Little Friends and Big Friends Explained: Complete Formula Guide

Having little friends and big friends explained clearly demystifies the core arithmetic formulas of the Japanese soroban. Little Friends are complement pairs that add up to 5, used when lower beads are full. Big Friends are complement pairs that add up to 10, used for carrying into the tens column.

Published: October 4, 2026 · 10 min read · Formula Mastery

Why Does the Abacus Need "Friend" Formulas?

On a modern soroban, each rod has only four lower beads (worth 1 each) and one upper bead (worth 5). That means you can only represent numbers from 0 to 9 directly on any single rod.

When you want to add a number, but do not have enough physical beads available on that rod, you must exchange beads using complement arithmetic. The abacus community affectionately calls these complement pairs "Friends" because they help each other complete a target number (either 5 or 10).

What are Little Friends (Complements of 5)?

Little Friends formulas are used when you need to add or subtract on a rod where the upper 5-bead is available, but the lower beads are already occupied. Because the target base is 5, there are only two pairs of Little Friends:

4 and 1 are Friends

Because 4 + 1 = 5, when you need 4, you call upon 1. When you need 1, you call upon 4.

3 and 2 are Friends

Because 3 + 2 = 5, when you need 3, you call upon 2. When you need 2, you call upon 3.

The Little Friends Addition Formulas (+):

To add 4: +4 = +5 − 1 (Push down 5, push down 1)
To add 3: +3 = +5 − 2 (Push down 5, push down 2)
To add 2: +2 = +5 − 3 (Push down 5, push down 3)
To add 1: +1 = +5 − 4 (Push down 5, push down 4)

The Little Friends Subtraction Formulas (−):

When subtracting a small number, but not enough lower beads are active, borrow from the 5-bead:

To subtract 4: −4 = −5 + 1 (Lift up 5, push up 1)
To subtract 3: −3 = −5 + 2 (Lift up 5, push up 2)
To subtract 2: −2 = −5 + 3 (Lift up 5, push up 3)
To subtract 1: −1 = −5 + 4 (Lift up 5, push up 4)

Example Calculation: 4 + 3 = 7

Start with 4 on the rod (all four lower beads up). To add 3, you have no lower beads left. Look to the Little Friend of 3, which is 2. The formula is: +5 − 2. Pull the upper 5-bead down, and push 2 lower beads down. The rod now shows one 5-bead and two 1-beads. The answer is 7!

Figure 1: Result of 4 + 3. The 5-bead is down and two 1-beads remain active, making 7.

What are Big Friends (Complements of 10)?

Big Friends formulas apply when a rod cannot hold the full sum (the sum reaches or exceeds 10). Here, we borrow or carry with the tens rod to the left. The target base is 10, creating five complement pairs:

9 and 1: 9 + 1 = 10
8 and 2: 8 + 2 = 10
7 and 3: 7 + 3 = 10
6 and 4: 6 + 4 = 10
5 and 5: 5 + 5 = 10

The Big Friends Subtraction Formulas (−):

To subtract 9: −9 = −10 + 1
To subtract 8: −8 = −10 + 2
To subtract 7: −7 = −10 + 3
To subtract 6: −6 = −10 + 4
To subtract 5: −5 = −10 + 5

Figure 2: Example 8 + 5 = 13. Tens rod has 1 bead, ones rod has 3 beads.

Example Calculation: 8 + 5 = 13

Start with 8 on the ones rod (5-bead down + three 1-beads up). You want to add 5, but the 5-bead on this rod is already in use! Use the Big Friend of 5, which is 5. Formula: +10 − 5.

Push up 1 lower bead on the tens rod (+10).
Push up the 5-bead on the ones rod away from the beam (−5).
Read both rods: 1 on the tens rod, 3 on the ones rod. The result is 13!

How to Help Kids Memorize Friend Formulas

Rote recitation of formulas can bore young learners. Instead, use playful association:

High-Five Pairs: Ask your child, "Who is the high-five buddy of 3?" They quickly reply "2!"
Ten-Finger Pairs: Hold up 7 fingers and ask, "How many fingers are sleeping?" Three fingers!
Play Friend Dash: Abacus Buddy includes a dedicated rapid-fire game called Friend Dash that trains complement reflexes in 60-second bursts.

Master Friend Formulas Step by Step

Progress smoothly from Level 5 through Level 10 with interactive hints on Abacus Buddy.

Practice Friends Formulas Free