Addition on the abacus involves registering the numbers on the beads in the straight-forward left-to-right sequence they are written down in.

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## Addition

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 |

### 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 |