Scientific notation

Scientific notation, also known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is often favored by scientists, mathematicians and engineers, who work with such numbers.

In scientific notation all numbers are written like this:

a × 10b

("a times ten to the power of b"), where the exponent b is an integer, and the coefficient a is any real number (but see normalized notation below), called the significand or mantissa (though the term "mantissa" may cause confusion as it can also refer to the fractional part of the common logarithm). If the number is negative then a minus sign precedes a (as in ordinary decimal notation).

Ordinary decimal notation Scientific notation (normalized)
300 3×102
4,000 4×103
5,720,000,000 5.72×109
0.0000000061 6.1×10−9

Contents

Normalized notation

Any given number can be written in the form of a×10^b in many ways; for example 350 can be written as 3.5×102 or 35×101 or 350×100.

In normalized scientific notation, the exponent b is chosen such that the absolute value of a remains at least one but less than ten (1 ≤ |a| < 10). For example, 350 is written as 3.5×102. This form allows easy comparison of two numbers of the same sign in a, as the exponent b gives the number's order of magnitude. In normalized notation the exponent b is negative for a number with absolute value between 0 and 1 (e.g., minus one half is −5×10−1). The 10 and exponent are usually omitted when the exponent is 0. Note that 0 itself cannot be written in normalised scientific notation since the mantissa would have to be zero and the exponent undefined.

In many fields, scientific notation is normally in this way, except during intermediate calculations or when an unnormalised form, such as engineering notation, is desired. (Normalized) scientific notation is often called exponential notation — although the latter term is more general and also applies when a is not restricted to the range 1 to 10 (as in engineering notation for instance) and to bases other than 10 (as in 315× 2^20).

E notation

A calculator display showing the Avogadro constant in E notation

Most calculators and many computer programs present very large and very small results in scientific notation. Because superscripted exponents like 107 cannot always be conveniently represented on computers, typewriters and calculators, an alternative format is often used: the letter E or e represents times ten raised to the power of, thus replacing the × 10, followed by the value of the exponent. Note that the character e is not related to the mathematical constant e (a confusion that is less likely with capital E); and though it stands for exponent, the notation is usually referred to as (scientific) E notation or (scientific) e notation, rather than (scientific) exponential notation (though the latter also occurs).

Examples and alternatives

Engineering notation

Engineering notation differs from normalized scientific notation in that the exponent b is restricted to multiples of 3. Consequently, the absolute value of a is in the range 1 ≤ |a| < 1000, rather than 1 ≤ |a| < 10. Though similar in concept, engineering notation is rarely called scientific notation. This allows the numbers to explicitly match their corresponding SI prefixes, which facilitates reading and oral communication. For example, 12.5×10−9 m can be read as "twelve-point-five nanometers" or written as 12.5 nm, while its scientific notation counterpart 1.25×10−8 m would likely be read out as "one-point-two-five times ten-to-the-negative-eighth meters".

Use of spaces

In normalized scientific notation, in E notation, and in engineering notation, the space (which in typesetting may be represented by a normal width space or a thin space) that is allowed only before and after "×" or in front of "E" or "e" is sometimes omitted, though it is less common to do so before the alphabetical character.[4]

Motivation

Scientific notation is a very convenient way to write large or small numbers and do calculations with them. It also quickly conveys two properties of a measurement that are useful to scientists—significant figures and order of magnitude. Writing in scientific notation allows a person to eliminate zeros in front of or behind the significant digits. This is most useful for very large measurements in astronomy or very small measurements in the study of molecules. The examples below display this.

Examples

Significant figures

Ambiguity of the last digit in scientific notation

It is customary in scientific measurements to record all the significant digits from the measurements, and to guess one additional digit if there is any information at all available to the observer to make a guess. The resulting number is considered more valuable than it would be without that extra digit, and it is considered a significant digit because it contains some information leading to greater precision in measurements and in aggregations of measurements (adding them or multiplying them together).

Additional information about precision can be conveyed through additional notations. In some cases, it may be useful to know how exact the final significant digit is. For instance, the accepted value of the unit of elementary charge can properly be expressed as 1.602176487(40)×10−19 C,[5] which is shorthand for 1.602176487±0.000000040×10−19 C.

Order of magnitude

Scientific notation also enables simpler order-of-magnitude comparisons. A proton's mass is 0.0000000000000000000000000016726 kg. If this is written as 1.6726×10−27 kg, it is easier to compare this mass with that of the electron, given above. The order of magnitude of the ratio of the masses can be obtained by comparing the exponents instead of the more error-prone task of counting the leading zeros. In this case, −27 is larger than −31 and therefore the proton is roughly four orders of magnitude (about 10000 times) more massive than the electron.

Scientific notation also avoids misunderstandings due to regional differences in certain quantifiers, such as billion, which might indicate either 109 or 1012.

Using scientific notation

Converting

To convert from ordinary decimal notation to scientific notation, move the decimal separator the desired number of places to the left or right, so that the significand will be in the desired range (between 1 and 10 for the normalized form). If you moved the decimal point n places to the left then multiply by 10n; if you moved the decimal point n places to the right then multiply by 10−n. For example, starting with 1230000, move the decimal point six places to the left yielding 1.23, and multiply by 106, to give the result 1.23×106. Similarly, starting with 0.000000456, move the decimal point seven places to the right yielding 4.56, and multiply by 10−7, to give the result 4.56×10−7.

If the decimal separator did not move then the exponent multiplier is logically 100, which is correct since 100 = 1. However, the exponent part "× 100" is normally omitted, so, for example, 1.234×100 is just written as 1.234.

To convert from scientific notation to ordinary decimal notation, take the significand and move the decimal separator by the number of places indicated by the exponent — left if the exponent is negative, or right if the exponent is positive. Add leading or trailing zeroes as necessary. For example, given 9.5 × 1010, move the decimal point ten places to the right to yield 95000000000.

Conversion between different scientific notation representations of the same number is achieved by performing opposite operations of multiplication or division by a power of ten on the significand and the exponent parts. The decimal separator in the significand is shifted n places to the left (or right), corresponding to division (multiplication) by 10n, and n is added to (subtracted from) the exponent, corresponding to a canceling multiplication (division) by 10n. For example:

1.234×103 = 12.34×102 = 123.4×101 = 1234

Basic operations

Given two numbers in scientific notation,

x_0=a_0\times10^{b_0}
x_1=a_1\times10^{b_1}

Multiplication and division are performed using the rules for operation with exponential functions:

x_0 x_1=a_0 a_1\times10^{b_0+b_1}
\frac{x_0}{x_1}=\frac{a_0}{a_1}\times10^{b_0-b_1}

some examples are:

5.67\times10^{-5} \times 2.34\times10^2 \approx 13.3\times10^{-3} = 1.33\times10^{-2}
\frac{2.34\times10^2}{5.67\times10^{-5}} \approx 0.413\times10^{7} = 4.13\times10^6

Addition and subtraction require the numbers to be represented using the same exponential part, so that the significant can be simply added or subtracted. These operations may therefore take two steps to perform. First, if needed, convert one number to a representation with the same exponential part as the other. This is usually done with the one with the smaller exponent. In this example, x1 is rewritten as:

x_1 = c \times10^{b_0}

Next, add or subtract the significands:

x_0 \pm x_1=(a_0\pm c)\times10^{b_0}

An example:

2.34\times10^{-5} + 5.67\times10^{-6} = 2.34\times10^{-5} + 0.567\times10^{-5} \approx 2.91\times10^{-5}

See also

Notes and references

  1. ↑ Report on the Algorithmic Language ALGOL 60, Ed. P. Naur, Copenhagen 1960
  2. ↑ "Revised Report on the Algorithmic Language Algol 68". September 1973. http://www.springerlink.com/content/k902506t443683p5/. Retrieved April 30, 2007. 
  3. ↑ "SIMULA Standard As defined by the SIMULA Standards Group - 3.1 Numbers". August 1986. http://prosjekt.ring.hibu.no/simula/Standard/chap_1.htm. Retrieved October 6, 2009. 
  4. ↑ Samples of usage of terminology and variants: [1], [2], [3], [4], [5], [6]
  5. ↑ NIST value for the elementary charge

External links