When to use scientific notation...

Factual Questions
1

As a lover of nonfiction, I was thumbing through one of my reference books, and I came across a table of constants. You know - the speed to light, Pi, and so on. Being of a scientific nature, and involving some pretty big numbers, there was a lot of scientific notation in use.

But I have long questioned why people bother with scientific notation in some instances. Don’t get me wrong - scientific notation has its place, most notably to express numbers that would be too long to write out in decimal notation. For example, if you want to write the Avogadro constant, it’s much easier to write 6.022 x 10^23 than it would be to write the whole thing out. It saves space, while not bothering with some of the insignificant decimal values beyond the initial “.022”.

But I have wondered why people insist on using scientific notation in cases where it takes more time (time, space, characters, etc…) to write out the scientific notation than it would be to just write out the number in decimal form. For example, my reference book lists the following values:

149.59787 x 10^6 (16 characters, not counting the caret)
2.731500 x 10^2 (15 characters, not counting the caret)

The decimal notation form of these would be:

149,597,870 (only 11 characters)
273.15 (only 6 characters)

These examples, especially the second one, border on the scientifically pretentious. Even if you really wanted to list the exact value of my first example (an astronomical unit is 149,597,870.691 km) you’re still only at 15 characters, one less than the 16 characters it took to use scientific notation. The decimal notation is more accurate, and uses less space. So why bother with scientific notation?

The only thing I can think of is that scientific notation implies inaccuracy (or in some cases, accuracy). By expressing a number in scientific notation, the implied message is “insignificant figures have been removed from this number, so you’d better realize that we’re talking about a close approximation, and not an absolute value.” Is that why scientific notation is used if even it seems like it doesn’t need to be?

  • Peculiar.
2

Your 2nd example doesn’t exactly work though.

2.731500 x 10^2

Is not the same as:

273.15

You either have to write is as 2.7315E[sup]2[/sup]
or 273.1500

The trailing zeros imply the accuracy whether it’s in scientific notation or not. And if you’re typing, you can leave out the (X10^) and replace it with an E for brevity.

To answer your question, you use it whenever you deem it neccesary.

I’d write a thousand as 1,000 not 1E[sup]3[/sup]

A million? Maybe. It’s really up to you - unless it’s on a test on scientific notation - then it’s up to the instructor’s rules.

3

Scientific notation is also very useful if you’re doing quick calculations – you just group the orders of magnitude involved together and get some immediate ballpark estimate. Also, often, deciding whether or not to use it would take longer than just writing it one way, and since the scientific notation is better in more cases, you just default to that.

4

In the case of tables like that, there’s a combination of several factors. One which you already figured out and brewha confirmed, that it’s not the same to write 1.23E6 as 1,230,000.

The first one says, to me, “we’re sure of the 1.2, the x.x3 not so much and anything beyond that search us.” The second one promises 7-figure accuracy.
Another factor is contextual: for an accountant it makes sense to budget that new bridge at 25,454,768.12€ and then go 10% over budget (heck, 10% is nothing in some of those public works); I was trained to write only one more figure that I am sure of (as in the example above). To me, “25M€ with a 10% error margin” gives the same information as the previous number and it’s more honest; to my brother the finance manager it’s incomplete.
And yet another, is being consistant in format. If the whole table is being written in scientific notation, then the whole table is being written in scientific notation and with the underlying scientific assumptions about accuracy, not “only those numbers where the scientific notation is shorter than decimal notation.”

5

Note that when you use scientific notation in the form 1.23e4, you don’t make the 4 a superscript. Unfortunately there is a vastly different meaning of exp(4) that could be confused with it.

6

For me, scientific notation provides a quick take on the order of magnitude of a number. When numbers are all written in normalized scientific notation (the non-exponent part is between 0 and 10), you can quickly compare and assess orders of magnitude.

7

Good responses so far, and pretty much in line with what I was expecting. To comment on brewha’s reply:

I see exactly what you’re saying, and I implied the very same thing in my OP, about how scientific notation can imply accuracy as well as the possibility of inaccuracy. Also, I agree 100% that the trailing zeroes in a decimal expression will convey exactness. But while the trailing zeros in decimal notation states explicitly that this number is exact, is it also not customary that decimal numbers are implicitly exact?

For example, “273.15” is exactly two hundred seventy three and one hundred fifty thousandths. However, if you knew that the number you were expressing was not exactly 273.15, you would notate it differently. You might draw a line over the “15” to express that they repeat infinitely. Or you could write “273.15…” to convey that the decimal places continue on, either rationally or irrationally. Such conventions exist because any decimal number without such conventions is understood to be exact.

(Different contexts and usages apply, of course, and may change things. In response to comments about context, the reference table I was looking at used a mix of decimal and scientific notation, and for the most part, the logic behind their usage was obvious. But there were a few that stood out, such as the examples I gave.)

  • Peculiar.
8

I’m pretty sure that his point is that 273.15 means 273.15+/-.01, whereas 273.1500 means 273.1500+/-.0001. So the extra zeros imply a different margin of error.

9

Since when did E become common use other than in calculators? I can’t say I’ve ever, ever seen it used in any printed text. My natural instinct on a computer is to use ** 10^3* or x 10^3 because the caret means an exponent, and is a natural substitute when one can’t simply write ×10³.