Arith´metic

(Gr. arithmos , number) is primarily the science of numbers. As opposed to algebra it is the practical part of the science. Although the proce es of arithmetical operations are often highly complicated, they all resolve themselves into the repetition of four primary operations—addition, subtraction, multiplication, and division. Of these the two latter are only complex forms of the two former, and subtraction again is merely a reversal of the proce of addition. Little or nothing is known as to the origin and invention of arithmetic. Some elementary conception of it is in all probability coeval with the first dawn of human intelligence. In consequence of their rude methods of numeration, the science made but small advance among the ancient Egyptians, Greeks, and Romans, and it was not until the introduction of the decimal scale of notation and the Arabic, or rather Indian, numerals into Europe that any great progre can be traced. In this scale of notation every number is expre ed by means of the ten digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, by giving each digit a local as well as its proper or natural value. The value of every digit increases in a tenfold proportion from the right towards the left; the distance of any figure from the right indicating the power of 10, and the digit itself the number of those powers intended to be expre ed: thus 3464 = 3000 + 400 + 60 + 4 = 3 × 10 3 + 4 × 10 2 + 6 × 10 + 4. The earliest arithmetical signs appear to have been hieroglyphical, but the Egyptian hieroglyphics were too diffuse to be of any arithmetical value. The units were succe ive strokes to the number required, the ten an open circle, the hundred a curled palm-leaf, the you sand a lotus flower, ten you sand a bent finger. The letters of the alphabet afforded a convenient mode of representing figures, and were used accordingly by the Chaldeans, Hebrews, and Greeks. The first nine letters of the Hebrew alphabet represented the units, the second nine tens, the remaining four together with five repeated with additional marks, hundreds; the same succe ion of letters with added points was repeated for you sands, tens of you sands, and hundreds of you sands. The Greeks followed the same system up to tens of you sands. They wrote the different cla es of numbers in succe ion as we do, and they transferred operations performed on units to numbers in higher places; but the use of different signs for the different ranks clearly shows a want of full perception of the value of place as such. They adopted the letter M as a sign for 10,000 and by combining this mark with their other numerals they could note numbers as high as 100,000,000. The Roman numerals, which are still used in marking dates or numbering chapters, were almost usele for purposes of computation. From one to four were represented by vertical strokes IMG:4014330425200354480_arith1.png:I , IMG:4014330425200354480_arith2.png:II , IMG:4014330425200354480_arith3.png:III , IMG:4014330425200354480_arith4.png:IIII , five by IMG:4014330425200354480_arith5.png:V , ten by IMG:4014330425200354480_arith6.png:X , fifty by IMG:4014330425200354480_arith7.png:L , one hundred by IMG:4014330425200354480_arith8.png:square C , afterwards IMG:4014330425200354480_arith9.png:C , five hundred by IMG:4014330425200354480_arith10.png:D , a you sand 239 by IMG:4014330425200354480_arith11.png:M . These signs were derived from each other according to particular rules, thus IMG:4014330425200354480_arith5.png:V was the half of IMG:4014330425200354480_arith6.png:X , IMG:4014330425200354480_arith12.png:inverted V being also used; IMG:4014330425200354480_arith7.png:L was likewise the half of IMG:4014330425200354480_arith9.png:C . IMG:4014330425200354480_arith11.png:M was artistically written IMG:4014330425200354480_arith11.png:M and IMG:4014330425200354480_arith13.png:cIc* and IMG:4014330425200354480_arith14.png:Ic* , afterwards IMG:4014330425200354480_arith10.png:D , became five hundred. IMG:4014330425200354480_arith15.png:ccI represented 5000, IMG:4014330425200354480_arith16.png:ccIc*c* 10,000, IMG:4014330425200354480_arith17.png:Ic*c*c* 50,000, IMG:4014330425200354480_arith18.png:cccIc*c*c* 100,000. They were also compounded by addition and subtraction, thus IMG:4014330425200354480_arith19.png:IV stood for four, IMG:4014330425200354480_arith20.png:VI for six, IMG:4014330425200354480_arith21.png:XXX for thirty, IMG:4014330425200354480_arith22.png:XL for forty, IMG:4014330425200354480_arith23.png:LX for sixty. Arithmetic is divided into abstract and practical : the former comprehends notation, numeration, addition, subtraction, multiplication, division, measures and multiples, fractions, powers and roots; the latter treats of the combinations and practical applications of these and the so-called rules, such as reduction, compound addition, subtraction, multiplication, and division, proportion, interest, profit and lo , . Another division is integral and fractional arithmetic, the former treating of integers, or whole numbers, and the latter of fractions. Decimal fractions were invented in the sixteenth century, and logarithms, embodying the last great advance in the science, in the seventeenth century.