1.
Explanation
A normalized floating-point number is a representation of a real number in scientific notation within a computer's memory using a fixed number of bits. It consists of three main components: the sign bit, the exponent, and the fraction (or mantissa). Normalization ensures that the most significant bit of the mantissa is always 1, which allows for a compact and efficient representation of a wide range of real numbers.
The general format of a normalized floating-point number is:
\[(-1)^S \times M \times 2^E\]
Where:
- \(S\) is the sign bit, representing the sign of the number (0 for positive, 1 for negative).
- \(M\) is the mantissa (or fraction), which is a binary representation of the significant digits.
- \(E\) is the exponent, an integer that scales the value using powers of 2.
Now, let's explain various arithmetic operations with normalized floating-point numbers using examples:
Addition:
To add two normalized floating-point numbers, you need to align the exponents and then perform the addition on the mantissas. If the exponents are different, you must shift one of the mantissas to make them the same. Here's an example:
Consider two normalized floating-point numbers:
- Number 1: \(0 \ 10000101 \ 11000000000000000000000\) (Sign: 0, Exponent: 133, Mantissa: 1.75)
- Number 2: \(0 \ 10000011 \ 10100000000000000000000\) (Sign: 0, Exponent: 131, Mantissa: 1.625)
1. Align the exponents:
Both numbers have the same sign, so you just need to align the exponents by shifting the second number's mantissa to the right by 2 positions (131 to 133).
2. Add the mantissas:
\(1.10000000000000000000000\) + \(0.00101000000000000000000\) = \(1.10101000000000000000000\)
3. Normalize the result:
\(0 \ 10000101 \ 10101000000000000000000\) (Sign: 0, Exponent: 133, Mantissa: 1.8125)
Subtraction:
Subtraction is similar to addition. You align the exponents and then perform the subtraction on the mantissas. Here's an example:
Let's subtract the second number from the first number in the previous example:
1. Align the exponents:
Both numbers have the same sign, so you align the exponents by shifting the second number's mantissa to the right by 2 positions (131 to 133).
2. Subtract the mantissas:
\(1.10000000000000000000000\) - \(0.00101000000000000000000\) = \(1.01011000000000000000000\)
3. Normalize the result:
\(0 \ 10000101 \ 10101000000000000000000\) (Sign: 0, Exponent: 133, Mantissa: 1.6875)
Multiplication:
To multiply two normalized floating-point numbers, you multiply their mantissas and add their exponents. Here's an example:
Consider two normalized floating-point numbers:
- Number 1: \(0 \ 10000101 \ 11000000000000000000000\) (Sign: 0, Exponent: 133, Mantissa: 1.75)
- Number 2: \(0 \ 10000011 \ 10100000000000000000000\) (Sign: 0, Exponent: 131, Mantissa: 1.625)
1. Multiply the mantissas:
\(1.75 \times 1.625 = 2.84375\) (in decimal)
2. Add the exponents:
\(133 + 131 = 264\)
3. Normalize the result:
\(0 \ 10000111 \ 00111000000000000000000\) (Sign: 0, Exponent: 135, Mantissa: 1.4375)
Division:
To divide two normalized floating-point numbers, you divide their mantissas and subtract the divisor's exponent from the dividend's exponent. Here's an example:
Consider two normalized floating-point numbers:
- Number 1: \(0 \ 10000101 \ 11000000000000000000000\) (Sign: 0, Exponent: 133, Mantissa: 1.75)
- Number 2: \(0 \ 10000011 \ 10100000000000000000000\) (Sign: 0, Exponent: 131, Mantissa: 1.625)
1. Divide the mantissas:
\(1.75 / 1.625 = 1.07692307692\) (in decimal)
2. Subtract the exponents:
\(133 - 131 = 2\)
3. Normalize the result:
\(0 \ 10000101 \ 00110011001100110011001\) (Sign: 0, Exponent: 135, Mantissa: 1.07692307692)
These examples illustrate the arithmetic operations with normalized floating-point numbers. Note that rounding and normalization are essential steps to ensure that the result adheres to the precision and format of the floating-point representation used by the computer. Additionally, errors may accumulate in these operations, affecting the accuracy of the results.