Floating-point arithmetic

In <a href="/facts/Computing/Kghtm9fX">computing</a>, floating-point arithmetic (FP) is <a href="/facts/Arithmetic/hvHwkbMV">arithmetic</a> on subsets of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> formed by a <a href="/facts/Significand/JFCVyTz4">significand</a> (a <a href="/facts/Sign_(mathematics)/z0xollg1">signed</a> sequence of a fixed number of digits in some <a href="/facts/Radix/J5RGWRJF">base</a>) multiplied by an <a href="/facts/Integer_power/pac6QY2g">integer power</a> of that base.
Numbers of this form are called floating-point numbers.: 3 : 10 
For example, the number 2469/200 is a floating-point number in base ten with five digits:

2469
        
          /
        
        200
        =
        12.345
        =
        
        
          
            
              12345
              ⏟
            
          
          
            significand
          
        
        
        ×
        
        
          
            
              10
              ⏟
            
          
          
            base

−
                  3
                
              
              ⏞
            
          
          
            exponent
          
        
      
    
    {\displaystyle 2469/200=12.345=\!\underbrace {12345} _{\text{significand}}\!\times \!\underbrace {10} _{\text{base}}\!\!\!\!\!\!\!\overbrace {{}^{-3}} ^{\text{exponent}}}

However, 7716/625 = 12.3456 is not a floating-point number in base ten with five digits—it needs six digits.
The nearest floating-point number with only five digits is 12.346.
And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits.
In practice, most floating-point systems use <a href="/facts/Binary_number/a6JDSRPs">base two</a>, though base ten (<a href="/facts/Decimal_floating_point/1F9rG2Sn">decimal floating point</a>) is also common.
Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations by <a href="/facts/Rounding/rCLYXN00">rounding</a> any result that is not a floating-point number itself to a nearby floating-point number.: 22 : 10 
For example, in a floating-point arithmetic with five base-ten digits, the sum 12.345 + 1.0001 = 13.3451 might be rounded to 13.345.
The term floating point refers to the fact that the number's <a href="/facts/Radix_point/3axE09oT">radix point</a> can "float" anywhere to the left, right, or between the <a href="/facts/Significant_digits/RUfYtpgo">significant digits</a> of the number. This position is indicated by the exponent, so floating point can be considered a form of <a href="/facts/Scientific_notation/YgyYEmq8">scientific notation</a>.
A floating-point system can be used to represent, with a fixed number of digits, numbers of very different <a href="/facts/Orders_of_magnitude_(numbers)/uu2o1hk2">orders of magnitude</a> — such as the number of meters <a href="/facts/Orders_of_magnitude_(length)/YzL1bHGX">between galaxies</a> or <a href="/facts/Orders_of_magnitude_(length)/YzL1bHGX">between protons in an atom</a>. For this reason, floating-point arithmetic is often used to allow very small and very large real numbers that require fast processing times. The result of this <a href="/facts/Dynamic_range/zxcuI4Fn">dynamic range</a> is that the numbers that can be represented are not uniformly spaced; the difference between two consecutive representable numbers varies with their exponent.

Over the years, a variety of floating-point representations have been used in computers. In 1985, the <a href="/facts/IEEE_754/dD3e7zAl">IEEE 754</a> Standard for Floating-Point Arithmetic was established, and since the 1990s, the most commonly encountered representations are those defined by the IEEE.
The speed of floating-point operations, commonly measured in terms of <a href="/facts/FLOPS/6zYlH68F">FLOPS</a>, is an important characteristic of a <a href="/facts/Computer_system/SQe1aK05">computer system</a>, especially for applications that involve intensive mathematical calculations.
A <a href="/facts/Floating-point_unit/XWTW2AbF">floating-point unit</a> (FPU, colloquially a math <a href="/facts/Coprocessor/WsVXzAGo">coprocessor</a>) is a part of a computer system specially designed to carry out operations on floating-point numbers.

Floating-point arithmetic open-in-new

Floating-point arithmetic