Scientific notation

# Scientific notation

Overview
Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in standard decimal notation
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

. Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians, doctors, and engineers.
Discussion

Recent Discussions
Encyclopedia
Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in standard decimal notation
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

. 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 in the form of
(a times ten raised to the power of b), where the exponent
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

b is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

, and the coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

a is any real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

(however, see normalized notation below), called the significand
Significand
The significand is part of a floating-point number, consisting of its significant digits. Depending on the interpretation of the exponent, the significand may represent an integer or a fraction.-Examples:...

or mantissa
Mantissa
* The mantissa is the significand in a common logarithm or floating-point number.* Metaphorically, it is the part of the self that eludes conscious awareness or self-understanding.* An addition of little importance.Mantissa may also refer to:...

. The term "mantissa" may cause confusion, however, because it can also refer to the fractional
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

part of the common logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

. If the number is negative then a minus sign precedes a (as in ordinary decimal notation).
Standard decimal notation Normalized scientific notation
300
4,000
-53,000
6,720,000,000
0.000 000 007 51

## Normalized notation

Any given number can be written in the form of in many ways; for example, 350 can be written as or or .
It does not have to be a whole number then a decimal.

In normalized scientific notation, the exponent b is chosen such that the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

of a remains at least one but less than ten (1 ≤ |a| < 10). Following these rules, 350 is always written as . 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
Order of magnitude
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. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount...

. In normalized notation the exponent b is negative for a number with absolute value between 0 and 1 (e.g., negative one half is written as ). 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.

Normalized scientific form is the typical form of expression of large numbers for many fields, except during intermediate calculations or when an unnormalised form, such as engineering notation
Engineering notation
Engineering notation is a version of scientific notation in which the powers of ten must be multiples of three...

, is desired. (Normalized) scientific notation is often called exponential
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

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 ).

## E notation

Most calculators
Calculator
An electronic calculator is a small, portable, usually inexpensive electronic device used to perform the basic operations of arithmetic. Modern calculators are more portable than most computers, though most PDAs are comparable in size to handheld calculators.The first solid-state electronic...

and many computer programs
Computer program
A computer program is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute...

present very large and very small results in scientific notation. Because superscripted
Subscript and superscript
A subscript or superscript is a number, figure, symbol, or indicator that appears smaller than the normal line of type and is set slightly below or above it – subscripts appear at or below the baseline, while superscripts are above...

exponents like 107 cannot always be conveniently represented, the letter E or e
E
E is the fifth letter and a vowel in the basic modern Latin alphabet. It is the most commonly used letter in the Czech, Danish, Dutch, English, French, German, Hungarian, Latin, Norwegian, Spanish, and Swedish languages.-History:...

is often used to represent times ten raised to the power of (which would be written as "x 10b") and is followed by the value of the exponent. Note that in this usage the character e is not related to the mathematical constant e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

or the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

ex (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++
C++
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...

, FORTRAN
Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...

, MATLAB
MATLAB
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...

, Perl
Perl
Perl is a high-level, general-purpose, interpreted, dynamic programming language. Perl was originally developed by Larry Wall in 1987 as a general-purpose Unix scripting language to make report processing easier. Since then, it has undergone many changes and revisions and become widely popular...

, Java
Java (programming language)
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities...

and Python
Python (programming language)
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...

programming languages, 6.0221418E23 or 6.0221418e23 is equivalent to
Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in standard decimal notation
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

. 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 in the form of
(a times ten raised to the power of b), where the exponent
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

b is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

, and the coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

a is any real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

(however, see normalized notation below), called the significand
Significand
The significand is part of a floating-point number, consisting of its significant digits. Depending on the interpretation of the exponent, the significand may represent an integer or a fraction.-Examples:...

or mantissa
Mantissa
* The mantissa is the significand in a common logarithm or floating-point number.* Metaphorically, it is the part of the self that eludes conscious awareness or self-understanding.* An addition of little importance.Mantissa may also refer to:...

. The term "mantissa" may cause confusion, however, because it can also refer to the fractional
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

part of the common logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

. If the number is negative then a minus sign precedes a (as in ordinary decimal notation).
Standard decimal notation Normalized scientific notation
300 {{val|3|e=2}}
4,000 {{val|4|e=3}}
-53,000 {{val|-5.3|e=4}}
6,720,000,000 {{val|6.72|e=9}}
0.000 000 007 51 {{val|7.51|e=-9}}

## Normalized notation

Any given number can be written in the form of {{gaps|a|e=b}} in many ways; for example, 350 can be written as {{val|3.5|e=2}} or {{val|35|e=1}} or {{val|350|e=0}}.
It does not have to be a whole number then a decimal.

In normalized scientific notation, the exponent b is chosen such that the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

of a remains at least one but less than ten (1 ≤ |a| < 10). Following these rules, 350 is always written as {{val|3.5|e=2}}. 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
Order of magnitude
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. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount...

. In normalized notation the exponent b is negative for a number with absolute value between 0 and 1 (e.g., negative one half is written as {{val|-5|e=-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.

Normalized scientific form is the typical form of expression of large numbers for many fields, except during intermediate calculations or when an unnormalised form, such as engineering notation
Engineering notation
Engineering notation is a version of scientific notation in which the powers of ten must be multiples of three...

, is desired. (Normalized) scientific notation is often called exponential
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

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 {{gaps|315|base= 2|e=20}}).

## E notation

Most calculators
Calculator
An electronic calculator is a small, portable, usually inexpensive electronic device used to perform the basic operations of arithmetic. Modern calculators are more portable than most computers, though most PDAs are comparable in size to handheld calculators.The first solid-state electronic...

and many computer programs
Computer program
A computer program is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute...

present very large and very small results in scientific notation. Because superscripted
Subscript and superscript
A subscript or superscript is a number, figure, symbol, or indicator that appears smaller than the normal line of type and is set slightly below or above it – subscripts appear at or below the baseline, while superscripts are above...

exponents like 107 cannot always be conveniently represented, the letter E or e
E
E is the fifth letter and a vowel in the basic modern Latin alphabet. It is the most commonly used letter in the Czech, Danish, Dutch, English, French, German, Hungarian, Latin, Norwegian, Spanish, and Swedish languages.-History:...

is often used to represent times ten raised to the power of (which would be written as "x 10b") and is followed by the value of the exponent. Note that in this usage the character e is not related to the mathematical constant e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

or the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

ex (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++
C++
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...

, FORTRAN
Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...

, MATLAB
MATLAB
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...

, Perl
Perl
Perl is a high-level, general-purpose, interpreted, dynamic programming language. Perl was originally developed by Larry Wall in 1987 as a general-purpose Unix scripting language to make report processing easier. Since then, it has undergone many changes and revisions and become widely popular...

, Java
Java (programming language)
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities...

and Python
Python (programming language)
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...

programming languages, 6.0221418E23 or 6.0221418e23 is equivalent to
Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in standard decimal notation
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

. 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 in the form of
(a times ten raised to the power of b), where the exponent
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

b is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

, and the coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

a is any real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

(however, see normalized notation below), called the significand
Significand
The significand is part of a floating-point number, consisting of its significant digits. Depending on the interpretation of the exponent, the significand may represent an integer or a fraction.-Examples:...

or mantissa
Mantissa
* The mantissa is the significand in a common logarithm or floating-point number.* Metaphorically, it is the part of the self that eludes conscious awareness or self-understanding.* An addition of little importance.Mantissa may also refer to:...

. The term "mantissa" may cause confusion, however, because it can also refer to the fractional
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

part of the common logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

. If the number is negative then a minus sign precedes a (as in ordinary decimal notation).
Standard decimal notation Normalized scientific notation
300 {{val|3|e=2}}
4,000 {{val|4|e=3}}
-53,000 {{val|-5.3|e=4}}
6,720,000,000 {{val|6.72|e=9}}
0.000 000 007 51 {{val|7.51|e=-9}}

## Normalized notation

Any given number can be written in the form of {{gaps|a|e=b}} in many ways; for example, 350 can be written as {{val|3.5|e=2}} or {{val|35|e=1}} or {{val|350|e=0}}.
It does not have to be a whole number then a decimal.

In normalized scientific notation, the exponent b is chosen such that the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

of a remains at least one but less than ten (1 ≤ |a| < 10). Following these rules, 350 is always written as {{val|3.5|e=2}}. 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
Order of magnitude
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. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount...

. In normalized notation the exponent b is negative for a number with absolute value between 0 and 1 (e.g., negative one half is written as {{val|-5|e=-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.

Normalized scientific form is the typical form of expression of large numbers for many fields, except during intermediate calculations or when an unnormalised form, such as engineering notation
Engineering notation
Engineering notation is a version of scientific notation in which the powers of ten must be multiples of three...

, is desired. (Normalized) scientific notation is often called exponential
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

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 {{gaps|315|base= 2|e=20}}).

## E notation

Most calculators
Calculator
An electronic calculator is a small, portable, usually inexpensive electronic device used to perform the basic operations of arithmetic. Modern calculators are more portable than most computers, though most PDAs are comparable in size to handheld calculators.The first solid-state electronic...

and many computer programs
Computer program
A computer program is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute...

present very large and very small results in scientific notation. Because superscripted
Subscript and superscript
A subscript or superscript is a number, figure, symbol, or indicator that appears smaller than the normal line of type and is set slightly below or above it – subscripts appear at or below the baseline, while superscripts are above...

exponents like 107 cannot always be conveniently represented, the letter E or e
E
E is the fifth letter and a vowel in the basic modern Latin alphabet. It is the most commonly used letter in the Czech, Danish, Dutch, English, French, German, Hungarian, Latin, Norwegian, Spanish, and Swedish languages.-History:...

is often used to represent times ten raised to the power of (which would be written as "x 10b") and is followed by the value of the exponent. Note that in this usage the character e is not related to the mathematical constant e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

or the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

ex (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++
C++
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...

, FORTRAN
Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...

, MATLAB
MATLAB
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...

, Perl
Perl
Perl is a high-level, general-purpose, interpreted, dynamic programming language. Perl was originally developed by Larry Wall in 1987 as a general-purpose Unix scripting language to make report processing easier. Since then, it has undergone many changes and revisions and become widely popular...

, Java
Java (programming language)
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities...

and Python
Python (programming language)
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...

programming languages, 6.0221418E23 or 6.0221418e23 is equivalent to {{val. FORTRAN also uses "D" to signify double precision numbers.
• The ALGOL 60
ALGOL
ALGOL is a family of imperative computer programming languages originally developed in the mid 1950s which greatly influenced many other languages and became the de facto way algorithms were described in textbooks and academic works for almost the next 30 years...

programming language uses a subscript ten "10" character instead of the letter E, for example: {{nowrap begin}}6.02214151023{{nowrap end}}.
• The ALGOL 68
ALGOL 68
ALGOL 68 isan imperative computerprogramming language that was conceived as a successor to theALGOL 60 programming language, designed with the goal of a...

programming language has the choice of 4 characters: e, E, \, or 10. By examples: {{nowrap begin}}6.0221415e23{{nowrap end}}, {{nowrap begin}}6.0221415E23{{nowrap end}}, {{nowrap begin}}6.0221415\23{{nowrap end}} or {{nowrap begin}}6.02214151023{{nowrap end}}.

{{SpecialChars
| alt = Decimal Exponent Symbol
| special = Unicode 6.0 "Miscellaneous Technical" characters
| characters = something like "₁₀" (Decimal Exponent Symbol U+23E8 TTF)
}}
• Decimal Exponent Symbol is part of "The Unicode Standard 6.0" e.g. {{nowrap begin}}6.0221415⏨23{{nowrap end}} - it was included to accommodate usage in the programming languages Algol 60 and Algol 68.
• The TI-83 series
TI-83 series
The TI-83 series of graphing calculators is manufactured by Texas Instruments.The original TI-83 is itself an upgraded version of the TI-82. Released in 1996, it is one of the most used graphing calculators for students...

and TI-84 Plus series
TI-84 Plus series
The TI-84 Plus is a graphing calculator made by Texas Instruments which was released in early 2004. There is no original TI-84, only the TI-84 Plus and TI-84 Plus Silver Edition models. It is an enhanced version of the TI-83 Plus. The key-by-key correspondence is relatively the same, but the 84...

of calculators use a stylized E character to display decimal exponent and the 10 character to denote an equivalent Operator
Operator (programming)
Programming languages typically support a set of operators: operations which differ from the language's functions in calling syntax and/or argument passing mode. Common examples that differ by syntax are mathematical arithmetic operations, e.g...

• The Simula
Simula
Simula is a name for two programming languages, Simula I and Simula 67, developed in the 1960s at the Norwegian Computing Center in Oslo, by Ole-Johan Dahl and Kristen Nygaard...

programming language requires the use of & (or && for long
Double precision
In computing, double precision is a computer number format that occupies two adjacent storage locations in computer memory. A double-precision number, sometimes simply called a double, may be defined to be an integer, fixed point, or floating point .Modern computers with 32-bit storage locations...

), for example: {{nowrap begin}}6.0221415&23{{nowrap end}} {{nowrap begin}}(or 6.0221415&&23){{nowrap end}}.

## Engineering notation

{{Main|Engineering notation}}
Engineering notation
Engineering notation
Engineering notation is a version of scientific notation in which the powers of ten must be multiples of three...

differs from normalized scientific notation in that the exponent b is restricted to multiples
Multiple (mathematics)
In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for some integer n , which is called the multiplier or coefficient. If a is not zero, this is equivalent to saying that b/a is an integer...

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, {{val|12.5|e=-9|u=m}} can be read as "twelve-point-five nanometers" or written as {{val|12.5|u=nm}}, while its scientific notation counterpart {{val|1.25|e=-8|u=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
Space (punctuation)
In writing, a space is a blank area devoid of content, serving to separate words, letters, numbers, and punctuation. Conventions for interword and intersentence spaces vary among languages, and in some cases the spacing rules are quite complex....

(which in typesetting
Typesetting
Typesetting is the composition of text by means of types.Typesetting requires the prior process of designing a font and storing it in some manner...

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.

### Examples

• An electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

's mass is about {{gaps|0.000|000|000|000|000|000|000|000|000|000|910|938|22}} kg. In scientific notation, this is written {{val|9.1093822|e=-31|u=kg}}.
• The Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

's mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

is about {{gaps|5|973|600|000|000|000|000|000|000}} kg. In scientific notation, this is written {{val|5.9736|e=24|u=kg}}.
• The Earth's circumference is approximately {{gaps|40|000|000}} m. In scientific notation, this is {{val|4|e=7|u=m}}. In engineering notation, this is written {{val|40|e=6|u=m}}. In SI writing style
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

, this may be written "{{val|40|u=Mm}}" (40 megameters).
• An inch
Inch
An inch is the name of a unit of length in a number of different systems, including Imperial units, and United States customary units. There are 36 inches in a yard and 12 inches in a foot...

is {{gaps|25|400}} micrometers
Micrometre
A micrometer , is by definition 1×10-6 of a meter .In plain English, it means one-millionth of a meter . Its unit symbol in the International System of Units is μm...

. Describing an inch as {{val|2.5400|e=4|u=µm}} unambiguously states that this conversion is correct to the nearest micrometer. An approximated value with only three significant digits would be {{val|2.54|e=4|u=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 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 {{val|1.602176487|(40)|e=-19|ul=C}}, which is shorthand for {{val|1.602176487|0.000000040|e=-19|u=C}}.

### Order of magnitude

Scientific notation also enables simpler order-of-magnitude comparisons. A proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

's mass is {{gaps|0.000|000|000|000|000|000|000|000|001|6726}} kg. If this is written as {{val|1.6726|e=-27 |u=kg}}, it is easier to compare this mass with that of the electron, given above. The order of magnitude
Order of magnitude
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. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount...

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 {{gaps|10|000}} times) more massive than the electron.

Scientific notation also avoids misunderstandings due to regional differences in certain quantifiers, such as billion
Long and short scales
The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale...

, which might indicate either 109 or 1012.

### 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 10n. For example, starting with {{gaps|1|230|000}}, move the decimal point six places to the left yielding 1.23, and multiply by 106, to give the result {{val|1.23|e=6}}. Similarly, starting with {{val|0.000000456}}, move the decimal point seven places to the right yielding 4.56, and multiply by 10−7, to give the result {{val|4.56|e=-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, {{val|1.234|e=0}} 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 {{gaps|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 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:
{{val|1.234|e=3}} = {{val|12.34|e=2}} = {{val|123.4|e=1}} = 1234

### Basic operations

Given two numbers in scientific notation,

and

Multiplication
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....

and division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

are performed using the rules for operation with exponential functions:

and

Some examples are:
and

Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

and subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...

require the numbers to be represented using the same exponential part, so that the significant can be simply added or subtracted. :

Next, add or subtract the significants:

An example:

• Engineering notation
Engineering notation
Engineering notation is a version of scientific notation in which the powers of ten must be multiples of three...

• Binary prefix
Binary prefix
In computing, a binary prefix is a specifier or mnemonic that is prepended to the units of digital information, the bit and the byte, to indicate multiplication by a power of 2...

• Floating point
Floating point
In computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...

• ISO 31-0
ISO 31-0
ISO 31-0 is the introductory part of international standard ISO 31 on quantities and units. It provides guidelines for using physical quantities, quantity and unit symbols, and coherent unit systems, especially the SI...

• ISO 31-11
ISO 31-11
ISO 31-11 was the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology...

• Significant figure
• Scientific pitch notation
Scientific pitch notation
Scientific pitch notation is one of several methods that name the notes of the standard Western chromatic scale by combining a letter-name, accidentals, and a number identifying the pitch's octave...