you can see that the algebra and the geometry are the same argument.

The Problem

In Frederic Rosen's 1831 translation, al-Khwarizmi states the problem as "one square, and ten roots of the same, amount to thirty-nine dirhems" (al-Khwarizmi 1831, 8). A "root" is the unknown number, and the "square" is that number multiplied by itself. The dirhems, a unit of currency, here just count units.

In modern notation, with x standing for the unknown number, the problem is

You're asked to find the positive number x that makes this true. Al-Khwarizmi gives a rule in words and then two geometric demonstrations of why it works. The steps below follow his second demonstration, which he introduces with "The same may also be explained by another figure" (al-Khwarizmi 1831, 10).

Working It Through

Step 1: Draw a square for the unknown

Draw a square whose side is the unknown length x. Its area is x times x, which is x^2. This square stands for the "square" in the problem. You don't know how big it is yet; that's the point of the problem.

Al-Khwarizmi's figures work this way throughout. Numbers become lengths and areas, and an equation becomes a statement that two areas are equal.

Step 2: Attach the ten roots as two rectangles

The ten roots, 10x, also need to be drawn as an area. A rectangle that is 10 long and x wide has area 10x. Al-Khwarizmi splits it in half: two rectangles, each 5 long and x wide, so each has area 5x. He attaches one along the right side of the square and one along the bottom (al-Khwarizmi 1831, 10).

The square and the two rectangles together make an L-shaped figure. Its area is x^2 + 5x + 5x, which is x^2 + 10x. The problem says that this total is 39.

Figure: A square of side x, with a 5 \times x rectangle attached along its right side and another along its bottom, forming an L shape; the missing 5 \times 5 corner square at the bottom right is shaded and labeled 25, and the whole completed square is marked with side x + 5. Alt text: A square labeled x squared, with two rectangles labeled 5x on two sides and a shaded 5-by-5 corner square completing a larger square of side x plus 5.

Step 3: Find the missing corner

Look at the L shape. It is almost a square, but the corner where the two rectangles would meet is empty. That gap is a square whose sides are both 5 long, because it is bounded by the 5-long ends of the two rectangles. Its area is 5 \times 5 = 25. In Rosen's translation, "there wants only a square of five multiplied by five, or twenty-five" (al-Khwarizmi 1831, 10–11).

Notice where the 5 came from: it is half the number of roots. That is why al-Khwarizmi's rule in words begins "halve the number of the roots."

Step 4: Complete the square

Fill in the corner. The L shape had area 39, and the corner adds 25, so the completed large square has area 39 + 25 = 64.

You now know the area of a square, so you can find its side. The side of a square of area 64 is 8, because 8 \times 8 = 64. On the figure, that side is made of the unknown length x plus the 5 that the rectangle added.

Step 5: Remove what you added, and check

The side of the large square is x + 5, and it equals 8. So x = 8 - 5 = 3.

Check it in the original problem. The square is 3 \times 3 = 9. Ten roots are 10 \times 3 = 30. Together they make 9 + 30 = 39, as required. Al-Khwarizmi reaches the same result: the root is three, and the square itself is nine (al-Khwarizmi 1831, 11).

Step 6: Translate each step into modern algebra

Every move on the figure is a balancing move on the equation. Attaching the two rectangles means writing 10x as 5x + 5x. Filling the corner means adding 25 to both sides:

The left side is now the area of the completed square, whose side is x + 5. So the equation becomes

Here (x + 5)^2 means x + 5 multiplied by itself. Taking the positive square root of both sides gives x + 5 = 8, and subtracting 5 from both sides gives x = 3.

Key Considerations

The negative solution. The modern equation has a second solution. If (x + 5)^2 = 64, then x + 5 could also be -8, which gives x = -13. You can check it: (-13)^2 is 169, ten times -13 is -130, and 169 - 130 = 39. Al-Khwarizmi didn't count this solution. In his figure, x is the side of a square, and a length can't be negative. His algebra worked with positive quantities only, which is also why he needed six separate types of equation rather than one.

Why halve the roots. The rule's first instruction, halve the ten roots, can look arbitrary in words. In the figure it is forced: you split the rectangle in two so that the pieces fit on two sides of the square and leave a square-shaped gap. Split it any other way and the gap isn't a square.

The common mistake: treating completing the square as a symbolic trick. Many people meet "add the square of half the coefficient to both sides" as a rule to memorize. That misses where the rule came from. It began as a literal completion: an L-shaped region with a missing corner, filled in to make a square whose side you can find. The symbols record the picture. Knowing that makes the rule easier to reconstruct when you've forgotten it, and it explains the name.

The method's reach. The same figure works for any problem of the form "a square and some roots equal a number," as long as the number of roots and the number are positive. With 6 roots instead of 10, you'd attach two rectangles of width 3 and fill a corner of 3 \times 3 = 9. That generality is what makes the picture a method rather than a single answer.

Summary

Al-Khwarizmi solved "a square and ten roots equal thirty-nine" by drawing it: a square, two rectangles, and a missing corner of 25 that completes a square of area 64. Each geometric step matches one line of modern algebra.

Geometric step	Modern algebraic line
Square of side x	x^2
Two rectangles, each 5 by x, attached to two sides	x^2 + 5x + 5x = x^2 + 10x = 39
Missing corner, 5 by 5	25
Completed square has area 39 + 25	x^2 + 10x + 25 = 64
Side of completed square is 8	(x + 5)^2 = 64, so x + 5 = 8
Remove the added 5	x = 3

Solution: x = 3. Check: 3^2 + 10 \times 3 = 9 + 30 = 39.

1. The table's rows follow al-Khwarizmi's second figure in order.
2. The check uses the original problem, not a rearranged version, so it catches any slip along the way.
3. The modern equation also allows x = -13, which al-Khwarizmi's geometry excludes.

References

• al-Khwarizmi, Muhammad ibn Musa. 1831. The Algebra of Mohammed ben Musa. Edited and translated by Frederic Rosen. London: Oriental Translation Fund.
  • Free: Internet Archive