Scientific Notation
Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) 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.
| Decimal notation | Scientific notation |
|---|---|
| Template:Val | Template:Val |
| Template:Val | Template:Val |
| Template:Val | Template:Val |
| Template:Val | Template:Val |
| Template:Val | Template:Val |
| Template:Val | Template:Val |
| Template:Val | Template:Val |
| Template:Val | Template:Val |
In scientific notation, nonzero numbers are written in the form
- m × 10Template:Sup
or m times ten raised to the power of n, where n is an integer, and the coefficient m is a nonzero real number (usually between 1 and 10 in absolute value, and nearly always written as a terminating decimal). The integer n is called the exponent and the real number m is called the significand or mantissa.[1] The term "mantissa" can be ambiguous where logarithms are involved, because it is also the traditional name of the fractional part of the common logarithm. 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 (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.
Contents
Normalized notation
Template:Main Any given real number can be written in the form Template:Gaps in many ways: for example, 350 can be written as Template:Val or Template:Val or Template:Val.
In normalized scientific notation (called "standard form" in the UK), the exponent n is chosen so that the absolute value of m remains at least one but less than ten (Template:Nowrap begin1 ≤ |m| < 10Template:Nowrap end). Thus 350 is written as Template:Val. 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 separating the numbers. It is also the form that is required when using tables of common logarithms. 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 Template:Val). 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, is desired. Normalized scientific notation is often called exponential 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, Template:Gaps).
Engineering notation
Template:Main 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, which facilitates reading and oral communication. For example, Template:Val can be read as "twelve-point-five nanometres" and written as Template:Val, while its scientific notation equivalent Template:Val would likely be read out as "one-point-two-five times ten-to-the-negative-eight metres".
Significant figures
Template:Main 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 Template:Val 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 Template:Val would become Template:Val if it had five significant digits. If the number were known to six or seven significant figures, it would be shown as Template:Val or Template:Val. 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 can properly be expressed as Template:Val, which is shorthand for Template:Val.
E notation
Most calculators and many computer programs present very large and very small results in scientific notation, typically invoked by a key labelled Template:Button (for exponent), Template:Button (for enter exponent), Template:Button, Template:Button, Template:Button, or Template:Button 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 Template:Nowrap) 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.[2]
Resources
- Exponents and Scientific Notation, OpenStax
- Scientific Notation, Texas A&M University
- Scientific Notation, Math Is Fun
