How to use an Abacus

Diagram of a typical Chinese abacus set to zero - all heaven beads pushed up and all earth bead pushed down.

Heaven beads, each worth 5

Earth beads, each worth 1
Try the on-line Abacus

The Abacus utilizes a combination of two bases (base-2 and base-5) to represent decimal numbers. It is held horizontally with the smaller deck at the top. Each bead on the top deck has the value 5 and each bead on the lower deck has the value 1. The beads are pushed towards the central crossbar to show numbers. Working from right to left, the first vertical line represents units, the next tens, the next hundreds and so on.

So for example to show the number 9, on the first line, one heaven bead (top deck) would be moved down (representing 5 units) and 4 earth beads (bottom deck) would be moved up (each representing 4 units). To show the number 79, in addition to the beads in the first line used to make the number 9, one heaven bead would be moved down and two earth beads would be moved up on the second line, representing 5 tens and 2 tens respectively.

Addition on the abacus involves registering the numbers on the beads in the straight-forward left-to-right sequence they are written down in. As long as the digits are placed correctly, and the carry's noted properly, the answer to the operation immediately presents itself right on the abacus.

There are 4 approaches to performing additions, each applied to a particular situation. Each of these techniques is explained (!) in tabular form in the sections that follow.

Simple Addition

When performing the addition 6+2, one would move 1 bead from the upper deck down (value = 5) and one bead from the lower deck up (value = 1); this represents 6. Moving 2 beads from the lower deck (in the same column) up (value = 1 * 2 beads = 2) would complete the operation. The answer is then obtained by reading resultant bead positions.

Given the first number	To add	Move bead(s)
		in the lower deck	in the upper deck
0,1,2,3,4,5,6,7 or 8	1	+1
0,1,2,5,6, or 7	2	+2
0,1,5 or 6	3	+3
0, or 3	4	+4
0, 1,2 3, or 4	5		+1
0, 1,2 or 3	6	+1	+1
0, 1 or 2	7	+2	+1
0 or 1	8	+3	+1
0	9	+4	+1

An example illustrating the use of the Addition Table, adding 2 and 7:

1. move down the Given the first number column to a row containing 2 and;
2. move down the To add column to a row containing 7: the 7th row, then;
3. move across to the Move bead columns and perform the operations specified:
   • in the lower deck, count 2 beads;
   • in the upper deck, count 1 bead.
4. The answer then presents itself on the abacus.

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.

Combined Adding-up And Taking Off

When the original number registered on a rod is smaller than 5, but will become greater than 5 after the addition, one bead from the upper-deck is moved down (added on to the beam) and one or more beads from the lower deck removed from the beam.

When a sum greater than 10 occurs on a certain rod, beads are removed from either or both the upper and lower decks and 1 bead is added to the rod directly to the left. Example: When adding 9 (10-1) to 8, one bead from the lower deck is removed (-1) and one bead from the lower deck on the row directly to the left is added (+10).

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

Combined Taking-off And Place Advancement

When a sum greater than 10 occurs on a certain rod, beads are removed from either or both the upper and lower decks and 1 bead is added to the rod directly to the left. Example: when adding 9 (10-1) to 8, one bead from the lower deck is removed (-1) and one bead from the lower deck on the row directly to the left is added (+10).

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

Combined Adding-up, Taking-off And Place Advancement

There are 4 cases when beads are added to the lower-deck, removed from the upper-deck and one bead added to the adjacent rod. Example: When adding 6 (+1-5+10) to 7, one bead is added to the lower-deck, one bead removed from the upper-deck and one bead is added to the left rod (lower-deck).

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

Subtraction