# Performing Subtraction on the Abacus

Subtraction is performed by first registering the minuend and then subtracting, starting from the left, by removing beads from either or both the lower or upper decks.

(Note: Please refer to the scans of the the book from which these instructions were copied for the most accurate version. Also, Dave Bagley's Java abacus includes a built-in interactive tutorial that teaches counting, addition and subtraction.)

## Subtraction

Subtraction is performed by first registering the minuend and then subtracting, starting from the left, by removing beads from either or both the lower or upper decks. The final bead-positions represent the answer.

### Simple Taking-off

This is achieved by simply taking off one or more beads from the lower deck, or sometimes both. Example: When subtracting 7 (represented by -5-2= -7) from 9, remove 1 bead from the upper-deck (-5) and 2 beads from the lower deck (-2). The remaining 2 beads represent the result.

Given the first number To subtract Move bead(s)
in the lower deck    in the upper deck
1,2,3,4,5,6,7 or 8 1 -1
2,3,4,7 or 8 2 -2
3,4,8 or 9 3 -3
4 or 9 4 -4
5,6,7,8, or 9 5
-1
6,7,8 or 9 6 -1 -1
7,8 or 9 7 -2 -1
8 or 9 8 -3 -1
9 9 -4 -1

Note: The "-" symbol in the Move bead(s) columns represents moving the bead(s) away from the middle beam.

### Combined Adding-up And Taking Off

When the number of beads in the lower deck is less than the subtracter (the number being subtracted), one or more beads are added in the lower deck and 1 bead is removed from the upper-deck.

Example: When subtracting 4 (+1-5 = -4) from 7 (represented by 1 bead in the upper-deck and 2 beads in the lower deck (less than 4, the subtracter), one bead is added to the lower deck (+1) and 1 bead is removed from the upper-deck (-5) leaving 3 beads, representing the result.

Given the first number To subtract (formula) Move bead(s)
in the lower deck    in the upper deck
5 1 (-5 +4) +4 -1
5 or 6 2 (-5 +3) +3 -1
5,6 or 7 3 (-5 +2) +2 -1
5,6,7 or 8 4 (-5 +1) +1 -1

Note: The "+" symbol in the Move bead(s) columns represents moving the bead(s) towards the middle beam; the "-" symbol indicates that the bead(s) should be moved away from the middle beam.

### Taking-off From A Rod Of Higher Order And Adding-up

When a number on a specific rod is smaller than the subtrahend (4 is the subtrahend when performing 13 - 4; note that in the ones column, the 3 is less than the 4) one bead for the order of tens and one bead from the lower-deck has to be taken off, and one bead from the upper-deck is counted.

Given the first number To subtract Move bead(s)
lower deck, adjacent (right) column   in the lower deck    in the upper deck
0 1(+9 -10) -1 +4 +1
0 or 1 2 (+8 -10) -1 +3 +1
0, 1 or 2 3 (+7 -10) -1 +2 +1
0, 1,2 or 9 4 (+6 -10) -1 +1 +1
0, 1,2 3, or 4 5 (+5 -10) -1
+1
0 or 5 6 (+4 -10) -1 +4
0,5 or 6 7 (+3 -10) -1 +3
0,1,2,5,6, or 7 8 (+2 -10) -1 +2
0,1,2,3,5,6,7 or 8 9 (+1 -10) -1 +1

### Combined Taking-off From A Rod Of Higher Order, Adding-up in the Upper-deck and Taking-off in the Lower-Deck

This technique is called for when a number on a specific rod is smaller than the supposed subtrahend [I have no idea what this means], but only in such cases as exemplified by 12 - 6.

Given the first number To subtract Move bead(s)
lower deck, adjacent (right) column   in the lower deck    in the upper deck
1,2,3 or 4 6 (-1 +5 -10) -1 -1 +1
2,3 or 4 7 (-2 +5 -10) -1 -2 +1
4 or 4 8 (-3 +5 -10) -1 -3 +1
4 9 (-4 +5 -10) -1 -4 +1