Tochio's Computation on the Soroban/Mono-Division

Remember the following tables:—

First Division—(Division by 1).

Note.—The above table is never used in mono-division, since the quotient of a number divided by 1 is the number itself. But its use will be found in poly-division as we shall subsequently see.

Second Division (Division by 2).

Third Division (Division by 3).

​Fourth Division (Division by 4).

Fifth Division (Division by 5).

Sixth Division (Division by 6).

Seventh Division (Division by 7).

Eighth Division (Division by 8).

​Ninth Division (Division by 9) .

Note—In operation, the words in parenthesis in the above tables should be omitted.

The author at first prepared the division tables in fractions, for instance: 1/2 makes 5, 1/3rd makes 3 & 1, 2/3rds make 6 & 2, 1/4th make 2 & 2, 2/4ths make 5, 3/4ths make 7 & 2, etc. However, it was thought convenient in practice to read the Indicator first, so the above form was taken; but, fractions are also given in parenthesis on the left, and it is left to learners to read them in either way liked.

The word "make" was selected to distinguish the operation from that of "once, twice, thrice," etc. But, if one understands the distinction, the word "be" may be substituted, if desirable.

Remember.—Whenever we say "once, twice, thrice," etc., the dividend in the original column is cleared and 1, 2, 3, etc., is respectively placed in the column ahead. Whenever we say "Make——&suffix——," the original column is transformed to the first figure and the suffix figure is inserted, or added to the following column. When there is no suffix, the original dividend column only is changed.

"Subjoin——" means that the original dividend column undergoes no change but "——" is added to the following column.

Remark.—To a learner, the foregoing tables may seem to be a puzzle, and very hard to remember. But, if you know that they are all based on 10, or if you consider the dividend figure as being one order higher, the tables are easily recollected. For instance:—

In "2nd part 1 make 5," consider 1 as 10 and divide by 2, then the quotient is 5. In "3rd part 1 make 3 and 1," divide 10 by 3 and the quotient is 3 and fractional 1. In "3rd part 2 make 6 and 2," divide 20 by 3, and you get 6 and fractional 2. In "5th part 1 double 1," in which the 1 is doubled and made 2, divide 10 by 5, and it ​is evidently 2. In "6th part 1 subjoin 4," you will see that 10 ÷ 6 is equal to 1 and fractional 4; and all the other tables may be similarly analysed. When we say "once, twice, etc.," each shows how many times the divisor is contained in the dividend.

Rule 5.—Division.—Place dividend on the right and divisor on the left, and divide from the highest column.

Let us now see how the Division tables work.

No. 1.—Indicator.—The first figure, or the highest column, of the divisor is called the Indicator, because it denotes the relative division table.

In mono-division, the divisor itself is the indicator.

​Note.—If you divide the remainder 1 saying 2nd pt. 1 make 5, it will be decimal 5.

Note.—If you continue division-saying 4th pt. 1 make 2 and 2; 4th pt. 2 make 5-decimal 25 is obtained.

Note.—If you divide 4—saying 5th pt. 4 double 4, you obtain decimal 8.

Note—If you divide the remainder 3, decimal 5 is obtained.

Where unoperated parts appear in parenthesis, the remark under G, Fig. 18, will similarly apply.

If you divide the remainder 5, decimal 625 is obtained.

Where unoperated parts appear in parenthesis, the remark under G, Fig. 18, will similarly apply.

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No. 2.—It was already explained that the basis of division tables is 10; and, in going through the foregoing mono-divisions, you would find that, after division, unit is removed always one column higher in the quotient. For example:—in 6 ÷ 2 quotient 3 appears in the next higher column, and in 24 ÷ 3 quotient 8 appears in the original tens column.