How to Use the Suanpan (Chinese) Abacus — Tutorial

Understanding the Suanpan: Structure and Number Representation

Before performing calculations, you must understand the physical layout and how numbers are encoded.

Core Components

• Frame: The outer structure that holds everything together.
• Beam (横梁): The horizontal bar dividing upper and lower beads.
• Rods (Columns): Vertical wires, each representing a place value (units, tens, hundreds, etc.). The rightmost rod is the units place.
• Upper Beads (Heaven Beads): Two per rod; each = 5 when moved down to the beam.
• Lower Beads (Earth Beads): Five per rod; each = 1 when moved up to the beam.

Representing Numbers

A number is formed by moving beads toward the beam. The value on a rod is the sum of activated beads.

Examples:

• 3: Move 3 lower beads up → 1+1+1 = 3
• 7: Move 1 upper bead down (5) + 2 lower beads up (2) → 5+2 = 7
• 9: Move 1 upper bead down (5) + 4 lower beads up (4) → 5+4 = 9
• 0: All beads away from the beam (upper beads up, lower beads down)

Note: The extra lower bead (compared to the Japanese soroban’s 4) allows for more flexible intermediate states during complex operations.

Mastering Complements: The Key to Speed and Efficiency

The true power of the suanpan lies in using complements—pairs of numbers that sum to 5 or 10. This avoids tedious bead-by-bead counting and enables rapid calculation.

Complements to 5

Used when adding or subtracting within a single rod:

• 1 ↔ 4
• 2 ↔ 3

Rule: To add a number that would exceed 5, subtract its complement to 5 and add 1 heaven bead.
Example: To add 3 to 4 (which would be 7):

• 4 = 4 earth beads up
• Instead of adding 3, subtract 2 (complement of 3 to 5) → 2 earth beads remain
• Add 1 heaven bead (5) → total = 5 + 2 = 7

Complements to 10

Used when a rod overflows (e.g., adding to 9):

• 1 ↔ 9
• 2 ↔ 8
• 3 ↔ 7
• 4 ↔ 6
• 5 ↔ 5

Rule: To add a number that would exceed 9, subtract its complement to 10 from the current rod and carry 1 to the rod on the left.

Example: Add 6 to 8:

• 8 = 1 heaven + 3 earth beads
• Subtract 4 (complement of 6 to 10) → 8 – 4 = 4
• Carry 1 to the tens rod → result = 14

Step-by-Step Operations

Addition

1. Enter the first number from left to right.
2. Add the second number, rod by rod from right to left.
3. Apply complements as needed:
   • If sum ≤ 5: use earth beads.
   • If 5 < sum ≤ 9: use 1 heaven + earth beads.
   • If sum ≥ 10: use complement to 10, carry to next rod.

Subtraction

1. Enter the minuend (larger number).
2. Subtract the subtrahend, rod by rod from right to left.
3. Borrowing: If you can’t subtract, borrow 1 from the left rod (add 10 to current rod), then subtract.

Example: 102 – 57

• Enter 102
• Subtract 7 from units: can’t do 2 – 7 → borrow 1 from tens (but tens is 0!)
• Borrow from hundreds: hundreds becomes 0, tens becomes 10
• Now borrow from tens: tens becomes 9, units becomes 12
• 12 – 7 = 5 → units = 5
• 9 – 5 = 4 → tens = 4
• Result: 45

Multiplication (Partial Products Method)

1. Place the multiplicand on the right side.
2. For each digit of the multiplier (from right to left):
   • Multiply it by the entire multiplicand.
   • Add the partial product to the left side, shifting one rod left for each digit position.
3. Sum all partial products.

Example: 24 × 3

• Place 24 on rods D (tens) and E (units)
• Multiply 3 × 4 = 12 → add to rods C and D
• Multiply 3 × 2 = 6 → add to rods B and C
• Final result on rods B–D: 72

Division (Traditional Method)

1. Place the dividend on the right, divisor on the left.
2. Estimate the first digit of the quotient.
3. Multiply the divisor by this digit and subtract from the dividend.
4. Record the quotient digit in the middle.
5. Repeat with the remainder.

This mirrors long division but is streamlined by the abacus’s visual layout.

Suanpan vs. Soroban: Key Differences

| Feature | Suanpan (Chinese) | Soroban (Japanese) | |--------|------------------------|------------------------| | Beads per rod | 2 upper, 5 lower | 1 upper, 4 lower | | Flexibility | Higher (extra beads for intermediate steps) | Optimised for speed | | Learning curve | Slightly steeper | More streamlined | | Best for | Traditional methods, mental math depth | Speed competitions, modern education |