Get Started in Science with a Big Bang by Getting More Out of Less Writing
With scientific notation, it’s all about that base.
Ah. Writing. It’s fun, but it can be tedious, tiring, and monotonous. Like Bart Simpson on The Simpsons cartoon famously scrawling something like “I will not expose the ignorance of the faculty.” on a chalkboard, one of my punishments as a kid was to hand-write something a hundred or a thousand times.
Despite reams of paper wasted on this punishment, my hand-writing today as a scientist and engineer is atrocious. The upside of it all is that I prefer to type, and scientific journals prefer not to receive handwritten manuscripts! I’ve survived without pretty penmanship after all.
An Astronomical Waste of Writing Time
One of my graduate professors was months from retirement and had an office enshrined with floor-to-ceiling, wall-to-wall-to-wall shelving solidly packed with handwritten notes created over his illustrious decades-long career in astrophysics. He was visibly stressed when telling me he wasn’t sure how he’d move all his notes to his home. I suggested that he could have scanned them all into PDF, but I suspect this only created a new kind of anxiety for him.
I’m absolutely confident that not one of my professor’s notes had a number written on it longer than about 7 digits. Why? I’m confident he used scientific notation.
If you’re not familiar with writing numbers with scientific notation, you might not be surprised that an astrophysicist might write out the diameter of the Milky Way Galaxy, for example, in units of meters like this:
Diameter of Milky Way Galaxy: 950000000000000000000 meters.
If you had to do this for your job, however, you’d likely feel like a child being punished.
Scientific notation enters the room with great news in your hour of handwriting darkness.
Resistance is Human, Patterns are Divine
If you have ever watched an engineer type out such a number, you might have seen them type it out like this:
Diameter of Milky Way Galaxy: 950 000 000 000 000 000 000 meters.
It might have commas instead of spaces like this:
Diameter of Milky Way Galaxy: 950,000,000,000,000,000,000 meters.
By grouping the zeros in sets of three using spaces or commas, the number of zeros can be read rather quickly in increments of three: three, six, nine, twelve, fifteen, eighteen, plus one — or nineteen. This is sometimes called engineering notation.
It’s useful because you’re human. Humans can easily recognize simple patterns when objects are arranged in small groups, and then use this pattern to quickly count a large collection of objects if every group has the same number of objects.
A very familiar example of writing long numbers is tally marks. Tally marks, are often written in groups of five: five, ten, fifteen, twenty, twenty-five, thirty, and so on. It’s usually done to tally up the score of a game or to keep record of the number of weeks or months you’ve been in prison.
We’re only human, and one of our superpowers is instantly recognizing the number of objects in a small group. I personally think 5 is the upper limit of objects we can count in such an efficient way, but I’ll leave that for another article.
Also, if you’re one who likes to think about numbers in terms of billions, as Carl Sagan was tongue-in-cheek famous for doing, you could quickly see that this is “950 billion billion meters.” It’s kind of fun to say, isn’t it?
One billion is 1,000,000,000, or a one followed by 9 zeros. So a “billion billion” is a one followed by 18 zeros. Saying “billion” or “million” or “thousand” is just another way of simplifying the use of numbers in the engineering notation I mentioned earlier. That’s because one million is written 1,000,000 and one thousand is written 1,000.
Do you see the groups of three zeros? That’s the beauty of engineering notation. You could read a number like 1,000,000,000 as “one thousand thousand thousand” or “one thousand million” if you really wanted to.
Scientists love this feel-good way of reading numbers so much that it’s been simplified a little more.
Reading a number like “one thousand” is almost like including the units. It’s a bit clever, but as one of my former supervisors would say, “it’s too clever by half,” because “thousand” isn’t a unit.
Ah, but we can make it part of our system of units if we like.
For example, “five thousand meters,” is “five kilometers.” How clever of us to make it even simpler by replacing “thousand” with a simple prefix of the unit, “kilo”.
At this point, you’re no doubt truly amazed by this clever shorthand, in which we’ve made engineering notation even better by merely adding a few letters to the unit.
But wait. There’s More.
Not only are long numbers needed to talk about really big things, they’re also needed if you want to talk about very small things. The diameter of a hydrogen atom in units of meters, for example, would be written like this:
Diameter of hydrogen atom: 0.00000000011 meters.
In engineering notation, it’s written like this:
Diameter of hydrogen atom: 0.000 000 000 11 meters.
Being clever, you’d simplify this by adding a prefix to the unit, like this:
Diameter of hydrogen atom: 0.11 nanometers.
The prefix nano- is the equivalent of nine zeros when they appear after the decimal point.
We’ve simplified a long, boring number and made it into a simple number with a tidy way to write units. It’s like replacing a team of horses with a car engine.
Not only are we lazy as human beings, we’re also greedy.
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Now that we’ve got a car engine, we want more.
This is where we want to remind ourselves what science is all about: measurements.
The numbers we deal with in science typically reflect a measured quantity, like the diameter of a hydrogen atom or diameter of a galaxy. How precisely we measured the diameter is also important if we are to ever pass the information to someone else who wants to compare it with their measurement.
If you were an engineer, who wanted to “reverse engineer” a car engine, for example, if you don’t know the precision of the measurements in the blueprints of the engine, you’re likely to make parts that are either slightly too big or slightly too small, and your engine won’t work like the original. A mere fraction of a millimeter in the size of an important part could make all the difference in a million dollar engine that works or not.
Scientific Notation — The Bugatti of Science
Let’s look at this measured value:
Diameter of hydrogen atom: 0.11 nanometers.
Now, look at this number in scientific notation:
Diameter of hydrogen atom: 1.1 × 10⁻¹⁰ meters.
Nice. We got rid of that leading zero to the left of the decimal place by shifting the digits by one place, and then we replaced the prefix (nano-) with “× 10⁻¹⁰”.
Now we won’t need to think about prefixes like milli- (0.000), micro- (0.000 000), and nano- and we won’t have to remember all those in-between prefixes like deci- (0.1) and centi- (0.01). Let’s not get started with all the special names of units like 0.1 nanometer which is called an angstrom, written with its own special symbol: Å.
Scientific notation is looking pretty cool and sleek, eh? So much information and no need to remember silly prefixes.
What about precision? It’s built-in just like your favorite cupholder.
Look at this number again:
1.1 × 10⁻¹⁰ meters.
Those two digits, 1.1, immediately show you the precision of the measurement. This number is actually rounded down from the precision of the number used for this article. The full-precision number was
1.06 × 10⁻¹⁰ meters.
This second number indicates a measurement made to “two decimal places” of 10⁻¹⁰ of a meter. The 6 in the second decimal place represents a measurement that was made to 10⁻¹² meters. In the rounded number above, 1.1 represents a measurement made to an order of one magnitude less precision than 1.06. That is, it’s 10 times less precise.
Let’s Break It Down
Here is the approximate diameter of the Milky Way Galaxy again, written in scientific notation:
9.5 × 10²⁰ meters.
Let’s break this down.
The most important part of the number in the figure above is the base of 10.
Scientific notation is so convenient that once you’ve started using it for a while, you won’t even need to think about what base you’re using. Just like your body knows about breathing, you’ll simply look at the coefficient and exponent and immediately know the number.
The 9.5 is called the coefficient. You might call it a “significand” or “mantissa” but rest assured that most scientists and engineers just call it a coefficient.
The decimal in the coefficient is where all the action takes place. In standard scientific notation, only one digit appears to the left of the decimal point, and the number of digits appearing to the right of the decimal point represents the precision of the measurement. Pretty easy stuff, eh?
Wield your knowledge wisely. If you were to write this number as:
9.5000 × 10²⁰ meters
you’d have claimed one thousand times more precision in your measurement than you likely made.
All the exponent does is tell you the order of magnitude of the number. Thanks to the nature of the base of 10, if the exponent is a positive number, it indicates the number of positions you would move the decimal point to the right if you were to write it in the long-form discussed earlier.
If the exponent is a negative number, it indicates the number of positions you’d move the decimal point to the left if you were to write it out in long-form. That’s all there is to the exponent. It’s just a counting number that tells you how many positions to move the decimal point to the right or left.
It’s All About That Base
Base 10 makes your life easy. Why wouldn’t you want to use it?
- The coefficient is a number between 0.0 and 10.0.
- The exponent merely tells us how big the number is — its order of magnitude — resulting from moving the decimal point to the right or left. Every order of magnitude is a 10× multiplier or divider.
It’s all about base 10.
The base and exponent, like 10³, tells you to increase the coefficient by three orders of magnitude by moving the decimal point three points to the right. For example, 1.0 × 10³, tells you to increase the order of magnitude of the coefficient by 3, or 1,000.
If the number is 2.0 × 10³ , the exponent tells you to increase the order of magnitude of the coefficient by 3 while the coefficient scales the number by 2.0. That is:
a. The order of magnitude is given by an exponent of 3: 1,000
b. Coefficient of 2.0 scales (multiplies) 1,000 by 2.0: 2,000.
Once you realize it’s all about that base, there’s no trouble.
