Unit analysis with the chain-link method can also help us to answer difficult questions. For example, calculating how many heart beats an average person experiences during their lifetime seems daunting. With the chain-link method we can come up with an estimated value relatively quickly. A search of the internet finds that the average life expectancy in the U.S. is 78.8 years and that the typical value for adult heart rate is between 60 BPM and 100 BPM so let’s take the middle-range value and go from there.
That’s a big number! In fact, it’s over three billion beats. As it turns out, humans are quite special among animals in the number of heartbeats per lifetime we experience. Visit the website of the beats per lifetime project for more information and an interactive look at heart rate statistics for various species.
In the previous calculation we chose to use a heart rate of 80 BPM, which was an approximation rather than an actual measurement or calculation. Therefore, our answer is only an estimate. However, we don’t expect anyone who lives to adulthood will get anywhere near 10x more or 10x fewer beats than this, so our answer is within an order of magnitude of what most people experience. Combining several already known, easily found, or approximate values to get a general idea of how big an answer should be, as we just did for beats per lifetime, provides an order of magnitude estimation. Play with this simulation to practice estimating sizes using only visual cues.
Order of magnitude estimation often relies on approximate values, so order of magnitude estimate and approximation are often used interchangeably. Adding to confusion, approximation is often used interchangeably with assumption or uses approximation to describe a quick, rough measurement with a high degree of uncertainty. In order to maximize clarity this textbook will strive to stick to using terms as defined according to the following table.
|Assumption||Ignoring some compilation of the in order to simplify the analysis or proceed even though information is lacking. Scientists state assumptions, justify why they were needed, and estimate their possible impact on results.||My cotton clothes are completely soaked through, so I assume they are not providing any insulating effect against the cold water.|
|Act of coming up with a rough value using prior knowledge and assumptions, but not by making a measurement for the purpose of determining the value.||The water feels cold, but not shocking, similar to the 70 °F swimming lake, so the approximate water temperature is 70 °F.|
|Uncertainty (more about this later)||Amount by which a measured, calculated, or approximated value could be different from the actual value.||85 °F would feel comfortable like the 82 °F college pool and 55 °F feels very cold, so + 15 F° is my uncertainty from 70 °F.|
|Order of Magnitude Estimate||Result of combining assumptions, approximate values, and/or measurements with large uncertainty to calculate an answer with large uncertainty, but has the correct order of magnitude.||Using known data, I estimated my time to exhaustion or loss of consciousness to be 5 hours (less than 50 hours and more than 0.5 hours).|
Considering that our beats per lifetime answer is only an order of magnitude estimation, we should round our final answer to have fewer significant figures. Let’s make it 3,000,000,000 beats per lifetime (BPL), or three billion BPL. A bit later in the chapter we will define what we mean by and significant figures and also talk more about why, when, and how we have to do this kind of rounding. For now we notice that it’s a bit distracting and a bit annoying writing out all those zeros, so by counting that there are nine places before the first digit we can use scientific notation and instead write: BPL. Alternatively we can use a metric prefix. The prefix for 109 is Giga (G) so we can write: 3 GBPL (read as gigabeats per lifetime). The table below shows the common metric prefixes. For a much more comprehensive list of prefixes visit the NIST website. One advantage of using metric units is that the different size units are related directly by factors of ten. For example 1 meter = 100 cm rather than 1 foot = 12 inches.
|Prefix||Symbol||Value||Example (some are approximate)|
|exa||E||1018||exameter||Em||1018 m||distance light travels in a century|
|peta||P||1015||petasecond||Ps||1015 s||30 million years|
|tera||T||1012||terawatt||TW||1012 W||powerful laser output|
|giga||G||109||gigahertz||GHz||109 Hz||a microwave frequency|
|mega||M||106||megacurie||MCi||106 Ci||high radioactivity|
|kilo||k||103||kilometer||km||103 m||about 6/10 mile|
|hecto||h||102||hectoliter||hL||102 L||26 gallons|
|deka||da||101||dekagram||dag||101 g||teaspoon of butter|
|deci||d||10-1||deciliter||dL||10-1 L||less than half a soda|
|centi||c||10-2||centimeter||cm||10-2 m||fingertip thickness|
|milli||m||10-3||millimeter||mm||10-3 m||flea at its shoulders|
|micro||µ||10-6||micrometer||µm||10-6 m||detail in microscope|
|nano||n||10-9||nanogram||ng||10-9 g||small speck of dust|
|pico||p||10-12||picofarad||pF||10-12 F||small capacitor in radio|
|femto||f||10-15||femtometer||fm||10-15 m||size of a proton|
|atto||a||10-18||attosecond||as||10-18 s||time light crosses an atom|