Scientific notation
Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form, since to do so would require writing out an inconveniently long string of digits.
This base ten notation is commonly used by scientists, mathematicians, and engineers, in part because it can simplify certain arithmetic operations.
In scientific notation, nonzero numbers are written in the form or m times ten raised to the power of n, where n is an integer, and the coefficient m is a nonzero real number (usually between 1 and 10 in absolute value, and nearly always written as a terminating decimal).
If the number is negative then a minus sign precedes m, as in ordinary decimal notation.
In normalized notation, the exponent is chosen so that the absolute value (modulus) of the significand m is at least 1 but less than 10.
Decimal floating point is a computer arithmetic system closely related to scientific notation.
In normalized scientific notation (called "standard form" in the United Kingdom), the exponent n is chosen so that the absolute value of m remains at least one but less than ten (1 ≤ |m| < 10).
In normalized notation, the exponent n is negative for a number with absolute value between 0 and 1 (e.g. 0.5 is written as 5×10−1).
Engineering notation allows the numbers to explicitly match their corresponding SI prefixes, which facilitates reading and oral communication.
While common in computer output, this abbreviated version of scientific notation is discouraged for published documents by some style guides.
The ALGOL 60 (1960) programming language uses a subscript ten "10" character instead of the letter "E", for example: 6.0221023.
[8][9] This presented a challenge for computer systems which did not provide such a character, so ALGOL W (1966) replaced the symbol by a single quote, e.g. 6.022'+23,[10] and some Soviet ALGOL variants allowed the use of the Cyrillic letter "ю", e.g. 6.022ю+23[citation needed].
[11] The ALGOL "10" character was included in the Soviet GOST 10859 text encoding (1964), and was added to Unicode 5.2 (2009) as U+23E8 ⏨ DECIMAL EXPONENT SYMBOL.
[13] Mathematica supports the shorthand notation 6.022*^23 (reserving the letter E for the mathematical constant e).
[14] To enter numbers in scientific notation calculators include a button labeled "EXP" or "×10x", among other variants.
The displays of pocket calculators of the 1970s did not display an explicit symbol between significand and exponent; instead, one or more digits were left blank (e.g. 6.022 23, as seen in the HP-25), or a pair of smaller and slightly raised digits were reserved for the exponent (e.g. 6.022 23, as seen in the Commodore PR100).
[17] In 1962, Ronald O. Whitaker of Rowco Engineering Co. proposed a power-of-ten system nomenclature where the exponent would be circled, e.g. 6.022 × 103 would be written as "6.022③".
Leading and trailing zeroes are not significant digits, because they exist only to show the scale of the number.
The number 1230400 is usually read to have five significant figures: 1, 2, 3, 0, and 4, the final two zeroes serving only as placeholders and adding no precision.
Thus, an additional advantage of scientific notation is that the number of significant figures is unambiguous.
First, move the decimal separator point sufficient places, n, to put the number's value within a desired range, between 1 and 10 for normalized notation.
To represent the number 1,230,400 in normalized scientific notation, the decimal separator would be moved 6 digits to the left and × 106 appended, resulting in 1.2304×106.
The number −0.0040321 would have its decimal separator shifted 3 digits to the right instead of the left and yield −4.0321×10−3 as a result.
Conversion between different scientific notation representations of the same number with different exponential values is achieved by performing opposite operations of multiplication or division by a power of ten on the significand and an subtraction or addition of one on the exponent part.
The decimal separator in the significand is shifted x places to the left (or right) and x is added to (or subtracted from) the exponent, as shown below.
[citation needed] In E notation, this is written as 1.001bE11b (or shorter: 1.001E11) with the letter "E" now standing for "times two (10b) to the power" here.
[26] This is closely related to the base-2 floating-point representation commonly used in computer arithmetic, and the usage of IEC binary prefixes (e.g. 1B10 for 1×210 (kibi), 1B20 for 1×220 (mebi), 1B30 for 1×230 (gibi), 1B40 for 1×240 (tebi)).
The FORTRAN Automatic Coding System for the IBM 704 EDPM: Programmer's Reference Manual (PDF).
New York: Applied Science Division and Programming Research Department, International Business Machines Corporation.
I'm going to begin using it in place of "exponent" which is technically incorrect, and the letter D to separate the "mantissa" from the decapower for typewritten numbers, as Jim also suggests.
