An important part of my scientific toolkit is how to think about things in the world over a wide range of magnitudes and time scales. This involves first understanding powers of ten; second, visualizing data over a wide range of magnitudes on graphs using logarithmic scales; and third, appreciating the meaning of magnitude scales such as the decibel (dB) scale for the loudness of sounds and the Richter scale for the strengths of earthquakes.
This toolkit ought to be a part of everyone thinking, but sadly I have found that even well educated nonscientists are flummoxed by log scales and can only vaguely grasp the difference between an earthquake on a Richter scale of 6 and 8 (a thousand times more energy released). Thinking in powers of 10 is such a basic skill that it ought to be taught along with integers in elementary school.
Scaling laws are found throughout nature. Galileo in 1638 pointed out that large animals have disproportionately thicker leg bones than small animals to support the weight of the animal. The heavier the animal, the more stout their legs need to be. This leads to a prediction that the thickness of the leg bone should scale with the 3/2 power of the length of the bone.
Another interesting scaling law is that between the volume of the cortical white matter, corresponding to the long-distance wires between cortical areas, and the gray matter, where the computing takes place. For mammals ranging over 5 orders of magnitude in weight from a pygmy shrew to an elephant, the white matter scales as the 5/4 power of the gray matter. This means that the bigger the brain, the disproportionately larger the fraction of the volume taken up by cortical wiring used for communication compared to the computing machinery.
I am concerned that students I teach have lost the art of estimating with powers of 10. When I was a student I used a slide rule to compute, but students now use calculators. A slide rule lets you carry out a long series of multiplications and divisions by adding and subtracting the logs of numbers; but at the end you need to figure out the powers of 10 by making a rough estimate. A calculator keeps track of this for you, but if you make a mistake in keying in a number you can be off by 10 orders of magnitude, which happens to students who don't have a feeling for orders of magnitude.
A final reason why familiarity with powers of 10 would improve everyone's cognitive toolkit is that it helps us comprehend our life and the world in which we live:
How many seconds are there in a lifetime? 109 sec
A second is an arbitrary time unit, but one that is based on our experience. Our visual system is bombarded by snapshots at a rate of around 3 per second caused by rapid eye movements called saccades. Athletes often win or lose a race by a fraction of a second. If you earned a dollar for every second in your life you would be a billionaire. However, a second can feel like a minute in front of an audience and a quiet weekend can disappear in a flash. As a child, a summer seemed to last forever, but as an adult, summer is over almost before it begins. William James speculated that subjective time was measured in novel experiences, which become rarer as you get older. Perhaps life is lived on a logarithmic time scale, compressed toward the end.
What is the GDP of the world? $1014
A billion dollars was once worth a lot, but there is now a long list of multibillionaires. The US government recently stimulated the world economy by loaning several trillion dollars to banks. It is difficult to grasp how much a trillion dollars ($10 ) represents, but several clever videos on YouTube (key words: trillion dollars) illustrate this with physical comparisons (a giant pile of $100 bills) and what you can buy with it (10 years of US response to 9/11). When you start thinking about the world economy, the trillions of dollars add up. A trillion here, a trillion there, pretty soon your talking about real money. But so far there aren't any trillionaires.
How many synapses are there in the brain? 1015
Two neurons can communicate with each other at a synapse, which is the computational unit in the brain. The typical cortical synapse is less than a micron in diameter (10[-6] meter), near the resolution limit of the light microscope. If the economy of the world is a stretch for us to contemplate, thinking about all the synapses in your head is mind boggling. If I had a dollar for every synapse in your brain I could support the current economy of the world for 10 years. Cortical neurons on average fire once a second, which implies a bandwidth of around 10 bits per second, greater than the total bandwidth of the internet backbone.
How many seconds will the sun shine? 1017 sec
Our sun has shined for billions of years and will continue to shine for billions more. The universe seems to be standing still during our lifetime, but on longer time scales the universe is filled with events of enormous violence. The spatial scales are also immense. Our space-time trajectory is a very tiny part of the universe, but we can at least attach powers of 10 to it and put it into perspective.