Digit by digit
For a bigger number, the same pieces find one digit of the root at a time: first the tens, then the units, and so on.
27 / ANTIQUITY TO THE RENAISSANCE
Jia Xian
Grow a cube a little along every edge. How many new pieces does it need? And how can that help you find a cube’s edge from its volume?
Explore the mathematics ↓
01 / A TRIANGLE FOR ROOTS
Count the pieces, find the root.
When a cube grows along its edges, the new pieces come in a fixed pattern: 3 slabs, 3 rods and 1 corner cube.
Jia Xian’s triangle lists these patterns for squares, cubes and higher powers. Each entry is the sum of the two above it.
Numbers written as:
Volume of the cube:
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THE IDEARow 3 of the triangle, 1 3 3 1, counts the pieces of a growing cube: (10 + b)³ = 1,000 + 3 × 100b + 3 × 10b² + b³. Try b until the new pieces use up the volume, and you have the root.
The person behind the ideas
- Worked in China during the Northern Song period.
- His own mathematical writings are lost.
- Yang Hui preserved descriptions of his methods in 1261.
More of the mathematics
An arithmetic shortcut
His additive-multiplicative method reaches the same numbers by repeated adding and multiplying. It is related to the process now called Horner’s method.