Mathematical skill ●●●○○

There are about 400\,000\,\mathrm m by car from Copenhagen to Aalborg in Denmark^{1} and 2\,600\,000\,\mathrm s in a month,^{2} while the thickness of a pencil stroke is 0.001\,\mathrm m and a lightning strike lasts 0.00003\,\mathrm s.^{[2]} Let’s rather write these in:

**kilo**metres,^{3}400\,\mathrm{\mathbf km},**mega**seconds,^{4}2.6 \,\mathrm{\mathbf Ms},**milli**metres,^{5}1 \,\mathrm{\mathbf mm} and**micro**seconds,^{6}30 \,\mathrm{\mathbf{\mu} m}.^{7}

Big or small numbers become easier to write with such a **prefix** representing a thousand, a thousandth, a million, a millionth etc.

Prefixes^{[4]} |
||||
---|---|---|---|---|

\vdots | ||||

Giga... | Billion | 1\,\mathrm G\ldots | = | 1000\,000\,000\ldots |

Mega... | Million | 1\,\mathrm M\ldots | = | 1000\,000\ldots |

Kilo... | Thousand | 1\,\mathrm k\ldots | = | 1000\ldots |

Milli... | Thousandth | 1\,\mathrm m\ldots | = | 0.001\ldots |

Micro... | Millionth | 1\,\mathrm \mu\ldots | = | 0.000\,001\ldots |

Nano... | Billionth | 1\,\mathrm n\ldots | = | 0.000\,000\,001\ldots |

\vdots |

Stepping up or down a prefix makes the number a thousand times larger or smaller. This significantly changes the number’s influence, its magnitude. Let us call each prefix step an **order of magnitude**. Saying ‘it is two orders of magnitude larger’ means something with *6 more* zeros; something a *million* times larger. It is a huge change.

If you on the other hand say ‘they are of the same order of magnitude’ then you are dealing with something that is less than a thousand times as large. It does *not* mean that the numbers are equal. One number can easily be double as large as the other since ‘double as large’ (meaning ‘two times as large’) is far from ‘a thousand times as large’.

The table below gives a feeling of scale in the case of size (represented by length) and time:

Size scale | |
---|---|

\vdots | |

Gigametres | The Sun^{[9]} |

Megametres | The Earth^{[8]} |

Kilometres | Mountains^{[7]} |

Millimetres | Insects |

Micrometres | Bacteria^{[6]} |

Nanometres | Molecules^{[5]} |

\vdots |

Time scale | |
---|---|

\vdots | |

Gigaseconds | A lifetime^{11} |

Megaseconds | A vacation^{10} |

Kiloseconds | A movie^{9} |

Milliseconds | Blink of an eye^{[11]} |

Microseconds | A lightning strike^{[2]} |

Nanoseconds | Computer memory^{8} |

\vdots |

It may sound odd to say ‘megametres’ and ‘kiloseconds’. We could also say for example ‘kilo-euros’. They are in principle perfectly fine words, we just are not used to them.

You have surely heard about other prefixes as well such as **deci**metre (\mathrm{dm}), **centi**metre (\mathrm{cm}) and maybe even **pico**second (\mathrm{pm}). Many more have been invented and collected into a standardised **international system of units**, abbreviated SI.^{12} A complete list of SI prefixes is found in the Prefixes resource.

References:

- ‘
**Route Copenhagen-Aalborg**’ (web page), Google Maps, goo.gl/maps/GyeGdhyS4o41SWFT7 (accessed May 2nd, 2019) - ‘
**An Explanation of Lightning**’ (web page),*Dennis Mollet & James Kistner*, Arizona State University, www.public.asu.edu/~gbadams/lightning/lightning.html (accessed May 2nd, 2019) - ‘
**Online Etymology Dictionary**’ (dictionary),*Douglas Harper*, www.etymonline.com - ‘
**The International System of Units (SI)**’ (book),*B. Inglis, J. Ullrich and others*, Bureau International des Poids et Mesures (BIPM), 9th ed., 2019, www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf, chapter 3 - ‘
**Biotechnology**’ (book),*P. David Clark & Nanette J. Pazdernik*, Newnes, 2nd ed., 2015, books.google.de/books?id=osqcBAAAQBAJ, ISBN 9780123850164 - ‘
**Small but Mighty: Cell Size and Bacteria**’ (article),*Petra Anne Levin (Department of Biology at Washington University) & Esther R. Angert (Department of Microbiology at Cornell University)*, Cold Spring Harbor Laboratory Press Biol, 2015, www.ncbi.nlm.nih.gov/pmc/articles/PMC4484965/pdf/cshperspect-CSZ-a019216.pdf, pages 2-3 - ‘
**Highest Mountain in the World**’ (web page), Geology.com, www.geology.com/records/highest-mountain-in-the-world.shtml (accessed May 3rd, 2019) - ‘
**How Big Is Earth?**’ (web page),*Tim Sharp*, Space.com, 2017, www.space.com/17638-how-big-is-earth.html (accessed May 3rd, 2019) - ‘
**The Sun**’ (web page), NASA, www.nasa.gov/sun (accessed May 3rd, 2019) - ‘
**RAM (Random Access Memory)**’ (web page),*Margaret Rouse, Stacey Peterson and others*, Tech Target: Search Storage, searchstorage.techtarget.com/definition/RAM-random-access-memory (accessed May 7th, 2019) - ‘
**Blink frequency and duration during perimetry and their relationship to test-retest threshold variability**’ (article),*Yanfang Wang (School of Biomedicine at Manchester University), Sonia S. Toor (School of Psychology at University of Reading) and others*, Investigative Ophthalmology and Visual Science, vol. 52, issue 7, 2011, iovs.arvojournals.org/article.aspx?articleid=2188061, DOI 10.1167/iovs.10-6553, pages 4546–4550 - ‘
**Guide for the Use of the International System of Units (SI)**’ (article),*Ambler Thompson & Barry N. Taylor*, National Institute of Standards and Technology, 2008, physics.nist.gov/cuu/pdf/sp811.pdf