Floating point numbers/single and double precision /class 9th new course computer sci

*Floating Point Representation (Example: 5.75 in Single vs Double Precision)* In computer systems, floating point numbers are stored using a format defined by the IEEE 754 standard. Let's explore how the number *5.75* is represented in both *single precision (32-bit)* and *double precision (64-bit)* formats. --- *1. Decimal to Binary Conversion of 5.75* 5 = `101` (binary) 0.75 = `0.11` (since 0.5 + 0.25 = 0.75) So, *5.75 = 101.11 (binary)* We convert this to **normalized binary form**: *1.4375 × 2²* --- *2. Single Precision (32-bit) Representation* *Sign bit:* `0` (since 5.75 is positive) *Exponent:* 2 + *bias (127)* = *129* → `10000001` (8 bits) *Mantissa:* Binary of *.4375* is `01110000000000000000000` (take 23 bits after decimal) *Final Representation:* ``` Sign | Exponent | Mantissa 0 | 10000001 | 01110000000000000000000 ``` --- *3. Double Precision (64-bit) Representation* *Sign bit:* `0` *Exponent:* 2 + *bias (1023)* = *1025* → `10000000001` (11 bits) *Mantissa:* Same fraction part (`.4375`) in binary: `011100000000...` (52 bits used) *Final Representation:* ``` Sign | Exponent | Mantissa 0 | 10000000001 | 0111000000000000000000000000000000000000000000000000 ``` --- Summary: The key difference is the **number of bits for exponent and mantissa**. Single precision uses 8 exponent bits and 23 mantissa bits, while double precision uses 11 exponent bits and 52 mantissa bits. This allows **double precision to store numbers with greater accuracy and range**.