Numbers and Computers

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The fractional bits continue the pattern set by the integer bits. This form of encoding cannot represent some values in binary. Even if more digits are used, an exact representation is impossible. While both unsigned and signed integers are used in digital systems, even a bit integer is not enough to handle all the range of numbers a calculator can handle, and that's not even including fractions. To approximate the greater range and precision of real numbers , we have to abandon signed integers and fixed-point numbers and go to a " floating-point " format.

In the decimal system, we are familiar with floating-point numbers of the form scientific notation :.

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We have a certain numeric value 1. If we have a negative exponent, that means the number is multiplied by a 1 that many places to the right of the decimal point. The advantage of this scheme is that by using the exponent we can get a much wider range of numbers, even if the number of digits in the significand, or the "numeric precision", is much smaller than the range.

Similar binary floating-point formats can be defined for computers. The IEEE standard specification defines a 64 bit floating-point format with:. Let's see what this format looks like by showing how such a number would be stored in 8 bytes of memory:. Once the bits here have been extracted, they are converted with the computation:. This scheme provides numbers valid out to about 15 decimal digits, with the following range of numbers:.

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The specification also defines several special values that are not defined numbers, and are known as NaNs , for "Not A Number". These are used by programs to designate invalid operations and the like. Some programs also use bit floating-point numbers. The most common scheme uses a bit significand with a sign bit, plus an 8-bit exponent in "excess" format, giving seven valid decimal digits.

Such floating-point numbers are known as "reals" or "floats" in general, but with a number of variations:. A bit float value is sometimes called a "real32" or a "single", meaning "single-precision floating-point value".

A bit float is sometimes called a "real64" or a "double", meaning "double-precision floating-point value". The relation between numbers and bit patterns is chosen for convenience in computer manipulation; eight bytes stored in computer memory may represent a bit real, two bit reals, or four signed or unsigned integers, or some other kind of data that fits into eight bytes. The only difference is how the computer interprets them.

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If the computer stored four unsigned integers and then read them back from memory as a bit real, it almost always would be a perfectly valid real number, though it would be junk data. Only a finite range of real numbers can be represented with a given number of bits. Arithmetic operations can overflow or underflow, producing a value too large or too small to be represented.

The representation has a limited precision. For example, only 15 decimal digits can be represented with a bit real. If a very small floating-point number is added to a large one, the result is just the large one. The small number was too small to even show up in 15 or 16 digits of resolution, and the computer effectively discards it. Analyzing the effect of limited precision is a well-studied problem. Estimates of the magnitude of round-off errors and methods to limit their effect on large calculations are part of any large computation project.


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The precision limit is different from the range limit, as it affects the significand, not the exponent. The significand is a binary fraction that doesn't necessarily perfectly match a decimal fraction. In many cases a sum of reciprocal powers of 2 does not matches a specific decimal fraction, and the results of computations will be slightly off. For example, the decimal fraction "0. Programming in assembly language requires the programmer to keep track of the representation of numbers.

Where the processor does not support a required mathematical operation, the programmer must work out a suitable algorithm and instruction sequence to carry out the operation; on some microprocessors, even integer multiplication must be done in software. High-level programming languages such as LISP and Python offer an abstract number that may be an expanded type such as rational , bignum , or complex.

Mathematical operations are carried out by library routines provided by the implementation of the language. A given mathematical symbol in the source code, by operator overloading , will invoke different object code appropriate to the representation of the numerical type; mathematical operations on any number—whether signed, unsigned, rational, floating-point, fixed-point, integral, or complex—are written exactly the same way.

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Some languages, such as REXX and Java , provide decimal floating points operations, which provide rounding errors of a different form. The initial version of this article was based on a public domain article from Greg Goebel's Vectorsite.

Why Binary Numbers Are Used By Computers?

From Wikipedia, the free encyclopedia. See also: Integer computer science. See also: Base Main article: Positional notation base conversion. This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources.

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Numbers and Computers Numbers and Computers
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