In the decimal system, fractions are represented as 0. abc. . . , where a, b, c, etc. denote any digit in between 0 and 9. The decimal point separates the integral part of the number from its fractional part. Consider the number (16.346)10. The expansion of this number is as follows:
Integral part: (16)10 = 1 ´ 101 + 6 ´ 100
Fraction: (0.346)10 = 3 ´ 10–1 + 4 ´ 10–2 + 6 ´ 10–3
= 3/10 + 4/100 + 6/1000
As shown above, in the case of fractions also, we are using the coefficients of the negative power of 10 to express fractional numbers. This means that the numbers after the point symbol are expanded by using negative powers of 10. The same concept can be used in all the number systems for representing fractions. For example, consider the binary number (11.101)2. This may be expanded as:
Integral part: (11)2 = 1 ´ 21 + 1 ´ 20 = (3)10
Fraction: (0.101)2 = 1 ´ 2–1 + 0 ´ 2–2 + 1 ´ 2–3 = 0.5 + 0.125 = (0.625)10
The decimal equivalent of the binary number (11.101)2 is thus obtained as:
(11.101)2 = 3 + 0.625 = (3.625)10
It may be noted that the point symbol in the binary system may be called as the binary point. In general, this may be designated as radix point.
Conversion from Decimal Fraction to Binary Fraction
Just as we divided a decimal whole number by the factor 2 to get its binary-equivalent whole number, to get the corresponding binary fraction we multiply decimal fraction also by 2. If the product of each number is greater than 1, we put a 1 below that number; otherwise we put a 0 as shown in Table 1.2.
Example 4: Convert decimal fraction 0.875 into its equivalent binary fraction.
Solution: Following the directions given above, we first draw the decimal fraction-to-binary fraction conversion table as illustrated in Table 1.2. One point to be remembered here is that multiplication of fractional part by 2 must be conducted from left-to-right (for the integral part, division by 2 has to be done from right-to-left as described in Example 3).
It can be seen that the multiplication operation stops at the third position on the right because the remainder at this point is 0. If the remainder is not 0, then multiplication by 2 must continue up to that point where the remainder becomes 0. However, if the remainder does not become 0, then we get a continued fraction. This is illustrated in the next example.
(0.875)10 ≡ (0.111)2
Example 5: Convert the decimal fraction 0.862 into its equivalent binary fraction.
Solution: Proceeding as explained in Example 4, we prepare Table 1.3, the decimal fraction-to-binary fraction table as shown. We find from Table 1.3 that the fraction does not converge; it continues as (110110…)2. We conclude that a binary fraction will converge only if and only if the last number in the expanded decimal fraction is 0.5, so that 0.5 ´ 2 = 1; otherwise the fraction will continue From Table 1.3, we find that
From Table 1.2, we find that
(0.862)10 ≡ (0.110110…)2
Conversion of Mixed Decimal Number-to-Mixed Binary Number
Example 6: Convert the decimal number (26.012) into equivalent binary number.
Solution: We first convert 26 into binary whole number, and then (0.012)10 into binary fraction. This is done as shown in Table 1.4.
From table 1.4, we obtain the binary equivalent of decimal number 26 as 11010. Next, we consider the fractional part. This is done as shown in Table 1.5. From Table 1.5, we find that the binary equivalent of decimal fraction 0.6875 is 0.1011.
Combining the integer and fractional parts, we get
(26.6875)10 = (11010.1011)2