The History of Algebra: From Al-Jabr to Modern Symbols

Today you can write \( x^2 + 10x = 39 \) in a few seconds. For most of history, the same problem would have taken a paragraph of words to state and a geometric diagram to solve. The story of algebra is the story of how people slowly invented the symbols and rules that make such problems routine.

Three Stages of Algebra

Historians often describe algebra as passing through three stages:

1. Rhetorical algebra: everything written in words, as in Babylonian and early Arabic texts.
2. Syncopated algebra: some abbreviations for unknowns and powers, as in Diophantus.
3. Symbolic algebra: full use of letters and signs, as developed in Europe from the 16th and 17th centuries.

These stages overlapped, but they help us see how gradually the familiar notation appeared.

Ancient Beginnings

Around 1800 BCE, Babylonian scribes solved quadratic problems about lengths and areas using recipes equivalent to completing the square, all written in words. Egyptian scribes solved linear "aha" problems, such as "a quantity and its seventh added together make 19", using a method of false position.

Around 250 CE in Alexandria, Diophantus wrote the Arithmetica, a collection of problems seeking rational or whole-number solutions. He used a special symbol for the unknown, abbreviations for its powers up to the sixth, and a sign for subtraction. This was a major step towards symbols, though he still had only one symbol for an unknown and wrote much in words.

Al-Khwarizmi and the Birth of Al-Jabr

Around 820 CE, at the House of Wisdom in Baghdad, Muhammad ibn Musa al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala (The Compendious Book on Calculation by Restoration and Balancing). The word al-jabr in the title became our word "algebra".

Al-Khwarizmi's book was rhetorical: no symbols, not even numerals in the text. But it was the first to treat solving equations as a subject in its own right, with general methods rather than isolated problems. He worked with three kinds of quantities: mal (the square, our \( x^2 \)), jidhr (the root, our \( x \)) and plain numbers.

Since negative numbers and zero coefficients were not accepted, he classified quadratic problems into six standard types, such as "squares equal roots" (\( ax^2 = bx \)) and "squares and numbers equal roots" (\( x^2 + c = bx \)). Two operations reduced any problem to a standard type:

• Al-jabr (restoration): removing a subtracted term by adding it to both sides.
• Al-muqabala (balancing): cancelling like terms that appear on both sides.

Worked Example 1: Al-jabr, al-muqabala and solving

Consider

$$ 4x^2 + 21 - 6x = 3x^2 + 4x $$

Step 1 (al-jabr): The left side has \( 6x \) subtracted. Restore it by adding \( 6x \) to both sides:

$$ 4x^2 + 21 = 3x^2 + 10x $$

Step 2 (al-muqabala): Squares appear on both sides. Balance by removing \( 3x^2 \) from each:

$$ x^2 + 21 = 10x $$

This is al-Khwarizmi's type "squares and numbers equal roots", and it is his own example. His rule says:

Step 3: Halve the number of roots: \( \tfrac{10}{2} = 5 \).

Step 4: Square it: \( 5^2 = 25 \).

Step 5: Subtract the number: \( 25 - 21 = 4 \).

Step 6: Take the root: \( \sqrt{4} = 2 \).

Step 7: Subtract from the half: \( 5 - 2 = 3 \), or add to it: \( 5 + 2 = 7 \).

So \( x = 3 \) or \( x = 7 \). Check: \( 9 + 21 = 30 = 10 \times 3 \) and \( 49 + 21 = 70 = 10 \times 7 \). Al-Khwarizmi explicitly noted that this type can have two solutions, a sophisticated observation.

Completing the square geometrically

For the type "squares and roots equal numbers", al-Khwarizmi's famous example is \( x^2 + 10x = 39 \). He justified the rule with a picture: draw a square of side \( x \), and attach rectangles of total area \( 10x \) as two strips of size \( x \times 5 \) on two sides. The missing corner is a \( 5 \times 5 \) square of area 25. Adding it completes a large square:

$$ x^2 + 10x + 25 = 39 + 25 = 64 $$

The large square has side \( x + 5 = 8 \), so \( x = 3 \). This is exactly where the phrase "completing the square" comes from.

Developments in the Islamic World

Al-Khwarizmi's successors extended algebra quickly. Abu Kamil (around 850–930) worked with irrational coefficients. Al-Karaji (around 953–1029) began treating algebra arithmetically, working with powers and polynomials and using an early form of proof by induction. Omar Khayyam (1048–1131) classified and geometrically solved all types of cubic equations. Latin translations of al-Khwarizmi by Robert of Chester (1145) and others carried these ideas into Europe.

Renaissance Italy: Solving the Cubic

In the early 1500s, Scipione del Ferro of Bologna found a formula for cubics of the form \( x^3 + px = q \) but kept it secret. Niccolo Tartaglia rediscovered it in 1535 and won a public problem-solving contest with it. Gerolamo Cardano persuaded Tartaglia to share the method, promising secrecy, then published it in his Ars Magna (1545), with credit to both, which caused a bitter quarrel. Cardano's student Lodovico Ferrari solved the quartic, also published in Ars Magna.

Worked Example 2: Cardano's example \( x^3 + 6x = 20 \)

For \( x^3 + px = q \), the formula is

$$ x = \sqrt[3]{\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} - \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} $$

Step 1: Here \( p = 6 \), \( q = 20 \). Compute

$$ \frac{q^2}{4} + \frac{p^3}{27} = \frac{400}{4} + \frac{216}{27} = 100 + 8 = 108 $$

Step 2: \( \sqrt{108} \approx 10.3923 \).

Step 3: The two cube roots:

$$ \sqrt[3]{10 + 10.3923} = \sqrt[3]{20.3923} \approx 2.7321 $$

$$ \sqrt[3]{-10 + 10.3923} = \sqrt[3]{0.3923} \approx 0.7321 $$

Step 4: Subtract: \( 2.7321 - 0.7321 = 2 \).

Check: \( 2^3 + 6(2) = 8 + 12 = 20 \). Correct. (In fact, the two cube roots are exactly \( \sqrt{3} + 1 \) and \( \sqrt{3} - 1 \).) In other cubics this formula forced mathematicians to take square roots of negative numbers, which eventually led to complex numbers.

The Arrival of Modern Symbols

The notation we use was assembled piece by piece:

• + and − first appeared in print in Johannes Widmann's arithmetic book of 1489.
• The equals sign = was introduced by the Welsh mathematician Robert Recorde in The Whetstone of Witte (1557), because "no two things can be more equal" than a pair of parallel lines.
• François Viète (1591) used letters systematically: vowels for unknowns and consonants for known quantities. This allowed general formulas, not just specific numbers.
• René Descartes, in La Géométrie (1637), introduced the convention still used today: \( x, y, z \) for unknowns, \( a, b, c \) for constants and exponents such as \( x^3 \).

Did You Know?

• In 1824, Niels Henrik Abel proved that there is no general formula using radicals for the quintic (degree 5) equation; Évariste Galois, who died at 20 in 1832, explained exactly when such formulas exist.
• The Spanish word algebrista once meant a bone-setter, because al-jabr meant restoring broken parts.
• Recorde's original equals sign was much longer than ours, like a pair of long parallel strokes.

Key Takeaways

• Algebra evolved from rhetorical (words) to syncopated (abbreviations) to symbolic notation.
• Al-Khwarizmi's book of around 820 CE made equation-solving a subject and gave algebra its name.
• Al-jabr means moving subtracted terms; al-muqabala means cancelling like terms.
• Cardano's Ars Magna (1545) published solutions of cubic and quartic equations.
• Modern symbols came from Widmann, Recorde, Viète and Descartes between 1489 and 1637.

If algebra, quadratic equations or polynomials are slowing you down, Amathify's experts and PhD scholars offer step-by-step solutions and assignment help, so reach us on WhatsApp or through the Ask a Question form.