1914

Floating-Point Numbers

Leonardo Torres y Quevedo (1852–1936), William Kahan (b. 1933)

Leonardo Torres y Quevedo was a Spanish engineer and mathematician who delighted in making practical machines. In 1906, he demonstrated a radio-controlled model boat for the king of Spain, and he designed a semirigid airship used in World War I.

Torres was also a fan of Babbage’s difference and analytical engines. In 1913, he published Essays in Automatics, which described Babbage’s work and presented the design for a machine that could calculate the value of the formula a(y–z)² for specified values of a, y, and z. To allow his machine to handle a wider range of numbers, Torres invented floating-point arithmetic.

Floating-point arithmetic extends the range of a numerical calculation by decreasing its accuracy. Instead of storing all of the digits in a number, the computer stores just a few significant digits, called the significand, and a much shorter exponent. The actual “number” is then computed using the formula: significand × base^exponent

For example, the gross domestic product of the United States in 2016 was 18.57 trillion dollars. Storing that number with a fixed-point representation requires 14 digits. But storing it in floating point requires just 6 digits: $18.57 trillion = 1.857 × 10¹³

Thus, with floating-point numbers, sometimes called scientific notation on modern calculators, a 10-digit register (a mechanical or electronic gadget that can store a number) that would normally be limited to storing numbers between 1 and 9,999,999,999 instead can be partitioned into an 8-digit significand and a 2-digit exponent, allowing it to store numbers as small as 0.0000001 × 10^–99 and as large as 9.9999999 × 10⁹⁹.

Modern floating-point systems use binary rather than decimal digits. Under the standard developed by Canadian mathematician William Kahan for the Intel® 8086 microprocessor and adopted in 1985 by the Institute of Electrical and Electronics Engineers (IEEE 754), single precision floating point uses 24 bits for the significand and 8 bits for the exponent.

For his work, Kahan won the Association for Computing Machinery’s (ACM) A.M. Turing Award in 1989.

SEE ALSO Binary Arithmetic (1703), Z3 Computer (1941), Binary-Coded Decimal (1944)

Portrait of Leonardo Torres y Quevedo by Argentinian cartoonist and illustrator Eulogia Merle (b. 1976).