This post covers everything you need to know about floating point numbers, their notation, and how they are represented in computers. In this article, we will teach you not only what floating point numbers are but also how they are stored and what makes a floating point number valid.
What is a Floating Point Number?
A floating point number is a numerical representation used for real numbers, especially those requiring a high degree of precision. Unlike integers, floating point numbers can represent both very large and very small values, including decimals. They are particularly useful in scientific computing, engineering, and any application requiring calculations with real numbers where precision is crucial.
A floating point number consists of two main parts: the mantissa (or significant) and the exponent. The mantissa holds the significant digits of the number, while the exponent scales the number by a power of a given base, usually 10 in decimal systems or 2 in binary systems.
What is Floating Point Notation?
Floating point notation is a way of representing real numbers that allows for a wide range of values. It is typically expressed in the form:
cssm × b^e
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Where:
- m is the mantissa (significant digits)
- b is the base (commonly 2 for binary, or 10 for decimal)
- e is the exponent, which determines the scale or position of the decimal point.
For example, in base 10, the number 5,200 can be written in floating point notation as 5.2 × 10^3. This form allows computers to efficiently handle both very large and very small numbers without loss of precision.
What is Meant by Floating Point Representation?
Floating point representation refers to the way real numbers are stored in a computer’s memory. In most computer systems, the IEEE 754 standard is used, which defines how numbers are structured into bits. The representation divides a floating point number into three parts:
- The sign bit, which determines if the number is positive or negative.
- The exponent, which scales the number.
- The mantissa, which holds the significant digits of the number.
This structure allows computers to store numbers with great precision and handle very large ranges of values, from tiny fractions to astronomical figures.
How are Floating Point Numbers Stored in the Computer?
Floating point numbers are stored in binary form, following the IEEE 754 standard. A floating point number is typically allocated 32 bits (single precision) or 64 bits (double precision) in memory. Here’s a breakdown of the storage format for a 32-bit floating point number:
- 1 bit for the sign (0 for positive, 1 for negative).
- 8 bits for the exponent, which is biased to allow for both positive and negative exponents.
- 23 bits for the mantissa, which represents the significant digits of the number.
When a number is stored, it’s normalized in binary format, and the decimal point “floats” depending on the value of the exponent, allowing for very high precision.
What is a Valid Floating Point Number?
A valid floating point number is one that conforms to the specified format and can be accurately represented in memory. It must have a mantissa and an exponent, and it should not overflow the limits set by the floating point standard (like exceeding the range allowed by single or double precision).
Special cases such as infinity, negative infinity, and “NaN” (Not a Number) are also considered valid representations within the floating point system, even though they don’t correspond to typical real numbers.
We hope this article helped you learn about floating point numbers, their notation, and how they are represented in computer systems. We believe this explanation clarified how floating point numbers are stored and what makes them valid in computation.
