Scientific notation makes life easier... it really does!

You have all probably performed many calculations using scientific notation, but have you ever wondered why there is such a notation for expressing numbers? Scientific notation was not developed merely for frustrating algebra students, but has a real-world application.

The world of the very large and very small

In the world of the sciences, you will encounter numbers that are much larger and smaller than you deal with in your everyday life outside of school. It is not difficult to express, "the temperature outside right now is 28.5 degrees Celsius." While temperature is a value requiring only reporting in the units, tens, and occasionally hundreds place, reporting the number of atoms in 12.0 grams of carbon would require the number of 6023 followed by 20 zeros! It would appear as:

     602,300,000,000,000,000,000,000 atoms

Since such a value takes up way too much space on paper, and is a mouthful to state, it is instead written in scientific notation:

     6.023 x 10^23 atoms

In scientific notation, a number is written as the product of two numbers: (1) a coefficient, and, (2) 10 raised to a power. In the above example, 6.023 is the coefficient, and 10 is raised to the 23rd power.  If you count how many times the decimal point was moved to the left, then, it is 23, the same value as the exponent.

When writing numbers out using scientific notation, the coefficient is always a number greater than or equal to one and less than ten.

Scientific notation not only makes huge numbers manageable, but small ones as well. Consider the mass of a a single hydrogen atom:

     0.000,000,000,000,000,000,000,327 gram

It would be much less cumbersome to report the mass, using scientific notation accordingly:

​     3.27 x 10^-22 gram

Note the negative sign in front of the exponent 22; whenever there are negative exponents, the value of the number is less than one; whenever exponents are positive, the value of the number is 10 or greater. 

Unit unification and continuity

Scientific notation not only makes numbers more manageable, but enables us to use the same unit whenever describing the very large or very small, by using standardized units. Consider this, if reporting the length of a paper clip, you might state, "1.25 inches." Alternatively, if reporting the length of your driveway, you'd state something like, "65 feet." In order to report a length very small you used a unit that differed from that of a length very large. If reporting how far your house is from school, you'd use yet another unit - that of miles. 

The benefits of scientific notation is that we can use just one unit whenever measuring the length of anything. Using meters, we could state ranges as diverse as the radius of a hydrogen atom (1.2 x 10^-10 m) to the diameter of the Milky Way galaxy (9.5 x 10^20 m). Of course scientists do like to insert prefixes in front of these units, such as kilo-, milli-, and centi-, but even then these prefixes themselves will have the same value meaning whenever placed in front of different units (such as those for mass, volume, and time). We will have much more to say on these units and prefixes in later modules.

Check-off List of Things to Do:

  1. Read this entire webpage
  2. Watch the first two minutes of Professor Dave Explains Scientific Notation and Dimensional Analysis video linked right.
  3. Complete Practice Set 1 - Scientific Notation worksheet 
  4. Check answers against the Answer Key
  5. Complete Practice Set 2 - Scientific Notation Calculations
  6. Check answers against the Answer Key
  7. If you need additional help with your calculator, watch the How to Use a Scientific Calculator video again, where applicable. 
  8. Take the Module 3 Quiz

Module 3 Resources