The fifth lower bead

A more efficient use of the abacus

Ancient abacuses, both Japanese style (soroban) and Chinese style (suanpan), used to have five beads on the lower deck even though this is unnecessary from the point of view of representing decimal numbers. 

What was the purpose of this additional fifth bead? 

We have to look for the answer in ancient books.

Pánzhū Suànfǎ illustration
showing the use of the 5th lower bead

The first book entirely dedicated to the abacus was the Computational Methods with the Beads in a Tray (Pánzhū Suànfǎ 盤珠算法) by Xú Xīnlǔ (徐心魯) published in 1573 (late Ming dynasty) and this is the book that contains a more exhaustive use of the fifth lower bead for the operations of addition and subtraction, having been repeatedly studied by scholars. I have summarized Xú Xīnlǔ's use of the 5th bead in a set of easy rules for addition and subtraction (which of course apply to everything that can be done with an abacus) and I have tried to analyze the advantage of doing things like that. If you are interested, please read the full article on jccAbacus Web, where you can also find other related resources.

Please be aware that using the lower 5th bead is easy and can be extremely addictive. You have been warned❗