Elements of arithmetic

by Augustus De Morgan

Here the dividend 734354 does not contain the divisor 780036, and we, therefore, write 0 as a root figure and make another contraction, or begin with

	54759	78003	648	734354(9
		78008	5	32277
		78013	4

At the next contraction the first column becomes |0054759, and is quite useless, so that the remainder of the process is the contracted division.

7801	34)	32277	(4137
		1072
		292
		58
		3

and the root required is 21·36094137.

I now write down the complete process for another equation, one root of which lies between 3 and 4: it is

x³ - 10x + 1 = 0

1	0				-10					-1(3·111039052073099	0796
	3				-1					2000
	6				1700					209000
	9	0			1791					19769000
	9	1			188300					743369000000
	9	2			189231					172311710273000
	9	30			19016300					991247447681
	9	31			19025631					39462875420
	9	32			1903496300	0	0			1391491559
	9	33	0		1903524299	0	9			58993123
	9	33	1		1903552298	2	7	0	0	1886047
	9	33	2		1903560698	0	5	9	1	172835
	9	33	30	0	1903569097	8	5	6	3	1515
	9	33	30	3	1903569144	5	2	2		183
	9	33	30	6	1903569191	1	8	8		12
	9	33	30	90	1903569193	0	6			1
	9	33	30	99	1903569194	9	3
	9	33	31	08
	09	33	31	17

The student need not repeat the rows of figures so far as they come under one another: thus, it is not necessary to repeat 190356. But he must use his own discretion as to how much it would be safe for him to omit. I have set down the whole process here as a guide.

The following examples will serve for exercise: