Numbers and algebra
c. AD 820
Algebra gets its name
Muhammad ibn Musa al-Khwarizmi
How do you solve an equation before anyone has invented the letter x?
Working in Baghdad’s House of Wisdom, al-Khwarizmi wrote a book on solving equations by “completion and balancing” (al-jabr, which became “algebra”). His own example: a square and ten of its roots make thirty-nine. His name became the word “algorithm”.
How they figured it out
Al-Khwarizmi wrote his algebra entirely in words: “a square and ten roots are equal to thirty-nine”. No symbols at all. He wrote it for practical people: merchants, surveyors, and judges dividing inheritances under Islamic law.
To solve it, he drew a picture. A square of side x. Ten roots become two rectangles, 5 by x, stuck on two sides. The shape is now a big square with one corner missing, a 5 by 5 piece. Add that 25 to both sides: the big square has area 64, so its side is 8, and x is 8 − 5 = 3. “Completing the square” was literal.
The equation
Do the same thing to both sides until the unknown stands alone.
See it in real life
Working back from the bill
“With a 10% tip the total came to ₹1,650. What was the bill?” You balance both sides: 1.1 × bill = 1,650, so the bill was ₹1,500. Spreadsheet “goal seek” and every calculator that solves for x is doing al-jabr.
Where engineers use it
Solving for unknowns
Every design calculation, from beam size to resistor value, rearranges an equation to isolate one quantity.
Algorithms
Step-by-step procedures, named after him, run everything from sat-navs to search engines.
Worked example: His own example, by completing the square
- Add to both sides:
- So , giving and .
Where it came from
- Zero becomes a number AD 628
Built on
Al-Khwarizmi’s arithmetic book carried Indian numerals, zero included, to the Islamic world and later to Europe.
What it led to
- Shapes become equations 1637
Built on
Descartes drew algebra’s equations as curves.