## Friday, July 12, 2019

### Scientific Notation

By Donald G. McGahan, Ph.D.

September 18, 2010

Chemistry quantities and units are more specialized, or more refined, than everyday usages. Numbers that are very large, or very small, are frequently expressed in scientific notation such as the value for 1 mole of atoms (6.022 x 10²³ atoms). Scientific notation utilizes the powers of 10 and this has advantages because writing out very large numbers is
cumbersome especially when many zeros are involved.

Sometimes the term “order of magnitude” is used. While this can be any base unit the term is typically used in a base 10 context.

The value for a mole (6.022 x 10²³) can be converted from scientific notation by moving the decimal 23 places to the right. The value 3.7 × 10¯⁹ m can be converted from scientific notation by moving the decimal place 9 places to the left (0.0000000037 m).

For 2 x 10¯³ the exponent is -3 and this is division becoming 2 ÷ (10 x 10 x 10) = 0.002. Note that if a zero is to the right of the decimal a zero must proceed the decimal.

For 2 x 10³ the exponent 3 is positive and multiplication becoming 2 x (10 x 10 x 10) = 2,000.

Two observations can be made about what is written above. First, we can sometimes write words rather than numbers. The example above, 2 x 10³, is could have been stated as two thousand. Many readers will readily know and be able to have a mental image of two thousand written out in Arabic Numerals. The example, 3.7 × 10¯⁹ m, is not so readily known as three point seven nanometers and also except-able as 3.7 nm. See how crafty I was to use the symbol for meter (m) in the example above? You probably knew the symbol for meter but may not have known the value for nanometer 10¯⁹ m.

Second, be aware that when the valence state of ions is written on elements we often use superscript(s) just as exponents are superscripts. The difference here is that the plus or minus symbol comes after the number representing the valence state of an element or compound that is greater than one (Fe³⁺). When the valence state is one, the number is implied and only the positive or negative symbol is used in the superscript location (Na⁺ or Cl¯).

Beyond convenience and efficiency, scientific notation offers a clean way to communicate information about significant figures (see section on significant figures).