The IEEE 754 standard (formally ANSI/IEEE Std 754-1985, later revised as IEEE 754-2008 and IEEE 754-2019) is a cornerstone of modern computing. It is the universally accepted technical standard for floating-point computation, defining consistent representations and arithmetic operations across diverse computer systems and programming languages. Its adoption ensures that floating-point calculations produce predictable and reproducible results, which is critical for portability and reliability in numerical software.

The IEEE 754 single-precision format uses a total of 32 bits to represent a floating-point number.
- Bit Allocation:
- Sign Bit (1 bit): This is the most significant bit (bit 31).
- 0 indicates a positive number.
- 1 indicates a negative number.
- Exponent Field (8 bits): These bits (from bit 30 down to bit 23) store the biased exponent.
- The bias for single-precision is 127.
- The actual value of the true exponent is calculated as: True_Exponent = Stored_Exponent - 127.
- Mantissa (Significand) Field (23 bits): These bits (from bit 22 down to bit 0) store the fractional part of the mantissa.
- Implied Leading 1: For normalized numbers (the vast majority of representable numbers), there is an implied leading 1 before the binary point. So, the actual mantissa value is 1.f_22f_21...f_0, where f_i are the bits stored in the mantissa field. This effectively gives a 24-bit precision (1 text implied bit + 23 text stored bits).

• Special Values (defined by reserved exponent values):
• Zero (±0.0): Represented by an exponent field of all zeros (00000000) and a mantissa field of all zeros. The sign bit distinguishes between +0.0 and -0.0, though they typically compare as equal.
• Infinity (±∞): Represented by an exponent field of all ones (11111111) and a mantissa field of all zeros. The sign bit indicates positive or negative infinity. Infinity results from operations like division by zero (e.g., 1.0/0.0).
• NaN (Not a Number): Represented by an exponent field of all ones (11111111) and a non-zero mantissa field. NaNs are used to represent the results of invalid or indeterminate operations, such as 0.0/0.0, ∞−∞, or sqrt(−1). NaNs are "sticky" – once a NaN is produced, most operations involving it will also result in a NaN. There are two types: Quiet NaN (QNaN) and Signaling NaN (SNaN). QNaNs propagate without signalling, while SNaNs typically raise an exception when accessed.
• Denormalized (or Subnormal) Numbers: Represented by an exponent field of all zeros (00000000) and a non-zero mantissa field. Unlike normalized numbers, these numbers have an implied leading 0 (i.e., 0.f_22f_21...f_0 times 2**True_Exponent_Min). They are used to represent numbers very close to zero that would otherwise "underflow" directly to zero. Denormalized numbers allow for "gradual underflow," meaning the precision gracefully degrades as numbers approach zero, which helps in preventing unexpected errors in certain algorithms. The smallest denormalized number is smaller than the smallest normalized number.

The IEEE 754 double-precision format uses 64 bits, offering a significantly wider range and much higher precision compared to single-precision.
- Bit Allocation:
- Sign Bit (1 bit): Bit 63.
- Exponent Field (11 bits): Bits 62-52.
- The bias for double-precision is 1023.
- Mantissa (Significand) Field (52 bits): Bits 51-0.
- Implied Leading 1: Similar to single-precision, there is an implied leading 1 for normalized numbers, resulting in an effective 53-bit mantissa (1 text implied bit + 52 text stored bits).

Floating-point arithmetic operations are considerably more involved and computationally intensive than integer arithmetic. This is due to the separate exponent and mantissa components, the need for alignment, normalization, and precise rounding. These operations are typically handled by a dedicated hardware unit called the Floating-Point Unit (FPU), which may be integrated into the main CPU or exist as a separate co-processor.

While indispensable, floating-point arithmetic introduces inherent limitations that must be understood to avoid common pitfalls in numerical computation:
- Finite Precision: Floating-point numbers represent a continuous range of real numbers using a finite number of bits. This means that only a discrete subset of real numbers can be represented exactly. Most real numbers...cannot be stored precisely.
- Rounding Errors: ...almost every arithmetic operation on floating-point numbers involves some degree of rounding. These small rounding errors, though tiny individually, can accumulate over a long sequence of computations.
- Loss of Significance: ... when two floating-point numbers of nearly equal magnitude are subtracted. ...the remaining bits may largely consist of accumulated rounding errors from prior operations.
- Non-Associativity: Floating-point arithmetic is not always strictly associative. This means that the order of operations can influence the final accuracy...