Scientific notation expresses a number as a·10b, where a has one digit
to the left of the decimal.
LEARNING OBJECTIVE [ edit ]
Express numbers in scientific notation and standard notation
KEY POINTS [ edit ]
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).
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").
TERM [ edit ]
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
EXAMPLE [ edit ]
The number 981 is actually 981., and it is followed by a decimalpoint. 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.
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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.
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To see how this is done, let us start with a somewhat smaller number such as 2480.
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.
The last form in 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 =
0.00059 =
0.00059 =
0.00059 =
0.0059
10
0.059
100
0.59
1000
5.9
10000
=
=
=
=
0.0059
10
1
0.059
10
0.59
10
3
5.9
10
= 0.059
2
4
⋅ 10−
1
= 0.0059
= 0.59
= 5.9
⋅ 10−
2
⋅ 10−
3
⋅ 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
therange 1 to 10 (as in engineering notation for instance) and tobases 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.