Scientific notation is used to make large numbers easier to write. For example, the distance from the earth to the the sun is

150000000 km or 1.5 x 10 8 km

This is a very important distance in astronomy and it has a special name, the Astronomical Unit (AU). Distances in the Solar System are usually measured in terms of AU.

Scientific notation can also be used to write very small numbers. The mass of one hydrogen atom is

0.00000000000000000000000000167 kg or 1.67 x 10 -27 kg

Such huge and small numbers occur in astronomy all the time.

In counting all those zeroes above, you might make a mistake, so it is also more reliable to use scientific notation. We know just by looking at the exponent how big, or small, the number is without counting the zeroes. The exponent is called the order of magnitude of the number. The distance to the sun is on the order of 108 km, and the mass of the hydrogen atom is on the order of 10-27 kg. When we look at the order of magnitude of a number, we are just looking at the exponent above the ten, and not at the numbers in front.
Use the following demonstration to become familiar with scientific notation. You can pick up the decimal place and move it through the large number. As you move the decimal point the exponent on the 10 will correct itself so that you will always have the correct abbreviated number in scientific notation. Pay attention to the exponent!
The equality above is always correct because as you lose a power of ten by moving the decimal point to the left you gain that power in the exponent on the ten. Even though all of the different places for the decimal point give you a correct answer the accepted way to do it right is to have the number to the left of the "x 10something" be between 1 and 10. So

14.959787 x 10 7 = 149,597,870

could be better and

0.14959787 x 10 9 = 149,597,870

could be better, too, but

1.4959787 x 10 8 = 149,597,870

is just right.

By making the number to the left of the "x 10something" between 1 and 10 you put all of the powers of ten into the exponent on the 10. That makes is very easy to see the order of magnitude of the number.

A useful abbreviation is to substitute " E " or " e " for " x 10 " as shown below

2.35 x 10 5 = 2.35e5 = 2.35E5

You can use this abbreviation to avoid writing a superscript exponent.
Calculators will usually express scientific notation in this way. Every calculator does scientific notation a little differently. If you have a fancy graphing calculator like a TI89 or HP48gII, then you will most likely have to set it to use scientific notation using a menu or mode option. For more simple calculators, there is often a button that switches modes. To enter numbers in scientific notation, look for a button marked EE or Exp. These buttons tell the calculator what the power of ten is.

EXAMPLE: To enter the number 9.989 x 1033 in your calculator, type the following:  "9.989 EE 33" or "9.989 EXP 33".

Be very careful not to confuse EE and Exp with the ex button. The ex button is for Euler's number (which is about 2.718) and tells the calculator that you want to raise 2.718 to some power. Typing "9.989 ex 33" means "9.989 x 2.71833."

Note that 100 is just 1. In fact, any number raised to the power zero is 1.

 

Multiplying and Dividing Numbers in Scientific Notation

To multiply numbers written in scientific notation, consider

10 x 10 x 10 = 10 1 x 10 1 x 10 1 = 1000 = 10 3

The exponents just add.  Likewise,

2 x 10 3 x 4 x 10 -3 = 2000 x 0.004 = 8 = 8 x10 0

which is found by multiplying 2 by 4 and adding the exponents 3 and -3.

Division works the same way, except you subtract the exponents, so

6 x 10 3 / 3 x 10 3 = 6000 / 3000 = 2 = 2 x10 0

which is found by dividing 6 by 3 and subtracting the exponents.

Taking the Tour

Clicking on the "Tour" button above will open a new window with an interactive exercise. You will see an image in the center and a scale bar to the right. Start by clicking the number 1. This will show you an old image of the Astronomy 101 Lab classroom.  Clicking larger numbers will take you further away from the classroom. Below the image you will see some numbers.  For images 0 through 6, these numbers are the distance from the classroom. For the rest of the images, these numbers are the size of the objects shown.  To the right of the image are the words "scientific notation" in green letters. If you click on this link, you will get the number under the image to scientific notation. Notice that the exponent on the ten and the scale number do not always match!

 

 

Questions:

List two reasons explaining why scientific notation is useful. In your own words, describe why these reasons are important.

1.
     

The following numbers are written in "true" scientific notation. Rewrite them below, using the "e" notation:

2. 2.998 x 108 =
     

3. 6.672 x 10-11 =
     


Write the following numbers in scientific notation (use the "e" notation - it is easier to type)

The mass of the Sun: 1,989,000,000,000,000,000,000,000,000,000 kg

4. The mass of the Sun:
     

Distance from the Sun to Earth: 149,600,000 km

5. Distance from the Sun to Earth:
     

Here are some questions to help you learn how to work with orders of magnitude.


Titan IVb

The Titan IVb rocket used in the United States space program is 62,200 millimeters tall (204 feet).

Courtesy NASA/JPL-Caltech

The Mars Spirit Rover is 1500 millimeters tall.

Write the height of the Titan IVb rocket and the height of the Mars Spirit Rover in scientific notation.

6. Titan IVb rocket -   millimeters
     

7. Mars Spirit Rover -   millimeters
     

What is the order of magnitude of

8. the height of the rocket? 
     

9. the height of the Rover? 
     

Now, just look at the orders of magnitude.

To the nearest order of magnitude, how many times taller is the rocket than the Rover?

10.
     

How does this compare to the exact answer of 62,200mm/1500mm = 41 times?

11.
     

Jupiter's mass is roughly 320 times the mass of Earth. By what order of magnitude do the two masses differ?

12.
     

The Sun's mass is about 343000 times the mass of Earth. By what order of magnitude do these two masses differ?

13.
     

Which one of the following is correct?

14. 1,800,000 g = 1.8E6 g

      2 Gm = 2E12 m

      13.0000 L = 1.3E-4 L

      46,000,000W = 4.6E6 W

How many times larger is the Solar System than the Earth?

15.
     

How many times larger is a cluster of galaxies than the Earth?

16.
     

Name one thing that you learned (about the relative scale of things in the universe) from the Powers of Ten film and slide show and explain why you think that it is significant.

17.