C. Algebra · Algebra: Expressions, Equations and Coding

The Story of Algebra: Ideas from Many Cultures

C1.1 A1 A2About 60 minutes

Learning goals

I can describe how algebra developed over thousands of years in several cultures, including Babylon, Egypt, China, India and Baghdad.
I can tell the story of one algebraic idea, where it came from and how it was used.
I can solve historical problems with today's symbols and compare the old and new methods.
I can explain why algebra still matters today, in school, at work and in technology.

Words to know

algebra
The part of mathematics that uses letters and symbols to stand for numbers, so we can describe general rules and solve for unknown values.

unknown
A value we do not know yet and want to find. Today we usually write it as a letter such as x.

al-jabr
Arabic for “restoring” or “completion”. In al-Khwarizmi's book it meant moving a subtracted amount to the other side of an equation by adding it to both sides. It gave us the word “algebra”.

al-muqābala
Arabic for “balancing”. Cancelling equal amounts that appear on both sides of an equation.

rhetorical algebra
Algebra written completely in words, with no symbols.

symbolic algebra
Algebra written with letters and symbols such as x, +, = and exponents, the way we write it today.

method of false position
An old strategy: guess a convenient answer, see how far off the result is, then scale the guess to fix it.

Remember from Grade 8

In Grade 8 you used variables in patterns and expressions, added binomials with algebra tiles, and solved equations with integers and decimals, including equations with variables on both sides. In Grade 9 you will do all of this with more confidence and more kinds of expressions. But first, a question most students never ask: where did algebra come from?

Algebra before the letter x

Imagine solving an equation with no symbols at all. No x, no + sign, not even an equals sign. Everything is written in sentences.

That is how algebra worked for most of its history. People in many parts of the world solved problems about land, trade, taxes, building and sharing goods. Over thousands of years, they invented methods, and then, much later, the short symbols we use today.

Algebra was not invented by one person or one culture. It grew, piece by piece, in many places.

Some milestones in the story of algebra (spacing not to scale). Ideas travelled between cultures through trade, translation and teaching.

A few chapters of the story

Babylon (in present-day Iraq), about 3800 years ago. Scribes pressed problems into clay tablets. Some ask for unknown lengths of fields from facts about their area, which we would now solve with equations. They used a base-60 number system. We still use base 60 for time: 60 seconds in a minute and 60 minutes in an hour.
Egypt, about 3500 years ago. The Rhind Papyrus, copied by a scribe named Ahmes, includes “aha” problems such as “A quantity and its quarter added together become 15.” Egyptian scribes often solved them by the method of false position.
China, by about 2000 years ago. The Nine Chapters on the Mathematical Art collected problems and methods. It solved systems of equations by arranging counting rods on a board, in a way very close to a method still taught today. In Liu Hui's famous commentary on the book (263 CE), rods of two colours (red and black) stand for positive and negative quantities.
India, 628 CE. The astronomer and mathematician Brahmagupta wrote rules for calculating with zero and negative numbers, which he described as “fortunes” and “debts”. For example: “the product of two debts is a fortune” (a negative times a negative is positive).
Baghdad, about 820 CE. At the House of Wisdom, Muhammad ibn Musa al-Khwarizmi wrote a book about solving equations by al-jabr and al-muqābala. The word algebra comes from al-jabr, and the word algorithm comes from his name. He still wrote everything in words.
Europe, 1500s and 1600s. Symbols arrived. In 1557 the Welsh mathematician Robert Recorde introduced the equals sign. In 1637, René Descartes in France used letters near the end of the alphabet (x, y, z) for unknowns, which is the habit we still follow.

The two moves in al-Khwarizmi's title. Al-jabr restores a subtracted amount by adding it to both sides. Al-muqābala cancels matching amounts on both sides. You still use both moves every time you solve an equation.

Worked example

Example 1: A Nine Chapters problem

Here is a problem from The Nine Chapters: some people buy an item together. If each person pays 8 coins, there are 3 coins too many. If each pays 7 coins, they are 4 coins short. How many people are there, and what is the price?

Let n be the number of people. Paying 8 each gives 8n coins, which is 3 more than the price. So the price is 8n − 3.
Paying 7 each gives 7n coins, which is 4 less than the price. So the price is also 7n + 4.
Both expressions describe the same price: 8n − 3 = 7n + 4.
Balance (al-muqābala): subtract 7n from both sides to get n − 3 = 4.
Restore (al-jabr): add 3 to both sides to get n = 7.
Price: 8(7) − 3 = 56 − 3 = 53. Check with the other plan: 7(7) + 4 = 49 + 4 = 53. ✓

There are 7 people and the item costs 53 coins.

Worked example

Example 2: The method of false position

Solve the Egyptian-style problem “A quantity and its quarter added together become 15” the way a scribe might, then with modern algebra.

False position: pick a guess that makes the quarter easy. Guess 4. Then 4 + (quarter of 4) = 4 + 1 = 5.
We wanted 15, not 5. Since 15 ÷ 5 = 3, the result is 3 times too small, so the answer must be 3 times the guess: 4 × 3 = 12.
Modern algebra: let x be the quantity. x + x/4 = 15.
Write x as 4x/4: 4x/4 + x/4 = 5x/4, so 5x/4 = 15.
Multiply both sides by 4: 5x = 60. Divide by 5: x = 12.
Check: 12 + 12/4 = 12 + 3 = 15. ✓

The quantity is 12. False position works here because the result grows in proportion to the guess: triple the guess, triple the result.

Worked example

Example 3: From words to symbols

Al-Khwarizmi's book has a famous problem: “A square and ten roots are equal to thirty-nine.” (“Root” meant the unknown and “square” meant the unknown times itself.) Write it in symbols and check that 3 is a solution.

“A square” is x². “Ten roots” is 10x. “Are equal to thirty-nine” is = 39.
Symbols: x² + 10x = 39.
Substitute x = 3: 3² + 10(3) = 9 + 30 = 39. ✓

x² + 10x = 39, and x = 3 works. Notice how much shorter the symbolic version is. That is why symbols spread.

How to research a math idea

When you research a mathematical idea, look for:

Who developed or used it, and where (name the specific culture or community).
When, as closely as sources agree. Many ancient dates are approximate, so write “about” or “c.”.
What problem it solved for people at the time.
How it was written or calculated (words, rods, tablets, symbols).
Why it matters today.

Use reliable sources: museum and university websites, encyclopedias and library books. If two sources disagree about a date, say so.

Hands-on activity

Make a “story card” for an algebra idea

You need: paper or a poster board, markers, a library book or a device with internet access (ask a grown-up which sites are OK).

Choose one idea and one culture. Some choices: Babylonian clay-tablet problems, Egyptian false position, counting rods in The Nine Chapters, Brahmagupta's rules for negative numbers, al-Khwarizmi's al-jabr, Recorde's equals sign, Descartes' x and y. You may also explore a mathematical idea from a knowledge system you or your family are connected to, using sources from that community.
Use the five questions in the tip above. Write each answer in one or two sentences.
Include one problem from that time, solved two ways: the historical way and the modern way with symbols.
Finish with a sentence that starts “This still matters today because …”.
Present your card to someone at home in about two minutes.

What to look for: a specific culture (not just “ancient people”), an approximate date, a real problem worked out correctly, and a clear statement of relevance.

Just for fun

Before 1557, mathematicians had to write “is equal to” in words, over and over. Robert Recorde got tired of it and drew two parallel lines instead, explaining that no two things can be more equal. Talk about a time-saving invention.

Your turn

What did al-jabr mean in al-Khwarizmi's book? Show it on 4x − 7 = 9.

Nine Chapters style: if each person pays 9 coins there are 11 too many; if each pays 6 coins they are 16 short. Find the number of people and the price.

Use false position to solve: “A quantity and its third together make 24.”

Write in symbols and solve: “A thing, twice the thing, and 5 more make 35.”

Brahmagupta wrote that “a fortune times a debt is a debt”. Write this as a modern integer rule with an example.

Which person's name gave us the word algorithm?

Give two reasons symbolic algebra was a big improvement over writing in words.

More practice: warm-up

Who introduced the equals sign, and in what year?

Whose writing in 1637 made x, y, z the usual letters for unknowns?

Use al-muqābala on 6x + 2 = 2x + 14: what is left after you take 2x and 2 from both sides? Solve.

What base did the Babylonian number system use? Name one place we still see it.

In Chinese work on The Nine Chapters (such as Liu Hui's commentary), what did rods of two different colours show?

More practice: on your own

Nine Chapters style: if each pays 5 coins they are 45 short; if each pays 7 coins they are 3 short. How many people, and what is the price?

Solve “A quantity and its fifth make 30” two ways: by false position and with an equation.

Evaluate (−3) × (−5) and say the result in Brahmagupta's “fortune and debt” language.

Al-Khwarizmi also solved problems like “a square and twenty-one equal ten roots”. Write it in symbols and check whether x = 3 and x = 7 work.

Give one example of how algebra is used in a job or in technology today, and say what the variable stands for.

Challenge

Explain how al-Khwarizmi could solve x² + 10x = 39 with a picture: a square of side x plus two rectangles that are 5 by x. (Hint: what small square completes the picture?)

Show that for any Nine Chapters problem where paying a each gives an excess of e and paying b each gives a shortfall of d, the number of people is n = (e + d) ÷ (a − b). Test it on Example 1.

False position works for “a quantity and its quarter make 15”. Show that it fails for “a quantity and 3 more make 10”, and explain why.

Check your understanding

Quiz

The word algebra comes from:

Where did al-Khwarizmi work, around 820 CE?

In Liu Hui's commentary on The Nine Chapters on the Mathematical Art, red and black counting rods were used to show:

Who introduced the equals sign in 1557?

Solve: “A quantity and its half together make 18.”

Brahmagupta (India, 628 CE) is known for writing rules for calculating with:

Which statement about the history of algebra is accurate?

For parents and teachers

Big idea: MTH1W expectation C1.1 asks students to research an algebraic concept to tell a story about its development and use in a specific culture, and to describe its relevance. This lesson gives an overview and models the research, then the activity asks your teen to research one idea in depth.

Common difficulties:

Vague answers such as “ancient people did math”. Push for a specific culture, place and approximate date.
Over-precise dates. Many ancient dates are estimates; “about” is the honest word.
Treating history as separate from math. The historical problems are real equations. Ask your teen to solve each one with modern algebra and check it.

How to help: Listen to the two-minute “story card” presentation and ask “How do you know?” about one fact. If your family has a connection to a culture with its own mathematical traditions, this is a good chance to talk about it. This lesson also builds Strand A skills: making connections between mathematics, different knowledge systems and real life (A2).