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The Basics of Physics
Significant Figures and Order of Magnitude
Physics Textbooks Boundless Physics The Basics of Physics Significant Figures and Order of Magnitude
Physics Textbooks Boundless Physics The Basics of Physics
Physics Textbooks Boundless Physics
Physics Textbooks
Physics
Concept Version 11
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Scientific Notation

Scientific notation is a way of writing numbers that are too big or too small in a convenient and standard form.

Learning Objective

  • Convert properly between standard and scientific notation and identify appropriate situations to use it


Key Points

    • Scientific notation means writing a number in terms of a product of something from 1 to 10 and something else that is a power of 10.
    • In scientific notation all numbers are written in the form of $a\cdot 10^{b}$ (a times ten raised to the power of b).
    • Each consecutive exponent number is ten times bigger than the previous one; negative exponents are used for small numbers.

Terms

  • 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

  • exponent

    The power to which a number, symbol or expression is to be raised. For example, the 3 in $x^3$.


Full Text

Scientific Notation: A Matter of Convenience

Scientific notation is a way of writing numbers that are too big or too small in a convenient and standard form. Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians and engineers. In scientific notation all numbers are written in the form of $a\cdot 10^{b}$  ($a$ multiplied by ten raised to the power of $b$), where the exponent $b$ is an integer, and the coefficient $a$ is any real number.

Scientific Notation

There are three parts to writing a number in scientific notation: the coefficient, the base, and the exponent.

Most of the interesting phenomena in our universe are not on the human scale. It would take about 1,000,000,000,000,000,000,000 bacteria to equal the mass of a human body. Thomas Young's discovery that light was a wave preceded the use of scientific notation, and he was obliged to write that the time required for one vibration of the wave was "$\displaystyle \frac{1}{500}$ of a millionth of a millionth of a second"; an inconvenient way of expressing the point. Scientific notation is a less awkward and wordy way to write very large and very small numbers such as these.

A Simple System

Scientific notation means writing a number in terms of a product of something from 1 to 10 and something else that is a power of ten.

For instance, $32 = 3.2\cdot 10^{1}$

$320 = 3.2\cdot 10^{2}$

$3200 = 3.2\cdot 10^{3}$, and so forth...

Each number is ten times bigger than the previous one. Since $10^{1}$ is ten times smaller than $10^{2}$, it makes sense to use the notation $10^{0}$ to stand for one, the number that is in turn ten times smaller than $10^{1}$. Continuing on, we can write $10^{-1}$ to stand for 0.1, the number ten times smaller than $10^{0}$. Negative exponents are used for small numbers:

$3.2=3.2\cdot 10^{0}$

$0.32=3.2\cdot 10^{-1}$

$0.032=3.2\cdot 10^{-2}$

Scientific notation displayed calculators can take other shortened forms that mean the same thing. For example, $3.2\cdot 10^{6}$ (written notation) is the same as $3.2\text{E+6}$ (notation on some calculators) and $3.2^{6}$ (notation on some other calculators).

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