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 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: SCALES OF NOTATION 1013 - where Zl, Z2, . . ., a,-; are the integral parts and xl, 2 2 , . . ., z-, the fractional parts (in the b-scale) of the products x5, 215, . . ., ~-15, respectively. Then convert the integral parts to the b-scale, - @I) ci;) = i i - 1 , 6'2) 4) =ii-Z, . . ., @in) ci;) =a-n, and obtain - - - r=(0.a-la-2 . . . a_,)(;). (IV) 5-scale arithmetic. Convert b and a-l, a-2, . . ., a_, to the 5-scale and define, using arithmetic operations in the 5-scale, =a-,/b + a-n+i, ~-,+a=~-n+i/b+a-n+2, x - ~ =x2/b + a-l ; then x=x-l/b. Numerical Methods The examples are restricted to the scales of 2, 8, 10 because of their importance to electronic computers. Note that the octal scale is a power of the binary scale. In fact, an octal digit corresponds to a triplet of binary digits. Then, binary arithmetic may be used whenever a number either is to be converted to the octal scale or is given in the octal scale and is to be converted to some other scale. Decimal 1 2 3 4 5 6 7 8 9 10 Octal 1 2 3 4 5 6 7 10 11 12 Binary 1 10 11 100 101 110 111 1000 1001 1010 Example 1. Convert X= (1369)~~~) to the octal scale. By (I) we have b=10, z=8(lo) and so, using decimal arithmetic, 1369/8=171+1/8, 17 118 = 2 1 + 318, 2 118 =2 + 518, 218 = 0 + 2/8 ; then X= (2531)(8). By (11) we have b=(12)(8) and A3=1(8), A2 Hence, using octal =3(8), A1=6(8), Ao= (ll)(8). arithmetic, X2= 1 - 12 +3= (15) (a), XI= 15 12 4-6 = (210) (s), X=210.12+11= (2531)(8). Using binary arithmetic we have, by (11), b=(1010)(2) and A3=1(2), A2=(11)(2,, A1=(110)(2), A,( 1001) (2). Thus x2=1.1010+11=(1101)(2), X,=llOl. 1010+110=(10 001 000)(2), x=10 001 000.1010+1001=(10 101 011 001)(2), x= (2531)(8). whence, on converting to the octal scale, Example 2. Convert X=(2531)(8) to the decimal scale. By (I) we have 5= lo= (12)(s) and hence, using octal arithmetic, 2531/12=210+ 11/12 210/12=15+6/12 15/12=1+3/12 1/12=0+ 1/12 Thus, converting to the decimal scale, - - - - A,= (1 1) (8)=9, Al= 6(8) =6, A2= 3 (8) =3, A3= 1, and so X= (1369) (10). By (11) we have ?;=lo, and the octal digits of X are unchanged in the decimal scale. Hence, using decimal arithmetic, X2=2 * 8+5= (21) (lo), X1=21*8+3=(171)(lo), x= 17 1 * 8 + 1 = ( 1369) (10). Using binary arithmetic we have, by (11), b=8=(1000)(2) and Ao=1,Al=(11)(2~,A2=(101)(~~, A3=(10)(2). Then, x2= l O ~ l O O O + 101 = (10 101) (a), X,=lO 101.1000+11=(10 101 011)(2), X=lO 101 011.1000+1=(10 101 011 001)(2), whence, on converting to the decimal scale, X= (1369)(10). Observe that in both examples above, octal arithmetic is used as an intermediate step to convert, according to (11), the given number to the binary scale. If, instead, the given number is fist converted to the binary scale, then binary arithmetic may be applied directly to convert, according to (I), the given number from the binary scale to the scale desired. The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.