# Scientific Notation

## Scientific notation expresses a number as a·10b, where a has one digit to the left of the decimal.

#### Key Points

• Scientific notation is a way of writing numbers that are too big or too small to be conveniently written in decimal form.

• In normalized scientific notation, the exponent b is chosen so that the absolute value of a remains at least one but less than ten (1 ≤ |a| < 10). Following these rules, 350 would always be written as 3.5×102.

• Most calculators present very large and very small results in scientific notation. 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 "x 10b").

#### Terms

• A method of writing or of displaying real numbers as a decimal number between 1 and 10 followed by an integer power of 10

#### Examples

• The number 981 is actually 981., and it is followed by a decimal point. In integers, the decimal point at the end is usually omitted. 981=981.=9.81×102. The decimal point is now two places to the left of its original position, and the power of 10 is 2.

#### Figures

1. ##### Standard Form Versus Scientific Form

In standard form, the number is written out as you are accustomed to, the ones digit to the farthest to the right (unless there is a decimal), then the tens digit to the left of the ones, and so on. In scientific notation, a number in standard notation with one nonzero digit to the left of the decimal is multiplied by ten to some power, as shown.

## Standard Form to Scientific Form

Very large numbers such as 43,000,000,000,000,000,000 (the number of different possible configurations of Rubik’s cube) and very small numbers such as 0.000000000000000000000340 (the mass of the amino acid tryptophan) are extremely inconvenient to write and read. Such numbers can be expressed more conveniently by writing them as part of a power of 10.

To see how this is done, let us start with a somewhat smaller number such as 2480. See Figure 1.

The last form in Figure 1 is called the scientific form of the number. There is one nonzero digit to the left of the decimal point and the absolute value of the exponent on 10 records the number of places the original decimal point was moved to the left. If instead we have a very small number, such as 0.00059, we instead move the decimal place to the right, as in the following:

$0.00059=\frac {0.0059}{10}=\frac {0.0059}{10^1}=0.0059\cdot 10 ^{-1}$

$0.00059=\frac {0.059}{100}=\frac {0.059}{10^2}=0.059\cdot 10 ^{-2}$

$0.00059=\frac {0.59}{1000}=\frac {0.59}{10^3}=0.59\cdot 10 ^{-3}$

$0.00059=\frac {5.9}{10000}=\frac {5.9}{10^4}=5.9\cdot 10 ^{-4}$

There is one nonzero digit to the left of the decimal point and the absolute value of the exponent of 10 records the number of places the original decimal point was moved to the right.

## Writing a Number in Scientific Notation

To write a number in scientific notation:

• Move the decimal point so that there is one nonzero digit to its left.
• Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.

A number written in scientific notation can be converted to standard form by reversing the process described above.

## Normalized Scientific Notation

Any given number can be written in the form of a×10b in many ways; for example, 350 can be written as 3.5×102 or 35×101 or 350×100. In normalized scientific notation, the exponent b is chosen so that the absolute value of a remains at least one but less than ten (1 ≤ |a| < 10). Following these rules, 350 would always be written as 3.5×102. 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. 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 −5×10−1). The 10 and exponent are usually omitted when the exponent is 0. Note that 0 cannot be written in normalized scientific notation since it cannot be expressed as a×10b for any non-zero a. 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, is desired. Normalized scientific notation is often called exponential 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 315× 220).

### E Notation

Most calculators and many computer programs present very large and very small results in scientific notation. 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 "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 or the exponential function 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. The use of this notation is not encouraged by publications.

#### Key Term Glossary

absolute value
For a real number, its numerical value without regard to its sign; formally, -1 times the number if the number is negative, and the number unmodified if it is zero or positive.
##### Appears in these related concepts:
base
A number raised to the power of an exponent.
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constant
An identifier that is bound to an invariant value.
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e
The base of the natural logarithm, 2.718281828459045…
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exponent
The power to which a number, symbol, or expression is to be raised. For example, the 3 in $x^3$.
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exponential
Any function that has an exponent as an independent variable.
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exponential function
Any function in which an independent variable is in the form of an exponent; they are the inverse functions of logarithms.
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expression
An arrangement of symbols denoting values, operations performed on them, and grouping symbols. E.g. (2x+4)/2
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function
A relation in which each element of the input is associated with exactly one element of the output.
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integer
An element of the infinite and numerable set {...,-3,-2,-1,0,1,2,3,...}.
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point
An entity that has a location in space or on a plane, but has no extent
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range
The set of values (points) which a function can obtain.
##### Appears in these related concepts:
scientific notation
A method of writing or of displaying real numbers as a decimal number between 1 and 10 followed by an integer power of 10
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sign
positive or negative polarity.
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term
any value (variable or constant) or expression separated from another term by a space or an appropriate character, in an overall expression or table.
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zero
Also known as a root, a zero is an x value at which the function of x is equal to 0.