Convert the following binary numbers to hexadecimal and decimal: (a) 1.10010, (b) 110.010. Explain why the decimal answer in (b) is 4 times that in (a).
The Correct Answer and Explanation is:
Let’s first convert the binary numbers to decimal and hexadecimal.
(a) Binary: 1.10010
Decimal Conversion:
We split the binary number into its integer and fractional parts:
- Integer part:
1= 1×20=11 \times 2^0 = 1 - Fractional part:
.10010
=1×2−1+0×2−2+0×2−3+1×2−4+0×2−5= 1 \times 2^{-1} + 0 \times 2^{-2} + 0 \times 2^{-3} + 1 \times 2^{-4} + 0 \times 2^{-5}
=0.5+0+0+0.0625+0= 0.5 + 0 + 0 + 0.0625 + 0
=0.5625= 0.5625
Decimal Result: 1+0.5625=1.56251 + 0.5625 = 1.5625
Hexadecimal Conversion:
Group the binary digits from the point outward:
- Binary:
1.10010→ Pad to 4 bits after and before the point:0001.1001 0000
→ Integer part:0001= 1
→ Fractional part:1001= 9 (Hex)
→ (Next digit0000= 0, but not significant beyond 4 bits)
Hexadecimal Result: 1.9161.9_{16}
(b) Binary: 110.010
Decimal Conversion:
- Integer part:
110= 1×22+1×21+0×20=4+2+0=61 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 4 + 2 + 0 = 6 - Fractional part:
.010
=0×2−1+1×2−2+0×2−3=0+0.25+0=0.25= 0 \times 2^{-1} + 1 \times 2^{-2} + 0 \times 2^{-3} = 0 + 0.25 + 0 = 0.25
Decimal Result: 6+0.25=6.256 + 0.25 = 6.25
Hexadecimal Conversion:
- Binary:
110.010→ Pad to:0110.0100
→ Integer part:0110= 6
→ Fractional part:0100= 4
Hexadecimal Result: 6.4166.4_{16}
Why is (b) 4 times (a)?
From the decimal results:
- (a) = 1.5625
- (b) = 6.25
- 1.5625×4=6.251.5625 \times 4 = 6.25
This happens because of the structure of binary numbers.
Let’s analyze:
- (a):
1.10010 - (b):
110.010
If we shift all bits of (a) left by two places (i.e., multiply by 22=42^2 = 4), we get:
1.10010 × 2^2 = 110.010
This is the exact value of (b). So, shifting left by 2 bits (which multiplies by 4 in binary) shows that (b) is 4 times (a).
Conclusion: In binary, each left shift by one bit multiplies the number by 2. Shifting 1.10010 left two places gives 110.010, thus explaining why (b) = 4 × (a).
