Traditional Abacus and Bead Arithmetic/Addition and subtraction/Extending the 123456789 exercise

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Introduction

As we have seen in the previous chapter, the "123456789 exercise", consisting of adding that number nine times to a cleared abacus until reaching the number 1111111101 and then subtracting it nine times until the abacus is cleared again, has been used since ancient times to illustrate and practice addition and subtraction. It is a convenient exercise because:

• it is long enough to be a non-trivial exercise
• if we do not return to the initial value (zero) it is a sign that we have made a mistake
• we do not need a book or exercise sheet
• uses many of the elementary cases of addition or subtraction of a digit to/from another digit

but it also has a couple of drawbacks:

• it does not use all pairs of digits (ex. a 3 is never added to a 5)
• after repeating it several times, one begins to mechanically memorize the exercise, so that we are no longer practicing addition and subtraction

To avoid these two problems we can extend the exercise in several ways.

Using a background

Instead of using a cleared abacus, we can fill 9 columns with a digit (111111111, 222222222, etc.), this multiplies by 10 the number of exercises at our disposal. Now we are sure to use all possible cases of addition and subtraction digit by digit while mechanical memorization becomes harder.

The following table contains the intermediate values for reference. Such values are traversed from top to bottom in addition and from bottom to top in subtraction.

Table 1: The 123456789 exercise with background

+1..9	0	1	2	3	4	+1..9
0	000000000	111111111	222222222	333333333	444444444	0
1	123456789	234567900	345679011	456790122	567901233	1
2	246913578	358024689	469135800	580246911	691358022	2
3	370370367	481481478	592592589	703703700	814814811	3
4	493827156	604938267	716049378	827160489	938271600	4
5	617283945	728395056	839506167	950617278	1061728389	5
6	740740734	851851845	962962956	1074074067	1185185178	6
7	864197523	975308634	1086419745	1197530856	1308641967	7
8	987654312	1098765423	1209876534	1320987645	1432098756	8
9	1111111101	1222222212	1333333323	1444444434	1555555545	9

Table 1: The 123456789 exercise with background (continuation)

+1..9	5	6	7	8	9	+1..9
0	555555555	666666666	777777777	888888888	999999999	0
1	679012344	790123455	901234566	1012345677	1123456788	1
2	802469133	913580244	1024691355	1135802466	1246913577	2
3	925925922	1037037033	1148148144	1259259255	1370370366	3
4	1049382711	1160493822	1271604933	1382716044	1493827155	4
5	1172839500	1283950611	1395061722	1506172833	1617283944	5
6	1296296289	1407407400	1518518511	1629629622	1740740733	6
7	1419753078	1530864189	1641975300	1753086411	1864197522	7
8	1543209867	1654320978	1765432089	1876543200	1987654311	8
9	1666666656	1777777767	1888888878	1999999989	2111111100	9

The 987654321 exercise

Additionally, instead of using the number 123456789 we can think of using any permutations of these digits that we are able to comfortably remember; for example, 987654321, the only one we will consider here. This gives us 10 other independent exercises for addition and subtraction practice. The following table shows us the intermediate values of this new series of exercises using a background.

In total, we already have 20 different exercises.

Table 2: The 987654321 exercise with background starting with addition

+9..1	0	1	2	3	4	+9..1
0	000000000	111111111	222222222	333333333	444444444	0
1	987654321	1098765432	1209876543	1320987654	1432098765	1
2	1975308642	2086419753	2197530864	2308641975	2419753086	2
3	2962962963	3074074074	3185185185	3296296296	3407407407	3
4	3950617284	4061728395	4172839506	4283950617	4395061728	4
5	4938271605	5049382716	5160493827	5271604938	5382716049	5
6	5925925926	6037037037	6148148148	6259259259	6370370370	6
7	6913580247	7024691358	7135802469	7246913580	7358024691	7
8	7901234568	8012345679	8123456790	8234567901	8345679012	8
9	8888888889	9000000000	9111111111	9222222222	9333333333	9

Table 2: The 987654321 exercise with background starting with addition (continuation)

+9..1	5	6	7	8	9	+9..1
0	555555555	666666666	777777777	888888888	999999999	0
1	1543209876	1654320987	1765432098	1876543209	1987654320	1
2	2530864197	2641975308	2753086419	2864197530	2975308641	2
3	3518518518	3629629629	3740740740	3851851851	3962962962	3
4	4506172839	4617283950	4728395061	4839506172	4950617283	4
5	5493827160	5604938271	5716049382	5827160493	5938271604	5
6	6481481481	6592592592	6703703703	6814814814	6925925925	6
7	7469135802	7580246913	7691358024	7802469135	7913580246	7
8	8456790123	8567901234	8679012345	8790123456	8901234567	8
9	9444444444	9555555555	9666666666	9777777777	9888888888	9

Starting with subtraction

If you still do not have enough with the 20 previous exercises, you can count on another 20 independent exercises just start by subtracting 123456789 or 987654321 from the background nine times, after which we will return the abacus to its original state by adding the number nine times. In doing so, sooner or later we will find negative numbers that we can handle on "the other side" of the abacus; that is, by borrowing 1 from a real or imaginary column located further to the left. Before ending the exercise, that borrowed 1 will be returned with a carry to its real or imaginary column, and we will be able to finish the exercise with the abacus in its original state.

Table 3: The 123456789 exercise with background starting with subtraction

-1..9	0	1	2	3	4	-1..9
0	000000000	111111111	222222222	333333333	444444444	0
1	9876543211	9987654322	98765433	209876544	320987655	1
2	9753086422	9864197533	9975308644	86419755	197530866	2
3	9629629633	9740740744	9851851855	9962962966	74074077	3
4	9506172844	9617283955	9728395066	9839506177	9950617288	4
5	9382716055	9493827166	9604938277	9716049388	9827160499	5
6	9259259266	9370370377	9481481488	9592592599	9703703710	6
7	9135802477	9246913588	9358024699	9469135810	9580246921	7
8	9012345688	9123456799	9234567910	9345679021	9456790132	8
9	8888888899	9000000010	9111111121	9222222232	9333333343	9

Table 3: The 123456789 exercise with background starting with subtraction (continuation)

-1..9	5	6	7	8	9	-1..9
0	555555555	666666666	777777777	888888888	999999999	0
1	432098766	543209877	654320988	765432099	876543210	1
2	308641977	419753088	530864199	641975310	753086421	2
3	185185188	296296299	407407410	518518521	629629632	3
4	61728399	172839510	283950621	395061732	506172843	4
5	9938271610	49382721	160493832	271604943	382716054	5
6	9814814821	9925925932	37037043	148148154	259259265	6
7	9691358032	9802469143	9913580254	24691365	135802476	7
8	9567901243	9679012354	9790123465	9901234576	12345687	8
9	9444444454	9555555565	9666666676	9777777787	9888888898	9

Table 4: The 987654321 exercise with background starting with subtraction

-9..1	0	1	2	3	4	-9..1
0	000000000	111111111	222222222	333333333	444444444	0
1	9012345679	9123456790	9234567901	9345679012	9456790123	1
2	8024691358	8135802469	8246913580	8358024691	8469135802	2
3	7037037037	7148148148	7259259259	7370370370	7481481481	3
4	6049382716	6160493827	6271604938	6382716049	6493827160	4
5	5061728395	5172839506	5283950617	5395061728	5506172839	5
6	4074074074	4185185185	4296296296	4407407407	4518518518	6
7	3086419753	3197530864	3308641975	3419753086	3530864197	7
8	2098765432	2209876543	2320987654	2432098765	2543209876	8
9	1111111111	1222222222	1333333333	1444444444	1555555555	9

Table 4: The 987654321 exercise with background starting with subtraction (continuation)

-9..1	5	6	7	8	9	-9..1
0	555555555	666666666	777777777	888888888	999999999	0
1	9567901234	9679012345	9790123456	9901234567	12345678	1
2	8580246913	8691358024	8802469135	8913580246	9024691357	2
3	7592592592	7703703703	7814814814	7925925925	8037037036	3
4	6604938271	6716049382	6827160493	6938271604	7049382715	4
5	5617283950	5728395061	5839506172	5950617283	6061728394	5
6	4629629629	4740740740	4851851851	4962962962	5074074073	6
7	3641975308	3753086419	3864197530	3975308641	4086419752	7
8	2654320987	2765432098	2876543209	2987654320	3098765431	8
9	1666666666	1777777777	1888888888	1999999999	2111111110	9

Using the 5th lower bead

This is the most interesting proposal in the context of traditional methods. The forty exercises above can be performed using the lower 5th bead, as explained in detail in the previous chapter: Use of the 5th lower bead, which will allow you to master this traditional technique.

With this, we have a total of 80 exercises!

Using alternate operation

And finally, why not? Even if only for the pleasure of overcoming a different difficulty, we can combine the previous exercises with an alternating direction of operation, from left to right and from right to left, as explained in the introductory chapter to addition and subtraction.

Example

Abacus	Comment
ABCDEFGHIJ
	Cleared abacus
+1
+2
+3
+4
+5
+6
+7
+8
+9
123456789	First step completed
+9
+8
+7
+6
+5
+4
+3
+2
+1
246913578	Second step completed
etc.

With this, you could go one step further in your understanding of bead mechanics.

Conclusion

With the 160 exercises presented here, you no longer have an excuse, you can practice addition and subtraction for hours, without exercise sheets, while comfortably seated on your sofa with only your abacus resting on your knees.

This is a door to mastery!