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A variable-length quantity (VLQ) is a universal code that uses an arbitrary number of binary octets (eight-bit bytes) to represent an arbitrarily large integer. A VLQ is essentially a base-128 representation of an unsigned integer with the addition of the eighth bit to mark continuation of bytes. VLQ is identical to LEB128 except in endianness ...
Conversely, precision can be lost when converting representations from integer to floating-point, since a floating-point type may be unable to exactly represent all possible values of some integer type. For example, float might be an IEEE 754 single precision type, which cannot represent the integer 16777217 exactly, while a 32-bit integer type ...
The last two examples illustrate what happens if x is a rather small number. In the second from last example, x = 1.110111⋯111 × 2 −50 ; 15 bits altogether. The binary is replaced very crudely by a single power of 2 (in this example, 2 −49) and its decimal equivalent is used.
Here we can show how to convert a base-10 real number into an IEEE 754 binary32 format using the following outline: Consider a real number with an integer and a fraction part such as 12.375; Convert and normalize the integer part into binary; Convert the fraction part using the following technique as shown here
With the example in view, a number of details can be discussed. The most important is the choice of the representation of the big number. In this case, only integer values are required for digits, so an array of fixed-width integers is adequate. It is convenient to have successive elements of the array represent higher powers of the base.
The register width of a processor determines the range of values that can be represented in its registers. Though the vast majority of computers can perform multiple-precision arithmetic on operands in memory, allowing numbers to be arbitrarily long and overflow to be avoided, the register width limits the sizes of numbers that can be operated on (e.g., added or subtracted) using a single ...
Any real number can be written in the form m × 10 ^ n in many ways: for example, 350 can be written as 3.5 × 10 2 or 35 × 10 1 or 350 × 10 0. In normalized scientific notation (called "standard form" in the United Kingdom), the exponent n is chosen so that the absolute value of m remains at least one but less than ten (1 ≤ | m | < 10).
These schemes support very large numbers; for example one kilobyte of memory could be used to store numbers up to 2466 decimal digits long. A Boolean or Flag type is a type that can represent only two values: 0 and 1, usually identified with false and true respectively. This type can be stored in memory using a single bit, but is often given a ...