In this article, we deal with item 5 of the following key skills that build up to understanding the floating point system:
- Define and identify the most and least significant bits in a word. Count how many combinations are possible in a word given its bit length.
- Encode a positive integer in binary. Decode an unsigned integer to decimal.
- Encode an integer in two's complement notation. Decode a signed integer to decimal.
- Use sign extension to expand the number of bits in a word. Explain the mathematics.
- Convert a decimal fraction to binary. Explain why this does not require any particular encoding scheme, such as a floating point standard.
Expanding our idea
We use the same method as we used before to convert number representations.
Practice
QUESTION 1 Convert the decimal number 25.5 to unsigned binary.
Answer: 25.5 - 16 = 9.5
9.5 - 8 = 1.5
1.5 - 1 = 0.5
0.5 - 0.5 = 0
The powers of two used are: the 4th, 3rd, 0th, and negative 1st. Negative powers of two gets placed to the right of the binary point. Arrange: 0001 1001.1.
QUESTION 2 Convert the decimal number 16.1 to unsigned binary.
Answer: 16.1 - 16 = 0.1
0.1 - 0.0625 [ or 2^-4 ] = 0.0375
0.0375 - 0.0315 [ or 2^-5 ] = 0.00625
0.00625 - 0.00390625 [ or 2^-8 ] = 0.00234375
0.00234375 - 0.001953125 [ or 2^-9 ] = 0.000390625
0.000390625 - 0.000244140625 [ or 2^-12 ] = 0.000146484375
...
We see that the pattern .0001 1001 1001 ... is repeating. Arrange the bits: 0001 0000.0001 1001 1001 ....
QUESTION 3 Convert 1011.0101 to decimal.
The bits below the binary point (.) have the values of the negative powers of two.
The answer is 2^3 + 2^1 + 2^0 + 2^-2 + 2^-4 = 10.3125.
It's not that difficult!
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