Welcome to The STEM Sessions Podcast. I am Jarl Cody, your host and narrator.
While reviewing outlines and in-work scripts for upcoming episodes, I noticed more than a couple of them rely on orders of magnitude in some capacity, be it part of a mathematical explanation or a means of sanity checking your thought process. So I figure it’s worth dedicating a full episode to the concept on its own.
Orders of magnitude is one of those scientific terms whose meaning in common lexicon diverge from their technical meanings. In this particular case, it’s a dilution of meaning. I’m as guilty as anyone. I’m fact, I doubt I’ve ever used the term to it’s full meaning outside of technical discussions.
I’m not sure if that’s positive or negative, if I’m being frank. On one hand, the intent is correct. But on the other hand, should I feel badly for devaluing the term? Perhaps that’s a question best left to historical linguists.
This is The STEM Sessions Podcast – Episode Ten. Orders of Magnitude.
We frequently use the term “orders of magnitude” to compare quantities and qualities found in our daily lives. These comparisons range from the speeds of cars to the difficulties of homework assignments. For example, one may brag “my Mustang is orders of magnitude faster than your Prius” – or a student may seek sympathy by declaring “my fluid dynamics class is orders of magnitude harder than any other class this semester”. Yet, despite of how often it’s invoked, the usage rarely reflects the term’s full technical meaning.
Yes, we correctly realize it means one quantity is much bigger or much smaller than another, but we rarely assign powers of ten to the comparisons. This, I believe, can be partly attributed to our inability to conceptualize the scales that multiple orders of magnitude bring to the table.
Let’s look at a million and billion, for example. Ask a random person to describe the difference between the two, and a common answer is likely to be “a billion is bigger than a million”. Perhaps, a few will even answer, “a billion is a thousand times larger than a million”. But if you ask for a description or picture of the difference, you may not receive one, because we don’t have many examples in our daily lives from which to draw.
In scientific notation, a million is 1 x 10^6 which is a one followed by six zeros, and a billion is 1 x 10^9 which is a one followed by nine zeros. The delta between the two numbers is effectively the difference between the exponents; a factor of 1 x 10^3 which is a one followed by three zeros, or 1000. Thus, a billion is one thousand times larger than a million. In terms of orders of magnitude, since 1000 is 1 x 10^3, and an order of magnitude is 10^1 power (or 10) a billion is three orders of magnitude larger than a million.
Yet, picturing a thousand times increase in something can be difficult. To do so, we need to put the numbers in terms that are familiar to us. Let’s use time as an example
One million seconds is approximately 11.5 days. One billion seconds is approximately 31.7… years. Not days, not months. Years! When starting with a million seconds, a one thousand times, or three orders of magnitude, increase is the difference between two weeks and three decades.
Let’s try distance next. One million inches is 15.8 miles (or about 25.5 km). One billion inches is 15,783 miles (or about 25,400 km)! In other words, one million inches takes you from one side of Los Angeles, CA to the other, but one billion inches? That takes you from Los Angeles to Perth, Australia… if you travel east – the long way around the planet. A million inches is the scale of the city; a billion inches is the scale of our planet.
Let’s use the same three orders of magnitude, but now we’ll reduce the quantities. Starting at one million, a three orders of magnitude reduction results in a quantity of one thousand, or in scientific notation 1 x 10^3.
As stated earlier, one million seconds is 11.5 days. One thousand seconds is 0.7 days. So the three orders of magnitude takes us from a week and a half to less than a day.
What about distance? We know one million inches is 15.8 miles, but one thousand inches is only 0.015 miles or about 83 feet. So if one million inches takes you across a city like Los Angeles, one thousand inches doesn’t even get you to the end of the block.
Revisiting the Mustang-Prius comparison, a 2020 Mustang has top speeds of 155-186 mph depending on the trim level. A 2021 Toyota Prius has a top speed of 112 mph. If the Mustang was literally an order of magnitude faster than the Prius, its top speed would be a ridiculous 1120 mph. In reality, the Mustang isn’t even twice as fast, with the Mustang Shelby GT500 being only 1.7 times faster than the Prius; a far cry from a 10 times increase.
Those of us living along fault lines often experience a real world example in orders of magnitude, yet we don’t recognize it as such. The Richter Scale for comparing the size of earthquakes. It’s a logarithmic scale in which each whole number increase brings ten times the amplitude (and about 31 times the energy). So a 4.0 earthquake is ten times bigger than a 3.0; a 5.0 ten times bigger than a 4.0; and a 6.0 ten times bigger than a 5.0 quake. Thus a 6.0 is one thousand (or three orders of magnitude) bigger than a 3.0 earthquake.
After experiencing a few earthquakes, you gain an understanding of this scaling. We often sleep through 3.0 earthquakes or mistake then for a truck driving by. But a 6.0 rocks you out of bed and damages structures and roads. Now, technically, the United States Geological Survey no longer uses the Richter Scale, which is a local magnitude scale; it now uses a moment magnitude scale, but while the absolute numbers may diverge a bit from the traditional Richter Scale, it’s still logarithmic and the example stands.
Even though we understand the differences in earthquake sizes on paper, it’s still difficult to assign a quantification to what we feel. Is that really what a three orders of magnitude increase feels like? How would I describe it so a person who has never felt an earthquake understands, too?
It’s difficult to picture billions of something unless it can be translated into another something from our every day experience. This makes us prone to both unintentional hyperbole and accidental understatements when attempting to put these concepts into words. This isn’t due to a lack of education or critical thinking. It’s just how our brains work. We simply struggle to understand extreme quantities, both large and small.
Thank you for listening to The STEM Sessions Podcast.
This episode was researched, written, and produced by me, Jarl Cody.
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Until the next one, stay curious.