Scientific Notation
Scientific notation is used to express a very large or small number in the form
Learning Objective

Explain why scientific notation is useful in performing calculations with large or small numbers
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
$n$ is chosen so that the absolute value of$m$ remains at least 1 but less than 10$\left ( 1 \leq \left  m \right  < 10 \right )$ —i.e. so that$m$ has exactly one digit left of its decimal point.  When numbers written in scientific notation are involved in multiplication or division, the standard rules for operations with exponentiation apply. When addition or subtraction is involved, the numbers must first be rewritten so the exponents are the same.
 Most calculators 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 "$\cdot 10^b$ ").
Terms

scientific notation
A method of writing or of displaying real numbers as a decimal number between 1 and 10 multiplied by an integer power of 10.

normalized scientific notation
A number written in scientific notation
$m \cdot 10^n$ such that the absolute value of$m$ remains at least 1 but less than 10.
Full Text
Scientific notation, also known as "standard form," is a way to more conveniently write numbers that are very large or very small. This method is commonly used by mathematicians, scientists, and engineers.
For example, the numbers
Scientific notation is written as follows:
This is read "
How to Use 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.
For example, let's write the number 43,500 in scientific notation. There are four digits in this number, so the decimal should be moved 4 places to the left to leave one nonzero digit left of the decimal point:
The exponent is 4 because the decimal point was moved to the left (the exponent would be positive had the decimal been moved to the right) by exactly 4 places.A number written in scientific notation can also be converted to standard form by reversing the process described above. For example, let's write the number
To reverse the process, we move the decimal point three places to the left, adding leading zeroes where necessary.
Normalized Scientific Notation
Any given number can be written in the form of
In normalized scientific notation, also called exponential notation, the exponent
Following these rules, 350 would always be written as
Normalized scientific form is the typical form of expression for large numbers in many fields, except during intermediate calculations or when an unnormalized form, such as engineering notation, is desired.
Calculations involving Scientific Notation
When numbers written in scientific notation are multiplied or divided, the standard rules for operations with exponentiation apply. For example:
When numbers written in scientific notation are added to or subtracted from each other, the terms first must be rewritten so the exponents are the same. Then, the constant value, or
E Notation
Most calculators and many computer programs present very large and very small results in scientific notation. Because superscripted exponents like
Key Term Reference
 absolute value
 Appears in these related concepts: Piecewise Functions, Inequalities with Absolute Value, and Equations with Absolute Value
 base
 Appears in these related concepts: Temple Architecture in the Greek Orientalizing Period, Rules for Exponent Arithmetic, and Rational Exponents
 constant
 Appears in these related concepts: Graphing Quadratic Equations in Vertex Form, Inverse Variation, and Direct Variation
 e
 Appears in these related concepts: Natural Logarithms, Business Stakeholders: Internal and External, and The Number e
 exponent
 Appears in these related concepts: Logarithms of Products, Logarithms of Powers, and Logarithms of Quotients
 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: Interest Compounded Continuously, Graphs of Exponential Functions, Base e, and Basics of Graphing Exponential Functions
 exponentiation
 Appears in these related concepts: Introduction to Exponents, Logarithmic Functions, and Radical Functions
 expression
 Appears in these related concepts: Compound Inequalities, Sets of Numbers, and Simplifying, Multiplying, and Dividing
 function
 Appears in these related concepts: Visualizing Domain and Range, The Vertical Line Test, and Solving Differential Equations
 point
 Appears in these related concepts: Graphing Quadratic Equations In Standard Form, Graphing Equations, and Polynomial and Rational Functions as Models
 sign
 Appears in these related concepts: Polynomial Inequalities, The Intermediate Value Theorem, and The Rule of Signs
 term
 Appears in these related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Introduction to Variables
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