Adding and taking away
Big friends: the 10 complements
A rod can only hold 9. When a sum spills over, take off the big friend here and carry a 1 next door.
The pairs
Two numbers are big friends when they add up to 10:
| Number | Its friend | Together |
|---|---|---|
| 1 | 9 | 1 + 9 = 10 |
| 2 | 8 | 2 + 8 = 10 |
| 3 | 7 | 3 + 7 = 10 |
| 4 | 6 | 4 + 6 = 10 |
| 5 | 5 | 5 + 5 = 10 |
| 6 | 4 | 6 + 4 = 10 |
| 7 | 3 | 7 + 3 = 10 |
| 8 | 2 | 8 + 2 = 10 |
| 9 | 1 | 9 + 1 = 10 |
Adding: fix this rod, then carry
Rule: if adding a number would take the rod past 9, take its big friend away from this rod, then add 1 to the rod on the left. Finish the current rod first, then touch the one on its left.
Worked example: 3 + 7 = 10
Sometimes taking the friend off needs a combined move of its own. Here the friend of 4 is 6, and taking 6 away from 6 is a heaven-and-earth pinch:
Worked example: 6 + 4 = 10
Taking away: borrow first
Rule: if the rod does not have enough to take the number away, take 1 from the rod on the left first, then add the big friend to this rod. This order is the reverse of adding, and it is the detail people get wrong most often.
Worked example: 10 − 7 = 3
Worked example: 12 − 5 = 7
Why the order is different
When adding, you finish the current rod before carrying so that you never have to read a half-changed number. When taking away, you remove the borrowed ten first so the friend lands on a rod with room for it. Do it the other way round and you end up moving beads you have already set.
Chain reactions: 99 + 1
If the rod on the left is already 9 when you carry, the same trick simply happens again, one rod further along:
Worked example: 99 + 1 = 100
Chart: which subtractions need a big friend?
Every BF cell means borrow from the left, then add the friend here.
| Rod shows | −1 | −2 | −3 | −4 | −5 | −6 | −7 | −8 | −9 |
|---|---|---|---|---|---|---|---|---|---|
| 0 | BF | BF | BF | BF | BF | BF | BF | BF | BF |
| 1 | D | BF | BF | BF | BF | BF | BF | BF | BF |
| 2 | D | D | BF | BF | BF | BF | BF | BF | BF |
| 3 | D | D | D | BF | BF | BF | BF | BF | BF |
| 4 | D | D | D | D | BF | BF | BF | BF | BF |
| 5 | LF | LF | LF | LF | 5 | BF | BF | BF | BF |
| 6 | D | LF | LF | LF | 5 | C | BF | BF | BF |
| 7 | D | D | LF | LF | 5 | C | C | BF | BF |
| 8 | D | D | D | LF | 5 | C | C | C | BF |
| 9 | D | D | D | D | 5 | C | C | C | C |
- DDirect: just move the beads
- 5The 5-bead on its own
- C5 and a bit, together
- LFLittle friend (5 complement)
- BFBig friend (carry or borrow)
Common questions
What is a 10 complement on the abacus?
It is the number that makes 10 with another: 1 and 9, 2 and 8, 3 and 7, 4 and 6, or 5 and 5. It is how carrying and borrowing are done on a soroban.
When do you carry on an abacus?
Whenever adding a digit would take a rod past 9. Take that digit’s big friend away from the rod, then add 1 to the rod on its left.
Why do you borrow from the left rod first when subtracting?
Taking the borrowed 1 away first means the complement lands on a rod that has room for it. Doing the current rod first makes you undo beads you have already moved.