What is scientific notation?

The concept of very large or very small numbers is something that is difficult for many students to comprehend. In general, students have difficulty with two things when dealing with numbers that have more zeros (either before OR after the decimal point) than they are used to. They often do not understand:

1. that "big" and "small" are relative terms,
2. the concept of "order of magnitude"

Scientific notation is a way to assess the order of magnitude and to visually decrease the zeros that the student sees. It also may help students to compare very large (or very small numbers) But students still have little intuition about scientific notation. Teaching them to recognize that scientific notation is a short hand way to better understand big and small numbers can be useful to them in all aspects of their academic career.

• Geologic time/Deep time
• The metric system
• Radioactive decay/Radiometric dating

Because everyone has different ways of learning, mathematicians have defined a number of ways that quantitative concepts can be represented to individuals. Here are some ways that big numbers and scientific notation can be represented.

• Verbal representation
We represent large (and small) numbers with modifiers that tell us how many zeros are attached to the number. One million (1. x 10⁶) tells us that the decimal (after the 1) moves 6 places to the right (1,000,000.). One thousandth (1. x 10^-3) tells us that the decimal moves 3 spaces to the left (0.001).

Show list of numbers and their modifiers

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number	power of ten	modifier word
0.000001	10-6	millionths
0.0001	10-4
0.001	10-3	thousandths
0.01	10-2	hundreths
0.1	10-1	tenths
1	100
10	101
100	102	hundred
1,000	103	thousand
1,000,000	106	million
1,000,000,000	109	billion
1,000,000,000,000	1012	trillion

• Numerical representation
4.5 billion years looks like this:

4500000000 years or 4,500,000,000 years
That's a lot of zeros!
We can also write that number in scientific notation:

4.5 X 10⁹ yrsor (on a calculator) 4.5E9 yrs
(1 billion in scientific notation means
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10!)
What does that number actually mean? Can you actually imagine 4.5 billion of anything?
• Graphical representation

Although many topics in geoscience use scientific notation, graphical representations are often plotted on logarithmic scales as in the electromagnetic spectrum below.



Log scale of electromagnetic spectrum. Details




There are many reasons for this - most importantly, it takes up less space - but a log scale does not give a sense of true scale:



Bar illustrating true scale of powers of ten. Details

Geologic timeis a great context in which to talk about big numbers and to give a sense of scale. We have the geologic time scale to give students a sense of what parts of "time" we know the most about. And how big "time" really is.

Here is a geologic time scale that gives a sense of scale



Geologic time scale. Details


It is interesting to note that the time periods that we know the most about (the Cenozoic, Mesozoic and Paleozoic) only make up a small fraction of the whole of geologic time!
• Symbolic representation
There are a number of analogies for thinking about big numbers. Students (particularly in entry-level classes) often relate well to analogies with money:

Money analogies:

A local radio station is tring to collect a million pennies for a worthy cause. How many dollars is this?
Answer: $10,000.

You have just won a contest. The prize is that you will get $1 bills from a bank teller at the rate of 1 per second for as long as you can stand there taking them (no bathroom breaks, no napping, etc.). How long will you have to stand there to become a millionaire?
Answer: about 12 days.

A very rich man decides that he will help to pay off the national debt by paying $1 into the coffers of the US at the rate of one per second. How long will it take him to pay off the debt (provided no more debt accumulates). (You can check the debt (to the penny) at the Treasury Dept. - over 7.6 trillion dollars in Feb 2005)?
Answer: about 32,000 years per trillion dollars

• Calculators:
Scientific notation provides an excellent way for students to become familiar with their (probably new) calculators. Many students do not know what scientific notation is and have little intuition about it. Even fewer really know how to use their calculators to express it. Often I see students just ignoring "the little number in the upper right" and so their answer is off by orders of magnitude. Have students figure out how to make their calculators express scientific notation. A quick exercise to do in class is have them multiply some numbers in scientific notation on their calculator and then write the answer in numerical notation.
• Computers:
Many spreadsheets can be formatted to display numbers numerically and in scientific notation. Showing students a table of these data expressed both ways may help them to get a sense of what is happening "in their calculators" when numbers are displayed in scientific notation. These numbers can also be plotted so that students can get a sense of scale.

Mathematicians also indicate that students learn quantitative concepts better when they work in groups and revisit a concept on more than one day. Therefore, when discussing quantitative concepts in entry-level geoscience courses, have students discuss or practice the concepts together. Also, make sure that you either include problems that may be extended over more than one class period or revisit the concept on numerous occasions.

Very big and very small numbers come up over and over in introductory geoscience: Geologic time, radiometric dating, unit conversions (especially metric), etc. When each new topic is introduced, make sure to point out that this is not new material, it's just a matter of remembering.

• Jerry Johnson of University of Nevada at Reno has an excellent example of Understanding Big Numbers that applies to population growth and government spending. This file includes student exercises and instructor notes.
• In the Quantitative Skills site, Barb Tewksbury discusses Back of the Envelope Calculations and has a number of exercises that give students practice in quantitative skills and in estimating numbers.
• A View from the Back of the Envelope has a number of links to interesting ways to explain (and estimate) big numbers and orders of magnitude.

Geomathshas a short page on scientific notation (more info) of large numbers.

MathWorld has a more mathematical explanation (more info) of scientific notation with a link to significant digits (more info) .