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Scientific notation

Scientific notation
Ordinary decimal notation 300 4,000 5,720,000,000 −0.000 000 006 1 Scientific notation (normalised) 3×102 4×103 5.72×109 −6.1×10−9

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:

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", thus replacing the " × 10n". The character "e" is not related to the mathematical constant e (a confusion not possible when using 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).

("a times ten to the power of b"), where the exponent b is an integer, and the coefficient a is any real number, 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).

• In the FORTRAN programming language 6.0221415E23 is equivalent to 6.022 141 5 × 1023. • The ALGOL programming language also uses the E notation; alternatively—when available—either character ’₁₀’ or ’\’ can be used. For example: 6.0221415₁₀23 and 6.0221415\23.[1][2]

Normalized notation
Any given number can be written in the form a × 10b 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. In many fields, scientific notation is normalized in this way, except during intermediate calculations or when an unnormalized 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 × 220).

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. Numbers in this form are easily read out using magnitude prefixes like mega- (b = 6), kilo- (b = 3), milli- (b = −3), micro- (b = −6) or nano- (b = −9). For example, 12.5×10−9 m can be read as "twelve point five nanometers" or written as 12.5 nm.

Use of spaces
In normalized scientific notation, in E notation, and in engineering notation, the space (which in typesetting


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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" may be omitted, though it is less common to do so before the alphabetical character.[3]

Scientific notation
they should be underlined to explicity show that they are significant zeros). In decimal notation, zeros next to the decimal point are not necessarily significant numbers. I.e., they may be there only to show where the decimal point is. In scientific notation, however, this ambiguity is resolved, because any zeros shown are considered significant by convention. For example, using scientific notation, the speed of light in SI units is 2.99792458×108 m/s and the inch is 2.54×10−2 m; both numbers are exact by definition of the units "inches" per cm and "meters" in terms of the speed of light.[4] In these cases, all the digits are significant. A single zero or any number of zeros could be added on the right side to show more significant digits, or a single zero with a bar on top could be added to show infinite significant digits (just as in decimal notation).

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 well.

• An electron’s mass is about 0.000 000 000 000 000 000 000 000 000 000 910 938 22 kg. In scientific notation, this is written 9.109 382 2 × 10-31 kg. • The Earth’s mass is about 5,973,600,000,000,000,000,000,000 kg. In scientific notation, this is written 5.9736×1024 kg. • The Earth’s circumference is approximately 40,000,000 m. In scientific notation, this is written 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 25,400 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 3 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.

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 thru additional notations. For example, in some cases, it may be useful to know how exact the final significant digit is, as explained below: For instance, the accepted value of the unit of elementary charge can properly be expressed as 1.602176487(40)×10−19 C (Coulomb),[5] where the (40) indicates 40 counts of uncertainty in the last decimal place. If a number has been rounded off, it can be written in the form 2.5 (½) × 10 -2 to explicitly indicate that there is a half-count of uncertainty in the last digit.

Order of magnitude
Scientific notation also enables simpler order-of-magnitude comparisons. A proton’s mass is 0.000 000 000 000 000 000 000 000 001 672 6 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 errorprone 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 10,000 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.

Significant figures
One advantage of scientific notation is that it greatly reduces the ambiguity of number of significant digits. All digits in normalized scientific notation are significant by convention. But in decimal notation any zero or series of zeros next to the decimal point are ambiguous, and may or may not indicate significant figures (when they are


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Scientific notation
some examples are:

Using scientific notation
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 mantissa 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 1,230,000, 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 0.000 000 456, 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 is just written as 1.234 rather than 1.234×100. To convert from scientific notation to ordinary decimal notation, take the mantissa 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 95,000,000,000. 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 mantissa and the exponent parts. The decimal separator in the mantissa 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 cancelling multiplication (division) by 10n. For example:

Addition and subtraction require the numbers to be represented using the same exponential part, so that the mantissas 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:

Next, add or subtract the mantissas:

An example:

See also
• • • • • • Binary prefix Floating point ISO 31-0 ISO 31-11 SI prefix Significant figures

Notes and references
[1] [2] "GOST 10859: 10 to the power of encoding". stand.html#gost10859. Retrieved on June 5 2007. "ALCOR "lower 10"". english/codes/5tape.html#alcor. Retrieved on June 5 2007. Samples of usage of terminology and variants: [1], [2], [3], [4], [5], [6] NIST value for the speed of light NIST value for the elementary charge


Basic operations
Given two numbers in scientific notation,

[4] [5]

External links
Multiplication and division are performed using the rules for operation with exponential functions: • Scientific Notation in Everyday Life • An exercise in converting to and from scientific notation • MathAce » Scientific Notation — Basic explanation and sample questions with solutions.


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Scientific notation

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