Rational solutions
A rational number is a whole number or a ratio of whole numbers. Diophantus sought positive rational answers to many of his problems.
10 / ANTIQUITY TO THE RENAISSANCE
Diophantus of Alexandria
Two baskets hold a known total of apples. One holds a few more than the other. You can work out both amounts without guessing.
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01 / REASONING WITH AN UNKNOWN
Find what is inside two covered baskets
Two covered baskets hold 16 apples between them. The second holds 4 more than the first. How many are in each?
Diophantus wrote the unknown amount with a single symbol and reasoned with it step by step. Act out his steps with real apples, and watch the working appear.
How many apples are in each covered basket?
Together the baskets hold 16 apples. The second holds 4 more than the first. Act it out with 16 apples.
Total apples:
Basket 2 has:
Total
=The difference
=Each basket
THE IDEATake away the difference, share the rest equally, then add the difference back. Writing each step with a symbol for the unknown is algebra: x + (x + 4) = 16, so x = 6.
The person behind the ideas
- Probably worked in the third century CE; very little is certain about his life.
- Arithmetica collects problems solved by reasoning with unknown quantities.
- He accepted positive fractional solutions as well as whole-number ones.
More of the mathematics
More than one unknown
Arithmetica includes problems that connect multiple unknown quantities through equations. Modern Diophantine equations often ask specifically for integer solutions.