Scientific notation

Scientific 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 commonly used in calculators, and by scientists, mathematicians, doctors, and engineers.

In scientific notation all numbers are written like this:

$a \times 10^b$

("a times ten raised 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

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 or the exponential function e^x (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

• In the C++, FORTRAN, MATLAB, Perl, Java^[1] and Python programming languages, 6.0221418E23 or 6.0221418e23 is equivalent to 6.0221418×10²³. FORTRAN also uses "D" to signify double precision numbers.^[2]
• The ALGOL 60 programming language uses a subscript ten "₁₀" character instead of the letter E, for example: 6.02214151023.^[3]
• The ALGOL 68 programming language has the choice of 4 characters: e, E, \, or ₁₀. By examples: 6.0221415e23, 6.0221415E23, 6.0221415\23 or 6.02214151023.^[4]

• Decimal Exponent Symbol is part of "The Unicode Standard 6.0" e.g. 6.0221415⏨23 - it was included to accommodate usage in the programming languages Algol 60 and Algol 68.
• The TI-83 series and TI-84 Plus series of calculators use a stylized E character to display decimal exponent and the 10 character to denote an equivalent Operator[7].
• The Simula programming language requires the use of & (or && for long), for example: 6.0221415&23 (or 6.0221415&&23).^[5]

Engineering notation

Main article: 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.^[6]

Examples

• An electron's mass is about 0.00000000000000000000000000000091093822 kg. In scientific notation, this is written 9.1093822×10⁻³¹ kg.
• The Earth's mass is about 5973600000000000000000000 kg. In scientific notation, this is written 5.9736×10²⁴ kg.
• The Earth's circumference is approximately 40000000 m. In scientific notation, this is 4×10⁷ m. In engineering notation, this is written 40×10⁶ m. In SI writing style, this may be written "40 Mm" (40 megameters).
• An inch is 25400 micrometers. Describing an inch as 2.5400×10⁴ µm unambiguously states that this conversion is correct to the nearest micrometer. An approximated value with only three significant digits would be 2.54×10⁴ µm instead. In this example, the number of significant zeros is actually infinite (which is not the case with most scientific measurements, which have a limited degree of precision). It can be properly written with the minimum number of significant zeros used with other numbers in the application (no need to have more significant digits that other factors or addends). Or a bar can be written over a single zero, indicating that it repeats forever. The bar symbol is just as valid in scientific notation as it is in decimal notation.

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,^[7] 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,

$x_0=a_0\times10^{b_0}$

and

$x_1=a_1\times10^{b_1}$

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

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

and

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

Some examples are:

$5.67\times10^{-5} \times 2.34\times10^2 \approx 13.3\times10^{-3} = 1.33\times10^{-2}$

and

$\frac{2.34\times10^2}{5.67\times10^{-5}} \approx 0.413\times10^{7} = 4.13\times10^6$

Addition and subtraction require the numbers to be represented using the same exponential part, so that the significant can be simply added or subtracted. :$x_1 = c \times10^{b_0}$

Next, add or subtract the significants:

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

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

• Engineering notation
• Binary prefix
• Floating point
• ISO 31-0
• ISO 31-11
• Significant figure
• Scientific pitch notation

Notes and references

1. ^ http://download.oracle.com/javase/tutorial/java/nutsandbolts/datatypes.html
2. ^ http://www.math.hawaii.edu/lab/197/fortran/fort3.htm#double
3. ^ Report on the Algorithmic Language ALGOL 60, Ed. P. Naur, Copenhagen 1960
4. ^ "Revised Report on the Algorithmic Language Algol 68". September 1973. http://www.springerlink.com/content/k902506t443683p5/. Retrieved April 30, 2007. 
5. ^ "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. 
6. ^ Samples of usage of terminology and variants: [1], [2], [3], [4], [5], [6]
7. ^ NIST value for the elementary charge

External links

• Decimal to Scientific Notation Converter
• Scientific Notation to Decimal Converter
• Scientific Notation in Everyday Life
• An exercise in converting to and from scientific notation
• MathAce » Scientific Notation—Basic explanation and sample questions with solutions. [as of 20 Apr 2011, this link is bad (404 error)]
• Scientific Notation To Decimal Converter.
• Decimal To Scientific Notation Converter.