1.30: Scientific Notation
Most of the stars pictured in this view of the night sky are hundreds of light years from Earth. A light year is the distance that light can travel in a year, or about 6 trillion (6,000,000,000,000) miles. As this example shows, quantities in science may be very large. Many other quantities in science are very small. Both very large and very small numbers have many zeroes, so they are hard to read and write without making mistakes. That’s where scientific notation comes in.
What Is Scientific Notation?
Scientific notation is a way of writing very large or very small numbers that uses exponents. Numbers are written in this format:
\begin{align*} a \times 10^b\end{align*}
The letter a stands for a decimal number, and the letter b stands for an exponent, or power, of 10. For example, the number 300 is written in scientific notation as 3.0 × 10^{2}. The number 0.03 is written as 3.0 × 10^{-2}. Do you need a review of exponents? If so, watch the video at this URL:
http://www.youtube.com/watch?v=8htcZca0JIA (9:45)
Using Scientific Notation
It’s easier to convert numbers to and from scientific notation than you might think. Refer to the examples on the chalkboard below as you read the following steps. Follow the steps in the sequence listed here to convert a number to scientific notation. Follow the steps in the reverse order to convert a number from scientific notation.
- Move the decimal point left or right until you reach the last nonzero digit. This new decimal number is a in a × 10^{b}.
- Count how many places you moved the decimal point in Step 1. This number is b in a × 10^{b}.
- Did you move the decimal point left (first example in the Figure below)? If so, b is positive. Did you move the decimal point right (second example in the Figure below)? If so, b is negative.
These two examples show how to convert large and small numbers to scientific notation.
Q: Apply the steps above to write 450,000 in scientific notation.
A: The unwritten decimal point in this number follows the last zero. Move the decimal point from this position to the left and stop just before the last digit, giving you 4.5 for a. The decimal point was moved five places to the left, so b is 5. In scientific notation the number is 4.5 × 10^{5}.
Q: Apply the steps in reverse order to write the number that is expressed as 7.2 × 10^{4} in scientific notation.
A: Add zeroes to 7.2 as you move the decimal point four places to the right. This gives you the number 72,000.
Summary
- Scientific notation is a way of writing very large or very small numbers that uses exponents. Numbers are written in the format a × 10^{b}.
- Changing numbers to or from scientific notation is easy to do by following three simple steps.
Vocabulary
- scientific notation: Way of writing very large or very small numbers that uses exponents in the format a × 10_{b}.
Practice
Go to this URL to practice converting numbers to and from scientific notation. Be sure to check your answers.
http://janus.astro.umd.edu/astro/scinote/
Review
- What is scientific notation? Why are numbers written in scientific notation?
- Write 0.0045 in scientific notation.
- What number is written as 6.0 × 10^{6} in scientific notation?
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Scientific Notation
Scientific notation is a means of representing a number as a product of a number that is at least 1 but less than 10 and a power of 10.order of magnitude
Formally, the order of magnitude is the exponent in scientific notation. Informally it refers to size. Two objects or numbers are of the same order of magnitude are relatively similar sizes.Standard Form
As opposed to scientific notation, standard form means writing numbers in the usual way with all of the zeros accounted for in the value.Image Attributions
- State the purpose of scientific notation.
- Explain how to convert numbers to and from scientific notation.