# Scientific notation

#### Prerequisites

When we quantify biological systems, we often have to include a wide range of scales, both of distance, time, and mass.

Consider a wild dog as a sample organism.

**Distances** considered can vary greatly.

The dog might be 1 meter in length | 1 m |

The dog might wander in a day over a range of tens of kilometers. | 10,000 m |

Following the evolutionary history of the species might take one over distances of thousands of kilometers. | 1,000,000 m |

The dog's organs may be on the scale of centimeters. | 0.01 m |

The dog is made up of cells that are about 100 micrometers in size. | 0.0001 m |

Some of the structures within the dog's cells are about 10 nanometers in size | 0.00000001 m |

the forces and interactions that drive the biochemistry of the cell occur at distances of fractions of a nanometer | 0.0000000001 m |

**Times** in biology can also vary over a huge range, from the millions of years for species to evolve (100,000,000,000,000 s), to hundreds of milliseconds it takes neural signals to propagate (0.1 s), to the nanoseconds it takes ions to diffuse across a membrane (0.000000001 s).

**Masses** considered in biology can also range from the 100-ton mass of a dinosaur (100,000 kg), to a microgram dose of a powerful hormone (0.000000001 kg).

All these zeros are hard to keep track of and manipulating them is prove to error. We handle these incredible ranges of numbers much more effectively by using scientific notation — basically counting the number of zeros. You have probably seen this many times, but may not have bothered to take it seriously or learn to use it.

*In this class, scientific notation is no longer optional - it is required.*

We will be covering such a large range of scales that without using scientific notation, you won't be able to keep track of anything. If you're not used to using it, now's the time to start!

To remind you of how scientific notation works, here's a brief review.

Instead of writing all the zeroes out and counting them every time, we use a trick: we write them as exponents -- powers of ten. So we have, for example:

1,000,000 m = 10^{6} m

10,000 m = 10^{4} m

0.0001 m = 10^{-4} m

0.00000001 m = 10^{-8} m.

Notice that we are not actually counting zeros but how many figures we have to move the decimal point. In the last one, we have to move the decimal point 8 places in order to get 1.0 x 10^{-8}. Remember that 10^{6} means 10 times 10 six times, and 10^{-4} means 1 divided by 10 four times and you'll always be able to check.

When we have numbers that are not exact powers of ten we typically use a number with one digit before the decimal times some power of ten. So, for example:

4,630,000 = 4.63 x 10^{6}

0.000259 = 2.59 x 10^{-4}.

Our brains are not very good at making sense of very large and very small numbers. The way we handle this is to introduce marker points that we can rescale to think about. You can handle about a factor of 1000 pretty easily. Think of a bead; then a wire with 10 beads on it to make a line. Then line 10 of those up to form a square. Now pile 10 of those squares on top of each other. Ta Da! A mental image of 1000 beads! (These are actually used in Montessori classrooms to teach children what 1000 looks like.) You can visualize a million by now imagining 1000 of these cubes and building up a 10x10x10 cube of these thousand-bead cubes.

What we do in science is to rescale our minds to an appropriate point and think within a range around that point. When we need to go beyond, we jump by factors of 1000. The international standard is

10^{9} = giga

10^{6} = mega

10^{3} = kilo

1

10^{-3} = milli

10^{-6} = micro

10^{-9} = nano

10^{-12} = pico

You will be expected to know all of these prefixes and translate comfortably among them.

Joe Redish 7/18/11

Last Modified: May 15, 2019