Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, or standard form in the UK. This base ten notation is commonly used by scientists, mathematicians, and engineers, in part because it can simplify certain arithmetic operations. On scientific calculators it is usually known as "SCI" display mode.
In scientific notation, all numbers are written in the form
- m × 10n
or m times ten raised to the power of n, where the exponent n is an integer, and the coefficient m is any real number. The integer n is called the order of magnitude and the real number m is called the significand or mantissa. The term "mantissa" may cause confusion because it is the name of the fractional part of the common logarithm. If the significand 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 (modulus) of the significand m is at least 1 but less than 10.
Decimal floating point is a computer arithmetic system closely related to scientific notation.
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 because they exist only to show the scale of the number. Therefore, 1230400 usually has five significant figures: 1, 2, 3, 0, and 4; the final two zeroes serve only as placeholders and add no precision to the original number.
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 place holding zeroes are no longer required. Thus 1230400 would become 1.23041230400 would become 1.2304×106. However, there is also the possibility that the number may be known to six or more significant figures, in which case the number would be shown as (for instance) 1.23040×106. Thus, an additional advantage of scientific notation is that the number of significant figures is clearer.
It is customary in scientific measurements to record all the definitely known digits from the measurements, and to estimate at least one additional digit if there is any information at all available to enable the observer to make an estimate. The resulting number contains more information than it would without that extra digit(s), and it (or they) 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 notations. It is often useful to know how exact the final digit(s) are. For instance, the accepted value of the unit of elementary charge can properly be expressed as 1.6021766208<
Additional information about precision can be conveyed through additional notations. It is often useful to know how exact the final digit(s) are. For instance, the accepted value of the unit of elementary charge can properly be expressed as 1.6021766208(98)×10−19 C, which is shorthand for (1.6021766208±0.0000000098)×10−19 C.
Most calculators and many computer programs present very large and very small results in scientific notation, typically invoked by a key labelled EXP (for exponent), EEX (for enter exponent), EE, EX, E, or ×10x 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 "× 10n") and is followed by the value of the exponent; in other words, for any two real numbers m and n, the usage of "mEn" would indicate a value of m × 10n. In this usage the character e is not related to the mathematical constant e or the exponential function ex (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
- The E notation was already used by the developers of SHARE Operating System (SOS) for the IBM 709 in 1958.
- In most popular programming languages,
6.022e23) is equivalent to 6.022×1023, and proton's mass is 0.0000000000000000000000000016726 kg. If written as 1.6726×10−27 kg, it is easier to compare this mass with that of an electron, given below. 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 (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.
In physics and astrophysics, the number of orders of magnitude between two numbers is sometimes referred to as "dex", a contraction of "decimal exponent" (see f.e. Chemical abundance ratios). For instance, if two numbers are within 1 dex of each other, then the ratio of the larger to the smaller number is less than 10. Fractional values can be used, so if within 0.5 dex, the ratio is less than 100.5, and so on.
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" is sometimes omitted, though it is less common to do so before the alphabetical character.
Further examples of scientific notation
- An electron's mass is about 0.000000000000000000000000000000910938356 kg. In scientific notation, this is written 9.10938356×10−31 kg (in SI units).
- The Earth's mass is about 5972400000000000000000000 kg. In scientific notation, this is written 5.9724×1024 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 megametres).
- An inch is defined as exactly 25.4 mm. Quoting a value of 25.400 mm shows that the value is correct to the nearest micrometre. An approximated value with only two significant digits would be 2.5×101 mm instead. As there is no limit to the number of significant digits, the length of an inch could, if required, be written as (say) 2.54000000000×101 mm instead.
- Hyperinflation is a problem that is caused when too much money is printed with regards to there being too few commodities, causing the inflation rate to rise by 50% or more in a single month; currencies tend to lose their intrinsic value over time. Some countries have had an inflation rate of 1 million percent or more in a single month, which usually results in the abandonment of the country's currency shortly afterwards. In November 2008, the monthly inflation rate of the Zimbabwean dollar reached 79.6 billion percent; the approximated value with three significant figures would be 7.96×1010 percent.
Converting a number in these cases means to either convert the number into scientific notation form, convert it back into decimal form or to change the exponent part of the equation. None of these alter the actual number, only how it's expressed.
Decimal to scientific
First, move the decimal separator point sufficient places, n, to put the number's value within a desired range, between 1 and 10 for normalized notation. If the decimal was moved to the left, append
× 10n; to the right,
× 10−n. To represent the number 1,230,400 in normalized scientific notation, the decimal separator would be moved 6 digits to the left and
× 106 appended, resulting in 1.2304×106. The number −0.0040321 would have its decimal separator shifted 3 digits to the right instead of the left and yield −4.0321×10−3 as a result.
Scientific to decimal
Converting a number from scientific notation to decimal notation, first remove the
× 10n on the end, then shift the decimal separator n digits to the right (positive n) or left (negative n). The number 1.2304×106 would have its decimal separator shifted 6 digits to the right and become 1,230,400, while −4.0321×10−3 would have its decimal separator moved 3 digits to the left and be −0.0040321.
Conversion between different scientific notation representations of the same number with different exponential values is achieved by performing opposite operations of multiplication or division by a power of ten on the significand and an subtraction or addition of one on the exponent part. The decimal separator in the significand is shifted x places to the left (or right) and x is added to (or subtracted from) the exponent, as shown below.
- 1.234×103 = 12.34×102 = 123.4×101 = 1234
Given two numbers in scientific notation,