3.3: Scientific Notation in Chemistry

Astronomers are accustomed to very large numbers. While the moon is only $$406,697 \: \text{km}$$ from earth at its maximum distance, the sun is much further away $$\left( 150 \: \text{million km} \right)$$. Proxima Centauri, the star nearest the earth, is $$39,900,000,000,000 \: \text{km}$$ away—and this is just the start of long distances. On the other end of the scale, some biologists deal with very small numbers: a typical fungus could be as small as 30 micrometers $$\left( 0.000030 \: \text{meters} \right)$$ in length, and a virus might only be 0.03 micrometers $$\left( 0.00000003 \: \text{meters} \right)$$ long.

Scientific Notation

Scientific notation is a way to express numbers as the product of two numbers: a coefficient and the number 10 raised to a power. It is a very useful tool for working with numbers that are either very large or very small. As an example, the distance from Earth to the Sun is about 150,000,000,000 meters—a very large distance indeed. In scientific notation, the distance is written as $$1.5 \times 10^{11} \: \text{m}$$. The coefficient is the 1.5 and must be a number greater than or equal to 1, and less than 10. The power of 10, or exponent, is 11 because you would have to multiply 1.5 by $$10^{11}$$ to get the correct number. Scientific notation is sometimes referred as exponential notation. A summary of SI units is given in the table below.

Table $$\PageIndex{1}$$: SI Prefixes
Prefix Unit Abbreviation Exponential Factor Meaning Example
giga $$\text{G}$$ $$10^9$$ 1,000,000,000 1 gigameter $$\left( \text{Gm} \right) = 10^9 \: \text{m}$$
mega $$\text{M}$$ $$10^6$$ 1,000,000 1 megameter $$\left( \text{Mm} \right) = 10^6 \: \text{m}$$
kilo $$\text{k}$$ $$10^3$$ 1,000 1 kilometer $$\left( \text{km} \right) = 1,000 \: \text{m}$$
hecto $$\text{h}$$ $$10^2$$ 100 1 hectometer $$\left( \text{hm} \right) = 100 \: \text{m}$$
deka $$\text{da}$$ $$10^1$$ 10 1 dekameter $$\left( \text{dam} \right) = 10 \: \text{m}$$
$$10^0$$ 1 1 meter $$\left( \text{m} \right)$$
deci $$\text{d}$$ $$10^{-1}$$ 1/10 1 decimeter $$\left( \text{dm} \right) = 0.1 \: \text{m}$$
centi $$\text{c}$$ $$10^{-2}$$ 1/100 1 centimeter $$\left( \text{cm} \right) = 0.01 \: \text{m}$$
milli $$\text{m}$$ $$10^{-3}$$ 1/1,000 1 millimeter $$\left( \text{mm} \right) = 0.001 \: \text{m}$$
micro $$\mu$$ $$10^{-6}$$ 1/1,000,000 1 micrometer $$\left( \mu \text{m} \right) = 10^{-6} \: \text{m}$$
nano $$\text{n}$$ $$10^{-9}$$ 1/1,000,000,000 1 nanometer $$\left( \text{nm} \right) = 10^{-9} \: \text{m}$$
pico $$\text{p}$$ $$10^{-12}$$ 1/1,000,000,000,000 1 picometer $$\left( \text{pm} \right) = 10^{-12} \: \text{m}$$

When working with small numbers, we use a negative exponent. So 0.1 meters is $$1 \times 10^{-1}$$ meters, 0.01 is $$1 \times 10^{-2}$$ and so forth. The table above gives examples of smaller units. Note the use of the leading zero (the zero to the left of the decimal point). That digit is there to help you see the decimal point more clearly. The figure 0.01 is less likely to be misunderstood than .01, where you may not see the decimal.

Summary

• Scientific notation allows us to express very large or very small numbers in a convenient way.
• Scientific notation uses a coefficient (a number between 1 and 10) and a power of ten sufficient for the actual number.