Significand

A Texas Instruments TI-84 Plus calculator display showing the Avogadro constant in E notation

Part of a number in scientific notation or in floating-point representation, consisting of its significant digits.

- Significand

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Significant figures

Significant figures (also known as the significant digits, precision or resolution) of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something.

Digits in light blue are significant figures; those in black are not.

Eliminate ambiguous or non-significant zeros by using Scientific Notation: For example, 1300 with three significant figures becomes 1.3. Likewise 0.0123 can be rewritten as 1.23. The part of the representation that contains the significant figures (1.30 or 1.23) is known as the significand or mantissa. The digits in the base and exponent ( or ) are considered exact numbers so for these digits, significant figures are irrelevant.

Floating-point arithmetic

Arithmetic using formulaic representation of real numbers as an approximation to support a trade-off between range and precision.

An early electromechanical programmable computer, the Z3, included floating-point arithmetic (replica on display at Deutsches Museum in Munich).
Single-precision floating point numbers on a number line: the green lines mark representable values.
Augmented version above showing both signs of representable values
Leonardo Torres y Quevedo, who proposed a form of floating point in 1914
William Kahan. A primary architect of the Intel 80x87 floating-point coprocessor and IEEE 754 floating-point standard.

In general, a floating-point number is represented approximately with a fixed number of significant digits (the significand) and scaled using an exponent in some fixed base; the base for the scaling is normally two, ten, or sixteen.

Double-precision floating-point format

Computer number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point.

The format is written with the significand having an implicit integer bit of value 1 (except for special data, see the exponent encoding below).

IEEE 754

Technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE).

Precision of binary32 and binary64 in the range 10−12 to 1012

Finite numbers, which can be described by three integers: s = a sign (zero or one), c = a significand (or coefficient) having no more than p digits when written in base b (i.e., an integer in the range through 0 to bp − 1), and q = an exponent such that emin ≤ q + p − 1 ≤ emax. The numerical value of such a finite number is (−1)s × c × bq. Moreover, there are two zero values, called signed zeros: the sign bit specifies whether a zero is +0 (positive zero) or −0 (negative zero).

Offset binary

Method for signed number representation where a signed number n is represented by the bit pattern corresponding to the unsigned number n + K, K being the biasing value or offset.

Computer simulation, one of the main cross-computing methodologies.

The "characteristic" (exponent) took the form of a seven-bit excess-64 number (The high-order bit of the same byte contained the sign of the significand).

Casio FX-702P

Pocket Computer, manufactured by Casio from 1981 to 1984.

A 28-year-old FX-702P in working condition

A 10-digit mantissa is displayed (including minus sign) however internal calculations use a 12-digit mantissa.

Microsoft Binary Format

Format for floating-point numbers which was used in Microsoft's BASIC language products, including MBASIC, GW-BASIC and QuickBASIC prior to version 4.00.

Altair 8800 front panel
Radio Shack Tandy TRS-80 Model I System
VAX-11/780 minicomputer

The original version was designed for memory-constrained systems and stored numbers in 32 bits (4 bytes), with a 23-bit mantissa, 1-bit sign, and an 8-bit exponent.

Honeywell 800

The Datamatic Division of Honeywell announced the H-800 electronic computer in 1958.

The 48 bit word allowed a seven bit exponent and 40 bit mantissa.

R4200

Microprocessor designed by MIPS Technologies, Inc. that implemented the MIPS III instruction set architecture (ISA).

Texas Instruments TMS1000

A notable feature is the use of the integer datapath for performing arithmetic operations on the mantissa portion of a floating point number.

PA-7100LC

Microprocessor that implements the PA-RISC 1.1 instruction set architecture developed by Hewlett-Packard (HP).

A PA-7100LC microprocessor.
Die shot of PA-7100LC.
A PA-7300LC microprocessor.
Die shot of PA-7300LC.

Prominently, the floating-point unit multiplier was modified to take up less area by halving the tree of carry-save adders that summed the partial products of the mantissa.