Scientific notation is a way of expressing

numbers A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

that are too large or too small (usually would result in a long string of digits) to be conveniently written in

decimal form A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...

. It may be referred to as scientific form or standard index form, or standard form in the United Kingdom. This

base ten The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

notation is commonly used by scientists, mathematicians, and engineers, in part because it can simplify certain

arithmetic operations Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th cen ...

. On scientific calculators it is usually known as "SCI" display mode. In scientific notation, nonzero numbers are written in the form or ''m'' times ten raised to the power of ''n'', where ''n'' is an

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

, and the

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...

''m'' is a nonzero

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

(usually between 1 and 10 in absolute value, and nearly always written as a

terminating decimal A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if an ...

). The integer ''n'' is called the

exponent Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...

and the real number ''m'' is called the ''

significand The significand (also mantissa or coefficient, sometimes also argument, or ambiguously fraction or characteristic) is part of a number in scientific notation or in floating-point representation, consisting of its significant digits. Depending on ...

'' or ''mantissa''. The term "mantissa" can be ambiguous where logarithms are involved, because it is also the traditional name of the

fractional part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...

of the

common logarithm In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered i ...

. If the number is negative then a minus sign precedes ''m'', as in ordinary decimal notation. In normalized notation, the exponent is chosen so that the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

(modulus) of the significand ''m'' is at least 1 but less than 10.

Decimal floating point Decimal floating-point (DFP) arithmetic refers to both a representation and operations on decimal floating-point numbers. Working directly with decimal (base-10) fractions can avoid the rounding errors that otherwise typically occur when convert ...

is a computer arithmetic system closely related to scientific notation.

Normalized notation

Any given real number can be written in the form in many ways: for example, 350 can be written as or or . In ''normalized'' scientific notation (called "standard form" in the United Kingdom), the exponent ''n'' is chosen so that the

of ''m'' remains at least one but less than ten (1 ≤ , ''m'', < 10). Thus 350 is written as . This form allows easy comparison of numbers: numbers with bigger exponents are (due to the normalization) larger than those with smaller exponents, and subtraction of exponents gives an estimate of the number of

orders of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...

separating the numbers. It is also the form that is required when using tables of

s. In normalized notation, the exponent ''n'' is negative for a number with absolute value between 0 and 1 (e.g. 0.5 is written as ). The 10 and exponent are often omitted when the exponent is 0. Normalized scientific form is the typical form of expression of large numbers in many fields, unless an unnormalized or differently normalized form, such as

engineering notation Engineering notation or engineering form (also technical notation) is a version of scientific notation in which the exponent of ten must be divisible by three (i.e., they are powers of a thousand, but written as, for example, 106 instead of 1000 ...

, is desired. Normalized scientific notation is often called

exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above * Exponential decay, decrease at a rate proportional to value *Exp ...

notation—although the latter term is more general and also applies when ''m'' is not restricted to the range 1 to 10 (as in engineering notation for instance) and to bases other than 10 (for example, ).

Engineering notation

Engineering notation (often named "ENG" on scientific calculators) differs from normalized scientific notation in that the exponent ''n'' is restricted to multiples of 3. Consequently, the absolute value of ''m'' is in the range 1 ≤ , ''m'', < 1000, rather than 1 ≤ , ''m'', < 10. Though similar in concept, engineering notation is rarely called scientific notation. Engineering notation allows the numbers to explicitly match their corresponding

SI prefixes A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pre ...

, which facilitates reading and oral communication. For example, can be read as "twelve-point-five nanometres" and written as , while its scientific notation equivalent would likely be read out as "one-point-two-five times ten-to-the-negative-eight metres".

Significant figures

A significant figure is a digit in a number that adds to its precision. This includes all nonzero numbers, zeroes between significant digits, and zeroes indicated to be significant. Leading and trailing zeroes are not significant digits, because they exist only to show the scale of the number. Unfortunately, this leads to ambiguity. The number is usually read to have five significant figures: 1, 2, 3, 0, and 4, the final two zeroes serving only as placeholders and adding no precision. The same number, however, would be used if the last two digits were also measured precisely and found to equal 0 — seven significant figures. When a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but the placeholding zeroes are no longer required. Thus would become if it had five significant digits. If the number were known to six or seven significant figures, it would be shown as or . Thus, an additional advantage of scientific notation is that the number of significant figures is unambiguous.

Estimated final digits

It is customary in scientific measurement to record all the definitely known digits from the measurement and to estimate at least one additional digit if there is any information at all available on its value. The resulting number contains more information than it would without the extra digit, which may be considered a significant digit because it conveys 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 notation. It is often useful to know how exact the final digit is. For instance, the accepted value of the mass of the

proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...

can properly be expressed as , which is shorthand for .

E notation

Most

calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...

s and many

computer program A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer program ...

s present very large and very small results in scientific notation, typically invoked by a key labelled (for ''exponent''), (for ''enter exponent''), , , , or depending on vendor and model. Because superscripted exponents like 10⁷ cannot always be conveniently displayed, the letter ''E'' (or ''e'') is often used to represent "times ten raised to the power of" (which would be written as ) and is followed by the value of the exponent; in other words, for any real number ''m'' and integer ''n'', the usage of "''m''E''n''" would indicate a value of ''m'' × 10^''n''. In this usage the character ''e'' is not related to the mathematical constant ''e'' or the

exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...

''e''^''x'' (a confusion that is unlikely if scientific notation is represented by a capital ''E''). Although the ''E'' stands for ''exponent'', the notation is usually referred to as ''(scientific) E notation'' rather than ''(scientific) exponential notation''. The use of E notation facilitates data entry and readability in textual communication since it minimizes keystrokes, avoids reduced font sizes and provides a simpler and more concise display, but it is not encouraged in some publications.

Examples and other notations

* Since its first version released for the

IBM 704 The IBM 704 is a large digital mainframe computer introduced by IBM in 1954. It was the first mass-produced computer with hardware for floating-point arithmetic. The IBM 704 ''Manual of operation'' states: The type 704 Electronic Data-Pro ...

in 1956, the Fortran language has used E notation for floating point numbers. It was not part of the preliminary specification as of 1954. * The E notation was already used by the developers of

SHARE Operating System The SHARE Operating System (SOS) is an operating system introduced in 1959 by the SHARE user group. It is an improvement on the General Motors GM-NAA I/O operating system, the first operating system for the IBM 704. The main objective was to im ...

(SOS) for the

IBM 709 The IBM 709 was a computer system, initially announced by IBM in January 1957 and first installed during August 1958. The 709 was an improved version of its predecessor, the IBM 704, and was the third of the IBM 700/7000 series of scientific com ...

in 1958. * In most popular programming languages, (or ) is equivalent to , and would be written (e.g.

Ada Ada may refer to: Places Africa * Ada Foah, a town in Ghana * Ada (Ghana parliament constituency) * Ada, Osun, a town in Nigeria Asia * Ada, Urmia, a village in West Azerbaijan Province, Iran * Ada, Karaman, a village in Karaman Province, Tur ...

, Analytica,