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Scientific Notation
Scientific notation expresses a number as a·10^{b}, where a has one digit to the left of the decimal.
Learning Objective

Express numbers in scientific notation and standard notation
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).
 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

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
Full Text
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.
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:
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×10^{b} in many ways; for example, 350 can be written as 3.5×10^{2} or 35×10^{1} or 350×10^{0}. 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 3.5×10^{2}. 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×10^{b} for any nonzero 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 3^{15}×2^{20}).
E Notation
Most calculators and many computer programs present very large and very small results in scientific notation. Because superscripted exponents like 10^{7} 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 10^{b}") 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 e^{x} (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 Reference
 absolute value
 Appears in these related concepts: Piecewise Functions, Introduction to Absolute Value, and Equations with Absolute Value
 base
 Appears in these related concepts: Logarithms of Powers, The Role of the Kidneys in AcidBase Balance, and Biology: DNA Structure and Replication
 constant
 Appears in these related concepts: Inverse Variation, Quadratic Functions of the Form f(x) = a(xh)^2 + k, and Direct Variation
 e
 Appears in these related concepts: Derivatives of Exponential Functions, Natural Logarithms, and e
 exponent
 Appears in these related concepts: Integer Exponents, Logarithms of Products, and Scientific Notation
 exponential
 Appears in these related concepts: Exponential Growth and Decay, Solving General Problems with Logarithms and Exponents, and Population Growth
 exponential function
 Appears in these related concepts: Further Transcendental Functions, Graphs of Exponential Functions, Base e, and Basics of Graphing Exponential Functions
 expression
 Appears in these related concepts: Simplifying, Multiplying, and Dividing, Expressions and Sets of Numbers, and Bacterial Transformation
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Modal Mixture
 integer
 Appears in these related concepts: Exact Numbers, Scientific Notation, and Finding a Specific Term
 point
 Appears in these related concepts: Quadratic Functions of the Form f(x) = ax^2 + bx + c, Where a is not Equal to 0, Circles, and Graphing Equations
 range
 Appears in these related concepts: Range, The Derivative as a Function, and Visualizing Domain and Range
 sign
 Appears in these related concepts: The Intermediate Value Theorem, Sequences of Statements, and The Rule of Signs
 term
 Appears in these related concepts: Arithmetic Sequences, Basics of Graphing Polynomial Functions, and The 22nd Amendment
Sources
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Cite This Source
Source: Boundless. “Scientific Notation.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 22 Jul. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/thebuildingblocksofalgebra1/exponentsscientificnotationandtheorderofoperations10/scientificnotation664740/