Page 1036 - The Mechatronics Handbook

TABLE 36.8 Radix Multiply Number Convertion Method (Terminating Case)
Sample Conversion Between Bases
Formalism
Higher to
Algebraic Arithmetic Lower Lower to Higher
F = F − 1 F − 2 F − 3 ··· F − m F −1 F −2 F −3 ··· F −m 0.100101 2
×1010
× R
0.125 10 101 .11001
∗ ×2
R F = F −1 · F −2 F −3 ··· F −m F −1 · F −2 F −3 ··· F −m 0 .25 ×1010
= F − 1 · F (1) × R
×2 111 .1101
∗
R F (1) = F −2 · F − 3 F − 4 F −2 · F −3 ··· F −m 0 .5 ×1010
··· F − m = F − 2 ·F (2) ×R ×2 1000 .001
1.0 ×1010
·
1 .10
∗
·
R F (2) = F −3 F − 4 F −5 · · ×1010
··· F −m = F −2 ·F (3)
· 10 .1
·
· ×1010
∗
R F (m−2) = F − m+ 1 F −m F −m+ 1 · F −m 101 .
·
= F −m +1 ·F (m−1) × R
∗
R F (m−1) = F −m F −m · .125 10 =.001 2 .100101 2 =.578125 10
TABLE 36.9 Nonterminating
Fraction Conversion Example
0.1 10 = 0.000110011 . . . 2
× 2
0 .2 or more compactly
× 2
0 . 4 0.1 10 = 0.00011 2
× 2
0 . 8
× 2
1 . 6
× 2
1 . 2
× 2
0 . 4
× 2
0 8 .
× 2
1 . 6

Conversion of a fraction from one base to another can be done by successive multiplications of the
fraction by the radix of the number system to which the fraction is to be converted. Each multiplication
by the radix gives a product that has the digits shifted to the left by one position. This moves the most
significant digit of the fraction to the left of the radix point, placing that digit in the integer portion of
the product, thereby isolating it from the fraction. This process is illustrated in algebraic form in the left
column of Table 36.8 and in arithmetic form in the next column. Two sample numeric conversions are
shown in the next two columns of Table 36.8.
Table 36.8 deals only with terminating fractions, that is, the remaining fractional part vanishes
after a finite number of steps. For a nonterminating case the procedure is continued until a sufficient
number of digits have been obtained to give the desired accuracy. A nonterminating case is illustrated
in Table 36.9. A set of digits, which repeat ad infinitum, are designated by an underscore, as shown
in Table 36.9.

©2002 CRC Press LLC