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Encyclopedia > Scientific notation

Scientific notation, also known as standard form, is a notation for writing numbers that is often used by scientists and mathematicians to make it easier to write large and small numbers. A number that is written in scientific notation has several properties that make it very useful to scientists. Your maths teacher is mr macblain



The basic concept for the practical notations is this mathematical exponential expression using powers of ten :


wherein exponent b is an integer, and the coefficient a any real number, called the significand or mantissa (using "mantissa" may cause confusion as it can also refer to the fractional part of the common logarithm). In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... The integers are commonly denoted by the above symbol. ... In mathematics, a coefficient is a constant multiplicative factor of a certain object. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The significand (also coefficient or mantissa) is the part of a floating-point number that contains its significant digits. ... For the traditional use of the word mantissa in mathematics, see common logarithm. ... In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

Normalized notation

Written in the form a × 10b, exponent b is chosen such that the absolute value of a remains at least one but less than ten (1 le left | a right vert < 10!). Normal mathematics convention dictates a minus sign to precede the first of the decimal digits of a for a negative number; that of b for a number with absolute value between 0 and 1, e.g. minus one half is -5 × 10-1. There is no need to represent zero in normalized form, the digit 0 suffices. The normalized 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 mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... Look up zero in Wiktionary, the free dictionary. ... An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ...

It is implicitly assumed that scientific notation should always be normalized except during calculations or when an unnormalized form, such as engineering notation, is desired. (Normalized) scientific notation is often called exponential notation — though the latter term is more general as it applies as well, for instance, to engineering notation or any value for a and to bases other than 10 like 315 × 220. Engineering notation is scientific notation in which the powers of ten are limited to those where the exponent is a multiple of three, i. ... Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ... In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. ...

E notation

Most calculators and many computer programs present very large and very small results in scientific notation. Because exponents like 107 can't always be conveniently represented on computers, typewriters, and calculators, an alternate notation is often used: the letter "E" or "e" is used for "times ten raised to the power", thus replacing the "×10n" while the exponent is not superscripted but is left on the same level with the significand (e.g. a E−6 not a E−6). The sign is often given even if positive (e.g. a E+11 rather than a E11). For example, 6.0221415 E+23 or 6.0221415 e23 is the same as 6.0221415 × 1023(Avogadro's number). Note that this character "e" is not related to the mathematical constant e (a confusion not possible when using capital "E") ; and though it is short for exponent, the notation is referred to as (scientific) E notation or (scientific) e notation rather than (scientific) exponential notation though the latter also occurs. The E notation is a variant representation that strictly meets the mathematical requirements of the normalized notation. Examples: A calculator is a device for performing calculations. ... A computer program is a collection of instructions that describe a task, or set of tasks, to be carried out by a computer. ... This article is about the term superscript as used in typography. ... Avogadros number, also called Avogadros constant (NA), named after Amedeo Avogadro, is formally defined to be the number of carbon-12 atoms in 12 grams (0. ... e is the unique number such that the value of the derivative (slope of a tangent line) of f (x)=ex (blue curve) at the point x=0 is exactly 1. ...

  • In the FORTRAN programming language 6.0221415E23 is equivalent to 6.0221415 × 1023.
  • The ALGOL[1][2] programming language also uses the E notation, alternatively - when available - either character '₁₀' or '' can be used, for example: 6.0221415₁₀23 and 6.022141523.

Fortran (previously FORTRAN[1]) is a general-purpose[2], procedural,[3] imperative programming language that is especially suited to numeric computation and scientific computing. ... Algol (β Per / Beta Persei) is a bright star in the constellation Perseus. ...

Engineering notation

Engineering notation differs from the normalized exponential notation by restricting the exponent b to multiples of 3. Consequently, the absolute value of a can need to be equal or larger than 10 (ranging from 1 to limit 1,000) and thus engineering notation does not meet the conditions set for the normalized form, and 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. Engineering notation is scientific notation in which the powers of ten are limited to those where the exponent is a multiple of three, i. ... Multiple is a comic book superhero in the Marvel Comics universe. ... Look up one thousand in Wiktionary, the free dictionary. ... An SI prefix (also known as a metric prefix) is a name or associated symbol that precedes a unit of measure (or its symbol) to form a decimal multiple or submultiple. ... mega- (symbol M) is an SI prefix in the SI system of units denoting a factor of 106, i. ... Kilo (symbol: k) is a prefix in the SI system denoting 103 or 1000. ... Milli (symbol m) is an SI prefix in the SI system of units denoting a factor of 10-3, or 1/1,000. ... Look up micro- in Wiktionary, the free dictionary. ... For other uses, see Nano (disambiguation). ...

Usage of spaces

In scientific normalized exponential notation, in E notation, and in engineering notation, the space (which in typesetting may be represented with a normal width space or a thin space) that is allowed only before and after "×" or in front of "E" or "e", may as well be omitted though it is less common to disappear from before the alphabetical character.[3] A space is a punctuation convention for providing interword separation in some scripts, including the Latin, Greek, Cyrillic, and Arabic. ... This article or section is in need of attention from an expert on the subject. ...


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. Significant figures (also called significant digits and abbreviated sig figs or sig digs, respectively) is a method of expressing errors in measurements. ...


  • An electron's mass is about 0.000 000 000 000 000 000 000 000 000 000 910 938 26 kg. In scientific notation, this is written 9.1093826×10−31 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 (i.e. 4 followed by 7 zeroes). 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").

e- redirects here. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... Unsolved problems in physics: What causes anything to have mass? The U.S. National Prototype Kilogram, which currently serves as the primary standard for measuring mass in the U.S. Mass is the property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... Cover of brochure The International System of Units. ...

Significant figures

Scientific notation is useful for indicating the precision with which a quantity was measured. Including only the significant figures (the digits which are known to be reliable, plus one uncertain digit) in the coefficient conveys the precison of the value. In the absence of any statement otherwise, the value of a physical quantity in scientific notation is assumed to have been measured to at least the quoted number of digits of precision with the last potentially in doubt by half a unit. In the fields of science, engineering, industry and statistics, accuracy is the degree of conformity of a measured or calculated quantity to its actual (true) value. ... Rounding to n significant figures is a form of rounding. ... A physical quantity is either a quantity within physics that can be measured (e. ...

As an example, consider the Earth's mass as presented above in conventional notation. Since that representation gives no indication of the accuracy of the reported value, a reader could incorrectly assume from the twenty-five digits shown that it is known right down to the last kilogram! The scientific notation indicates that it is known with a precision of ± 0.00005×1024 kg, or ± 5×1019 kg.

Where precision in such measurements is crucial more sophisticated expressions of measurement error must be used. Measurement is the determination of the size or magnitude of something. ...

Order of magnitude

Scientific notation also enables simple 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 simply comparing the exponents rather than counting all the leading zeros. In this case, '−27' is larger than '−31' and therefore the proton is four orders of magnitude (about 10,000 times) more massive than the electron. In physics, the proton (Greek proton = first) is a subatomic particle with an electric charge of one positive fundamental unit (1. ... An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ...

Scientific notation also avoids misunderstandings due to regional differences in certain quantifiers, such as 'billion', which might indicate either 109 or 1012. The long and short scales are two different numerical systems used throughout the world: Short scale is the English translation of the French term échelle courte. ...

Using scientific notation


Multiplication and division by 10 are easy to perform in scientific notation.

At the mantissa, multiplication by 10 may be seen as shifting the decimal separator one position to the right (adding a zero if needed): 12.34×10=123.4. Division may be seen as shifting it to the left: 12.34/10=1.234 This article or section does not cite its references or sources. ...

In the exponential part multiplication by 10 results in adding 1 to the exponent: 102×10=103. Division by 10 results in subtracting 1 from the exponent: 102/10=101.

Also notice that 1 is multiplication's neutral element and that 100=1. In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

To convert between different representations of the same number, all that is needed is to perform the opposite operations to each part. Thus multiplying the mantissa by 10, n times is done by shifting the decimal separator n times to the right. Dividing by 10 the same number of times is done by adding −n to the exponent. Some examples:

123.4 = 123.4times10^0 = (123.4/10^2) times (10^0times10^2) = 1.234times10^2
0.001234 = 0.001234times10^0 = (0.001234times 10^3) times (10^0 / 10^3) = 1.234times10^{-3}

Basic operations

Given two numbers in scientific notation,


Multiplication and division are performed using the rules for operation with exponential functions: In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...

x_0 x_1=a_0 a_1times10^{b_0+b_1}

some examples are:

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

Addition and subtraction require the numbers to be represented using the same exponential part, in order to simply add, or subtract, the mantissas, so it may 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. Second, add or subtract the mantissas. 3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ...

x_1^star = a_1^star times10^{b_0}
x_0 pm x_1=x_0 pm x_1^star=(a_0pm a_1^star)times10^{b_0}

an example:

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


  1. ^ GOST 10859: 10 to the power of encoding. Retrieved on Jun 5, 2007.
  2. ^ ALCOR "lower 10". Retrieved on Jun 5, 2007.
  3. ^ Samples of usage of terminology and variants: [1], [2], [3], [4], [5], [6]

See also

An SI prefix is a prefix which can be applied to any unit of the International System of Units (SI) to give subdivisions and multiples of that unit. ... ISO 31-0 is the introductory part of international standard ISO 31 on quantities and units. ...

External links

  Results from FactBites:
Scientific notation - definition of Scientific notation in Encyclopedia (552 words)
Scientific notation (standard index notation) is floating-point notation with radix (base) 10.
In that case an advantage of scientific notation is that trailing zeros which are the result of rounding are not needed.
Most calculators and many computer programs present very large and very small results in scientific notation; the 10 is usually omitted and the letter E for exponent is used; for example, 1.56234 E+29.
Scientific pitch notation - Wikipedia, the free encyclopedia (361 words)
Scientific pitch notation in Western music is a method of naming the notes of the standard Western chromatic scale by combining a letter-name, accidentals, and an Hindu-Arabic numeral identifying the pitch's octave.
Scientific pitch notation is often used to specify the range of an instrument.
It provides an unambiguous means of identifying a note in terms of musical notation rather than frequency, while at the same time avoiding the transposition conventions that are used in writing the music for instruments such as the clarinet and guitar.
  More results at FactBites »



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