Floating point representaction.
Floating point representation is a method used to represent real numbers in a computer's memory. It is a way to express numbers with a fractional part or a large range of magnitudes. It is widely used in scientific and engineering applications that require a high level of precision.
In computers, floating point numbers are stored using a standardized format called the IEEE 754 floating point standard. This standard defines two types of floating point numbers: single precision and double precision. Single precision uses 32 bits to store a floating point number, while double precision uses 64 bits.
To understand how floating point representation works, let's take an example using single precision format. A single precision floating point number is divided into three parts: the sign, exponent, and significand (also known as the mantissa).
1. Sign bit: The sign bit determines whether the number is positive or negative. It takes up 1 bit in the memory to represent the sign.
2. Exponent: The exponent represents the power of 2 to scale the significand. It determines the range of the number that can be represented. In single precision, the exponent is 8 bits long.
3. Significand: The significand is the fractional part of the number. It contains the actual digits of the number. In single precision, the significand is 23 bits long.
To compute the actual value of a floating point number, you need to use the formula:
value = (-1)^sign * (1 + significand) * 2^(exponent - bias)
The bias is a predefined value that adjusts the exponent range. For single precision, the bias is set to 127.
To represent a number in floating point format, you need to follow these steps:
1. Determine the sign of the number: If the number is positive, set the sign bit to 0. If the number is negative, set the sign bit to 1.
2. Convert the number to binary: Convert the absolute value of the number to binary. Separate the integer and fractional parts.
3. Normalize the binary representation: Shift the binary representation to have a single non-zero digit to the left of the decimal point. Adjust the exponent accordingly.
4. Determine the exponent: Calculate the exponent value by adding the bias to the number of right shifts required for normalization.
5. Determine the significand: Take the binary digits to the right of the decimal point and pad zeros to the right until you have 23 bits.
6. Combine the sign, exponent, and significand: Arrange the bits in the correct order defined by the IEEE 754 standard.
By following these steps, you can convert a real number into its floating point representation. It is important to note that floating point representation has limitations in terms of precision and accuracy due to the finite size of the significand and exponent.