Elements of arithmetic

by Augustus De Morgan

The next fraction is ¹⁴⁷³²⁶/₁₀₀₀₀₀₀₀, which we find by the table to be

1	+	4	+	7	+	3	+	2	+	6
100		1000		10000		100000		1000000		10000000

In this, 1 is not divided by 10, but by 100; if, therefore, we put a point before the whole, the rule is not true, for the first figure on the left of the point has the denominator which, according to the rule, the second ought to have, the second that which the third ought to have, and so on. In order to keep the same rule for this case, we must contrive to make 1 the second figure on the right of the point instead of the first. This may be done by placing a cipher between it and the point, thus, ·0147326. Here the rule holds good, for by that rule this fraction is

0	+	1	+	4	+	7	+	3	+	2	+	6
10		100		1000		10000		100000		1000000		10000000

which is the same as the preceding line, since ⁰/₁₀ is 0, and need not be reckoned.

Similarly, when there are two ciphers more in the denominator than there are figures in the numerator, the rule will be true if we place two ciphers between the point and the numerator. The rule, therefore, stated fully, is this:

To reduce a decimal fraction to a whole number and more simple decimals, or to more simple decimals alone if it do not contain a whole number, mark off by a point as many figures from the numerator as there are ciphers in the denominator. If the numerator have not places enough for this, write as many ciphers before it as it wants places, and put the point before these ciphers. Then, if there be any figures before the point, they make the whole number which the fraction contains. The first figure after the point with the denominator 10, the second with the denominator 100, and so on, are the fractions of which the first fraction is composed.

135. Decimal fractions are not usually written at full length. It is more convenient to write the numerator only, and to cut off from the numerator as many figures as there are ciphers in the denominator, when that is possible, by a point. When there are more ciphers in the denominator than figures in the numerator, as many ciphers are placed before the numerator as will supply the deficiency, and the point is placed before the ciphers. Thus, ·7 will be used in future to denote ⁷/₁₀, ·07 for ⁷/₁₀₀, and so on. The following tables will give the whole of this notation at one view, and will shew its connexion with the decimal notation explained in the first section. You will observe that the numbers on the right of the units’ place stand for units divided by 10, 100, 1000, &c. while those on the left are units multiplied by 10, 100, 1000, &c.

The student is recommended always to write the decimal point in a line with the top of the figures or in the middle, as is done here, and never at the bottom. The reason is, that it is usual in the higher branches of mathematics to use a point placed between two numbers or letters which are multiplied together; thus, 15. 16, a. b, (a + b). (c + d) stand for the products of those numbers or letters.

I.	123·4 stands for	1234	or 123	4	or 123 +	4
		10		10		10
	12·34	1234	or 12	34	or 12 +	3	+	4
		100		100		10		100
	1·234	1234	or 1	234	or 1 +	2	+	3	+	4
		1000		1000		10		100		1000
	·1234	1234			or	1	+	2	+	3	+	4
		10000				10		100		1000		10000
	·01234	1234			or			1	+	2	+	3	+	4
		100000						100		1000		10000		100000
	·001234	1234			or					1	+	2	+	3	+	4
		1000000								1000		10000		100000		1000000