Official Soroban Arithmetic Reference

Complete Abacus Formulas Cheat Sheet

The definitive rulebook for Small Friends (Base 5), Big Friends (Base 10), and Combination Formulas. Master the exact physical bead fingerings required for tournament-speed mental arithmetic.

Small Friends (5s) Big Friends (10s) Combinations (Family) Interactive Drills

The Golden Rule: Why Formulas Exist on the Soroban

On a standard 1:4 Japanese Soroban, each vertical rod contains exactly one Upper Heaven Bead (value 5) and four Lower Earth Beads (value 1 each). A single rod can physically display numbers from 0 to 9.

Direct Addition

Free beads exist on the current rod. You simply push lower beads up with thumb or pull heaven bead down with index.

Small Friends (5s)

Not enough earth beads to finish the number, but the 5-bead above the beam is free. You trade through the 5-value bead.

Big Friends (10s)

The current rod is full (exceeds 9). You clear the companion bead on the units rod and carry 1 to the left tens rod.

5

Small Friends Formulas (Base 5)

Pairs that sum to 5: (1 & 4) and (2 & 3)

What are Small Friends in Abacus? Small Friends are complementary pairs of numbers that add up to 5: (1 and 4) and (2 and 3). On a Japanese Soroban, they are used when adding or subtracting single digits (1 to 4) on an active rod where lower earth beads are exhausted, but the upper 5-value heaven bead is free. The calculation borrows from 5 using the formula: +Number = +5 − Friend.

Small friend formulas come into play during single-digit addition and subtraction when you need to add beads to a rod that already contains 1, 2, 3, or 4 lower beads, but the upper 5-bead is still unactivated. The core mechanical fingering is Hasami-komi (the pinch motion): bringing the 5-bead down while simultaneously dropping the unwanted lower beads.

+ Addition (+ Partner)

Rule: +5 − Friend
+1 = +5 − 4

Example: 4 + 1 = 5. Lower 5-bead, remove 4 earth beads.

Pinch 5 & −4
+2 = +5 − 3

Example: 3 + 2 = 5. Lower 5-bead, remove 3 earth beads.

Pinch 5 & −3
+3 = +5 − 2

Example: 2 + 3 = 5. Lower 5-bead, remove 2 earth beads.

Pinch 5 & −2
+4 = +5 − 1

Example: 1 + 4 = 5. Lower 5-bead, remove 1 earth bead.

Pinch 5 & −1

− Subtraction (− Partner)

Rule: −5 + Friend
−1 = −5 + 4

Example: 5 − 1 = 4. Push 5-bead up, raise 4 earth beads.

Push 5 & +4
−2 = −5 + 3

Example: 5 − 2 = 3. Push 5-bead up, raise 3 earth beads.

Push 5 & +3
−3 = −5 + 2

Example: 5 − 3 = 2. Push 5-bead up, raise 2 earth beads.

Push 5 & +2
−4 = −5 + 1

Example: 5 − 4 = 1. Push 5-bead up, raise 1 earth bead.

Push 5 & +1
Practice Tip: Never hesitate during small friends. Drill instant subconscious reflex using our timed complement trainer.
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10

Big Friends Formulas (Base 10)

Pairs that sum to 10: (9 & 1, 8 & 2, 7 & 3, 6 & 4, 5 & 5)

What are Big Friends in Abacus? Big Friends are complementary pairs of numbers that add up to 10: (9 and 1), (8 and 2), (7 and 3), (6 and 4), and (5 and 5). They are used during addition and subtraction when rod capacity (9) is exceeded. Instead of calculating directly on one rod, the student subtracts the 10-complement friend on the active rod and carries 1 bead to the tens rod on the left: +Number = −Friend + 10.

Big Friends represent carry-over (addition) and borrowing (subtraction) between adjacent rods. Whenever adding a number exceeds 9 on a single column, we subtract the number's 10-complement from the current column and add 1 bead to the column on the left.

+ Addition (+10 Carry)

Rule: −Friend + 10
+9 = −1 + 10 Clear 1 lower, Carry 1 left
+8 = −2 + 10 Clear 2 lower, Carry 1 left
+7 = −3 + 10 Clear 3 lower, Carry 1 left
+6 = −4 + 10 Clear 4 lower, Carry 1 left
+5 = −5 + 10 Push 5-bead up, Carry 1 left
+4 = −6 + 10 Clear 6 (Pinch open), Carry 1 left
+3 = −7 + 10 Clear 7 (Pinch open), Carry 1 left
+2 = −8 + 10 Clear 8 (Pinch open), Carry 1 left
+1 = −9 + 10 Clear 9 (Pinch open), Carry 1 left

− Subtraction (−10 Borrow)

Rule: −10 + Friend
−9 = −10 + 1 Drop 1 left, Raise 1 lower
−8 = −10 + 2 Drop 1 left, Raise 2 lower
−7 = −10 + 3 Drop 1 left, Raise 3 lower
−6 = −10 + 4 Drop 1 left, Raise 4 lower
−5 = −10 + 5 Drop 1 left, Lower 5-bead
−4 = −10 + 6 Drop 1 left, Pinch 6 closed
−3 = −10 + 7 Drop 1 left, Pinch 7 closed
−2 = −10 + 8 Drop 1 left, Pinch 8 closed
−1 = −10 + 9 Drop 1 left, Pinch 9 closed
Visual Step-by-Step Breakdown: Watch beads animate in real time as 10-complements carry across columns on our Step Solver.
See Big Friends in Step Solver →

Combination Formulas (Family / Mixed)

Used for numbers 6, 7, 8, 9 when 5-bead is locked

What are Combination Formulas in Abacus? Combination formulas (also called Mixed Friends or Family formulas) apply to digits 6, 7, 8, and 9 when neither direct addition nor simple Big Friends can be executed because both the 5-bead and earth beads are partially locked. The calculation combines both 5-complements and 10-complements in a single compound movement: add the earth partner, subtract 5, and carry 10 to the left rod (e.g., +6 = +1 − 5 + 10).

Combination formulas occur when you must add 6, 7, 8, or 9, but the rod does not have enough lower beads to apply simple Big Friends directly because the 5-bead is already resting on the reckoning beam. This requires a synchronized 3-step action: Add the small friend partner, remove 5, and carry 10 to the left.

Combination Addition Formulas (+6, +7, +8, +9)

+6 = +1 − 5 + 10

Raise 1 earth bead with thumb, push 5-bead up with index, carry 1 to left rod with thumb.

Example: 5 + 6 = 11
+7 = +2 − 5 + 10

Raise 2 earth beads with thumb, push 5-bead up with index, carry 1 to left rod with thumb.

Example: 5 + 7 = 12
+8 = +3 − 5 + 10

Raise 3 earth beads with thumb, push 5-bead up with index, carry 1 to left rod with thumb.

Example: 5 + 8 = 13
+9 = +4 − 5 + 10

Raise 4 earth beads with thumb, push 5-bead up with index, carry 1 to left rod with thumb.

Example: 5 + 9 = 14

Combination Subtraction Formulas (−6, −7, −8, −9)

−6 = −10 + 5 − 1

Drop 1 on left rod, pull 5-bead down to beam, pull 1 earth bead down.

Example: 11 − 6 = 5
−7 = −10 + 5 − 2

Drop 1 on left rod, pull 5-bead down to beam, pull 2 earth beads down.

Example: 12 − 7 = 5
−8 = −10 + 5 − 3

Drop 1 on left rod, pull 5-bead down to beam, pull 3 earth beads down.

Example: 13 − 8 = 5
−9 = −10 + 5 − 4

Drop 1 on left rod, pull 5-bead down to beam, pull 4 earth beads down.

Example: 14 − 9 = 5
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Want to See These Formulas in Action?

Use our interactive Step-by-Step Abacus Solver. Type in any arithmetic problem, and the tool highlights the exact bead fingerings and formulas applied at each stage.

Launch Step Solver Open Virtual Abacus
Advanced Operations

How to Do Multiplication on Abacus →

Master the (m + n) rod positioning formula and left-to-right partial product accumulation with zero scrap paper.
Advanced Operations

How to Do Division on Abacus (Kato Method) →

Learn quotient placement rules, trial quotient revision, and clean step-by-step subtraction without panicking.

Frequently Asked Questions

What are Small Friends in abacus?
Small Friends are pairs of numbers that add up to 5: (1 and 4, 2 and 3). They are used when adding or subtracting single digits (1 to 4) on a single rod when there are not enough lower earth beads available, but the upper 5-value heaven bead is free.
What are Big Friends in abacus?
Big Friends are pairs of numbers that add up to 10: (9 and 1, 8 and 2, 7 and 3, 6 and 4, 5 and 5). They are used when adding or subtracting causes a carry-over or borrow to the adjacent rod on the left.
What is a combination formula in abacus?
Combination formulas (also known as Mixed Friends or Family formulas) are used for numbers 6, 7, 8, and 9 when neither direct addition nor simple Big Friends can be executed directly because the 5-bead is already active. For example: +6 = +1 - 5 + 10.
Which finger moves which bead according to Japanese Soroban rules?
In authentic Japanese Soroban pedagogy, the right thumb is reserved exclusively for moving lower earth beads UP toward the reckoning beam. The right index finger operates all other movements: pulling lower earth beads DOWN, pulling the upper heaven bead DOWN (+5), and pushing the heaven bead UP (-5).