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Home » + Sales & Service » HP Heritage » HPCalc Timeline » Abacus »
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How to use an AbacusDiagram 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 AdditionWhen 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: - move down the Given the first number column to a row containing 2 and;
- move down the To add column to a row containing 7: the 7th row, then;
- 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.
- 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 OffWhen 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 AdvancementWhen 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 AdvancementThere 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 |
SubtractionSubtraction is performed by first registering the minuend and then subtracting, starting from the left, by removing beads form either or both the lower or upper decks. The final bead-positions represent the answer. Simple Taking-offThis 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 OffWhen 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-upWhen 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 |
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Combined Taking-off From A Rod Of Higher Order, Adding-up in the Upper-deck and Taking-off in the Lower-DeckThis 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 |
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In reply to Joe:
@Bill, @Einstein;
The first paragraph DOES NOT discuss the order one performs calculations in. It states, "Working from right to left, the first vertical line represents units, the next tens, the next hundreds and so on." this sentence is merely describing the value of each column. In other words, look at the right column, those are units, look to that column´s left, those are the 10s, etc... Yes Einstein... you need to read closer
Addition and subtraction is performed from LEFT to RIGHT! Look on YouTube if you don´t believe it. This article does state this in the 3rd paragraph, but judging by the comments it does not do so clearly enough :
@
nnnnnoooo
2/4/2010 11:30:32 PM - PK
| reply
u have to make it first got o youtube.com and type make abacus with that tacnic make it :_D loooser
In reply to Abacus Help Person:
I need to know how to use an abacus because we are doing a project and i don´t understand! Please help
fanie
2/4/2010 11:28:46 PM - PK
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actually, our numbering system goes right to left:
01
05
10
15
20
25
30
35
(incrementing starts at the right and ends at the left with each place having ten possible values (0 - 9))
In reply to -:
Dude the fifth sentence says "Working from right to left"...anyway...this is interesting.
connor
1/24/2010 5:26:21 PM - US
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@Bill, @Einstein;
The first paragraph DOES NOT discuss the order one performs calculations in. It states, "Working from right to left, the first vertical line represents units, the next tens, the next hundreds and so on." this sentence is merely describing the value of each column. In other words, look at the right column, those are units, look to that column´s left, those are the 10s, etc... Yes Einstein... you need to read closer
Addition and subtraction is performed from LEFT to RIGHT! Look on YouTube if you don´t believe it. This article does state this in the 3rd paragraph, but judging by the comments it does not do so clearly enough :
@
Joe
1/19/2010 8:30:35 AM - US
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The first paragraph says "working right to left". Read closer.
In reply to Allan:
This article leaves out one important thing; No make that two things that can make this article great. The first is that it does not tell you that you start calculating numbers on that abacus from right to left. This is obvious to someone that has used that abacus at least once but I came to this site not knowing anything at all about abaci. I had to go to three websites to figure out you go from right to left. In america everything goes from left to right. The second thing: visuals.
Einstein
1/12/2010 12:56:32 PM - US
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Correction Sophiya,
It does state in the directions, the first paragraph, that you work from right to left.
My wife gave me an Abaci for Christmas, and I had no idea how to work it, looking up this article was a great find.
Thanks for a great lesson.
In reply to Sophiya:
Good this is a best way to teach calculation to the younger ones without calculator. best idea.
Bill
12/25/2009 5:44:29 PM - US
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Good this is a best way to teach calculation to the younger ones without calculator. best idea.
Sophiya
12/16/2009 9:32:28 AM - UK
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Dude the fifth sentence says "Working from right to left"...anyway...this is interesting.
In reply to Allan:
This article leaves out one important thing; No make that two things that can make this article great. The first is that it does not tell you that you start calculating numbers on that abacus from right to left. This is obvious to someone that has used that abacus at least once but I came to this site not knowing anything at all about abaci. I had to go to three websites to figure out you go from right to left. In america everything goes from left to right. The second thing: visuals.
Anonymous
12/6/2009 12:57:44 PM - US
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Good Day,
I purchased an abicus in Russian in early 90´2.....all the shops were using it. I am attempting now to learn how to use it but unable to get the concept. Wonder if anyone can assist. I would like to know only the adding and substacting.
joan
5/26/2009 6:59:29 PM - US
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i have entered a workshop about da abacus and its really interesting...but is it possible to do multiplication and division?if yess in which level?i have just finished the first level
Fafi
3/18/2009 11:56:05 AM - LB
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nice quick way to learn the basics. thanks
bb
3/2/2009 5:19:04 PM - US
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Basics were very informative, but is it possible to do multiplication and or division? Some of the higher addition techniques were a little bit confusing at times.
cnd
6/11/2008 4:05:45 PM - US
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Sir,
Where Do I get 13 rods Abacus
P.K.Gupta
P.K.Gupta
8/21/2007 2:27:29 AM - IN
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i need to know about multiplacation
Leigh
5/24/2007 4:40:59 PM - US
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This article leaves out one important thing; No make that two things that can make this article great. The first is that it does not tell you that you start calculating numbers on that abacus from right to left. This is obvious to someone that has used that abacus at least once but I came to this site not knowing anything at all about abaci. I had to go to three websites to figure out you go from right to left. In america everything goes from left to right. The second thing: visuals.
Allan
10/22/2006 2:00:07 PM - US
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This is dumb
weirdo
9/8/2006 10:37:21 PM - US
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What about multiplication and division, or adding and subtracting really big numbers? Is it possible to do things like this on an abacus?
chisai_kame
8/13/2006 12:11:58 PM - US
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Try
http: webhome.idirect.com ~totton abacus
In reply to help in abacus person:
Can you show me how to use an abacus please in easy terms? please im doind an ancient inventions of China in our class and i picked the abacus. please help me!!!!!!!
DataRyder
3/31/2006 10:24:55 AM - SG
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As a Librarian Assistant needing this information I thank you for sharing it!
Marilyn
2/1/2006 5:19:33 AM - US
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Can you show me how to use an abacus please in easy terms? please im doind an ancient inventions of China in our class and i picked the abacus. please help me!!!!!!!
help in abacus person
3/24/2005 12:40:11 PM - US
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I like your web page very much. I hope you wouldn´t mind my pointing out a fly in the ointment. The abacus shown in the upper left corner of the page displays an incorrect representation of a number. Mathematically there´s nothing wrong with it as far as the abacus is concerned, but it just violates the decimal system of number representation. Furthermore, it can confuses and mislead the novice Chinese abacus user. Thank you.
Julian Loui
Virginia Beach, VA, U.S.A.
Julian Loui
2/23/2005 12:13:45 PM - US
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I know of a very recent book on the basic fundamentals of abacus. I find the book has no competition :)
Do send me a message if you need more info about this book.
In reply to Abacus Help Person:
I need to know how to use an abacus because we are doing a project and i don´t understand! Please help
shali
1/26/2005 2:03:53 PM - MY
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I need to know how to use an abacus because we are doing a project and i don´t understand! Please help
Abacus Help Person
1/26/2005 4:29:46 AM - US
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Hi... This is a very good page.
Bob
11/22/2004 2:19:16 AM - US
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