Floating-point arithmetic

In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a significand (a signed sequence of a fixed number of digits in some base) multiplied by an integer power of that base.

Numbers of this form are called floating-point numbers.

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 base two, though base ten (decimal floating point) is also common.

Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations by rounding any result that is not a floating-point number itself to a nearby floating-point number.

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 radix point can "float" anywhere to the left, right, or between the significant digits of the number. This position is indicated by the exponent, so floating point can be considered a form of scientific notation.