The Art of Number Visualization: Making Sense of Scales
An essential skill in an Engineer's toolkit
Last Sunday evening, I decided to take a break from screens around me and delve back into physical book reading. On hitting my small bookshelf, I selected revisiting "Wings of Fire" by APJ Abdul Kalam as the read for the evening.
Midway through the book, there was a text regarding development of an intermediate range missile and intercontinental missile whose range was described as 2780km. Now personally, numbers only make sense to me when they’re put in context of things that I can visualize. So, I decided to deduce what would 2780km look like, to make it interesting, do it without using internet or calculator. Visualizing scale of city (50km) is simple, but anything above which I tried to visualize, I couldn’t put a number to the scale. Another thing which I could visualize was the map of Earth, so I decided to take that as a scale and derive the rest, maybe as a scale of north to south length of India.
Now, I didn’t knew the numbers corresponding to size of country but I did remember from high school physics, thanks to the chapter on gravitation that the radius of Earth is around 6500km. Okay, so circumference would be 2*pi*r, North to South pole distance would be half of it and North Pole to Equator would reduce that by half further. So, pi*r/2 or 1.5 times 6500km → around 10000km. This bifurcated into 2 thoughts, one India is around less than 1/3rdish the length from Pole to Equator so around 3300km which was so surprisingly close to the actual answer (3214km) that it blew my mind. Second thought, this number came surprisingly close to round number, which on giving some thought reminded me from a documentary on measurement systems that this was no coincident; traditionally, the meter was defined as 1/10,000,000 of distance from North Pole to Equator. Which now made absolute sense of the closeness of these figures. Also, helped me put the range of the missile in perspective.
Expanding from the incident, made me revitalize the importance of visualizing numbers and scales of measurements, which are so different across domains. We can’t exaggerate the role this skill play, specially in the toolbox of an Engineer and Scientist.
There are so many vivid parameters one could measure or deduce. Few of these parameters like distance, time, speed are more vivid for a common person whereas others like voltage, wavelength are more abstract. Each parameter is important to a certain group of professionals or other. A Mechanical Engineer might be interested to know, calculate, measure, design parameters such as lengths, area, forces, volume, speed or weights. Electronics Engineer could be interested in voltage, current, capacitance, resistance, frequency or time. Physicist might be playing around with wavelengths, size, forces, energy and so on. Whereas accountants would be interested in monetary values, growth rate and such.
Even for a single parameter, the scales could be so so wide in sizes. A single number could be big or small depending on the frame of reference being considered. If we just take distance as parameter to be measured, a parameter so common and physical that anyone can relate to it.
A Mechanical Engineer designing enclosure of a everyday product, could be visualizing things in centimeters (or foot).
A Civil Engineer designing highways would be visualizing it in kilometers
An Astrophysicist would be measuring distance in lightyears
A particle physicist would be measuring atoms in nanometers and measuring diameter of proton in femtometers
If you can’t visualize these figures and scales, don’t worry. They’re really really hard to visualize. But let me give you some references to appreciate the scales of these. Note, these measurements are highly approximate and are only meant to visualize scale.
Well, we have to start somewhere, so let’s consider your arm length. A single arm length from center of your body to finger tip would be somewhere around 0.75 to 1 meter.
A pea, is of range 1cm. That means, if you close your eyes and lay 75 to 100 peas next to each other, it would span your arm length.
A tip of ball point pen is approximately 1mm. That is, you could have 10 such pen points across a pea
A soccer field may have a length of range 100 meters i.e. 50 individuals can lie next to each other fingertip to fingertip with arms stretched on a soccer field. (Yes, I do realize that’s not a standard American unit of measurement which would be American Football field)
And 10 such fields, length to length would give you 1km
Number of ballpoint pen tips which you could lay next to each other in 10 football fields (One million or 1,000, 000 or 10^6) is of the same magnitude as number of atoms next to each other in a ball point tip
And if atom was of size of a football field, nucleus would be of size of a pea kept on center point.
We can also go in the similar fashion to the other direction and so big. But, here is a better way to visualize the same. I recommend watching the video at half the speed to contemplate the size.
So, I leave you with this thought. Take your field, take any measurement you work with on a daily basis and try to extrapolate it to other fields. Whether you’re estimating computational resources needed for your application or amount, or force needed to move a car or how many iPhones can you simultaneously charge from your simple house plug in parallel. If the activity doesn’t apply to your profession, you can do the same with your income and spending. For e.g. Warren Buffet is so rich that him spending 700, 000 USD is equivalent to an average American buying a can of sprite. Frame your thoughts in these kind of analogies and I’d love to hear what you can come up with.
Till next time, happy thinking!