MHS Chemistry
Scientific Notation Rules

Measurements in science are often of very tiny amounts (like the radius of an atom) or very large amounts (like the radius of the Earth's orbit). These numbers can be represented in several ways. One way is to use measuring units that are about the correct size. In that case, the radius of a neon atom is about 0.35 angstroms, and the radius of the Earth's orbit is 1 Astronomical Unit. But what's an angstrom? And what is an Astronomical Unit? And how would we work with these two numbers in the same problem?

The solution is to measure them based on the same units. In that case, the radius of the neon atom is about 0.000 000 000 035 meters, and the radius of the Earth's orbit is about 150 000 000 000 meters. This at least lets you compare the two numbers, but it's still not a very convenient way to write them.

The solution to that is to use scientific notation. In scientific notation, the significant digits are always all shown, with the decimal point after the first one. This number is shown multiplied by 10 raised to the appropriate power to give the number the correct value. For the radius of the Earth's orbit, this would be 1.5x1011 m.

For a number smaller than one, the exponent is a negative number, to show that the significant part of the numer is divided by 10 a certain number of times. The radius of a neon atom is therefore written as 3.5x10-11 m.

For ease of typing (and writing) these can also be written using the letter "p" for "times 10 to the positive" and "n" for "times 10 to the negative." The radius of the Earth's orbit would be 1.5p11 m and the radius of the neon atom would be 3.5n11 m.

To change the appearance of a number that is in scientific notation to floating point notation ("normal numbers"), use the following rules:

• If the exponent is positive, move the decimal place to the right that many times. Add zeroes if necessary, but if you do remember not to write the decimal point.
• If the exponent is negative, move the decimal place to the left that many times. Add zeroes if necessary, but make sure you do write the decimal point.

Extra
One of the nice things about scientific notation is that the exponents have common names in the SI (metric) system. For instance, any time the exponent is +3, you can use the prefix kilo-. For a complete list of metric prefixes, click here.

Also, many calculators have an option besides scientific or floating point (normal) notation called "engineering notation." This way of writing numbers is very similar to scientific notation, except that instead of requiring exactly one number before the decimal point, up to three digits are allowed as long as the exponent is a multiple of three. This way, the exponent can always be translated instantly to a metric prefix.