If you've spent a reasonable amount of time on the internet lately, you've probably at least seen gameplay of Balatro. If you've spent a reasonable amount of time watching videos of (or playing) Balatro, you've probably seen people get such unreasonably high scores that the game switches to scientific notation (e.g. 1.43e15).
What does that mean?
I won't spend any significant amount of time going over it, but scientific notation is just a way to write large numbers without having to write out all the digits. It is important to note that this does lose quite a lot of precision, but we do this all the time; when I say the world population is "8.2 billion", I obviously don't mean exactly 8,200,000,000. Do you really need to know the other 8 digits of that number? By abbreviating it, it communicates that the other digits of the number don't really matter.
Scientific notation is a more mathematical way of doing this sort of communication; instead of converting the number into a name, you simply represent the name as a power of 10. For instance, 103 is one thousand, 106 is 1 million, and so on. Note that they don't have to be multiples of 3; 108 is 100 million, for example. This power of 10 is then multiplied by some real number between 1 and 10 (excluding 10 itself)*This is analogous to picking a number between 1 and 1000 to put in front of a number name; the reason we exclude 10 is the same reason why we don't say "1000 million" to represent 1 billion..
Put bluntly, a number in scientific notation takes the form of [mantissa] x 10[exponent]. Many text-based computer programs are unable to properly render the exponent, however, so you may see it in the form of [mantissa]e[exponent]*You may also see the exponent written out with an explicit plus sign in front of it (e.g. 1.23e+4). Again, it means the same thing.. There is no difference between the two other than they way they're written. The mantissa can be any real number between 1 and 10, and the exponent can be any integer. Those specific constraints ensure that each number has a "canonical" (unique) representation in scientific notation.
There is one more slightly unconventional concept I'd like to bring up, which is engineering notation. It's very similar to scientific notation, but with two key differences:
So, for instance, 31.41 billion is represented in engineering notation as 31.41 x 109. It's crucial to realize that this lines up with how you would write down the number in English; the mantissa remains the same, and the actual number word maps 1:1 with an exponent value*For instance, an exponent of 3 means "thousand"; 6 means "million"; and so on. Hopefully the pattern becomes clear in engineering notation..
Converting a number from scientific to engineering notation is easy; if the exponent is not already a multiple of 3*If you're doing this in your head, a useful shortcut is that a number is divisible by 3 if all of its digits add up to any multiple of 3., subtract 1 from it and multiply the mantissa by 10. If the exponent still isn't divisible by 3, perform that step again (you only ever have to do it at most twice).
Converting a number the other way, from engineering to scientific, is even easier: if the mantissa is 10 or greater, divide it by 10 and increment the exponent. Repeat this step again if necessary. Here are a few example conversions:
| Scientific | Engineering |
|---|---|
| 6.02 x 1023 | 602 x 1021 |
| 8 x 1013 | 80 x 1012 |
| 4.2 x 1069 | 4.2 x 1069 |
Note that the last one is the same in both forms; the exponent is already a multiple of 3, and needs no conversion.
It's very unlikely that you've used any number larger than a trillion in your day-to-day life, but they do in fact have names. You might have heard of them:
| Exponent | Name |
|---|---|
| 3 | thousand |
| 6 | million |
| 9 | billion |
| 12 | trillion |
| 15 | quadrillion |
| 18 | quintillion |
| 21 | sextillion |
| 24 | septillion |
| 27 | octillion |
| 30 | nonillion |
| 33 | decillion |
These numbers are very big. I need you to understand that. A lot of these numbers are already hitting the limits of human comprehension. For instance:
But is it possible to go higher? What would we name those numbers? Do the names go on forever? Well, the answer to all of those questions*Yes, all of them is yes!
If you take a close look at the aforementioned table of names, you might find a few of those prefixes familiar. Those prefixes (bi-, tri-, quad-, etc.) are Latin numeral prefixes, and they show up all the time in English. (They're the reason why bicycles and tricycles have those names, for one). And, sure enough, there's prefixes past 10, too!
Forming these prefixes is actually quite easy. Simply break down the value that the prefix represents ("quad" represents 4, for instance) into its individual ones, tens, and/or hundreds places. Then, take the corresponding value/position combination, and just string them all together into one long prefix. Note that they're combined in reverse order than they're written in; for instance, the prefix representing 32 is "duotrigin".
Here is the table that shows all of the individual components to form a Latin prefix, up to 999.
| Value | Ones | Tens | Hundreds |
|---|---|---|---|
| 1 | un- | deci- | centi- |
| 2 | duo- | vigin- | ducen- |
| 3 | tre- | trigin- | trescen- |
| 4 | quattor- | quadragin- | quadringen- |
| 5 | quin- | quinquagin- | quingen- |
| 6 | sex- | sexagin- | sescen- |
| 7 | septen- | septuagin- | septingen- |
| 8 | octo- | octagin- | octingen- |
| 9 | novem- | nonagin- | nongen- |
Now, you might immediately notice a few things:
These aren't 1:1 with actual Latin prefixes.
You can see that a billion should be called a "duotillion" according to this chart. This is because the common set of prefixes used today largely comes from the Conway-Guy system, which shuffles around a few of the prefixes to make them less ambiguous or confusing. In fact, the table here is slightly modified from the Conway-Guy system; the table shown here represents the more common shortenings found in some dictionaries, such as "quindecillion" instead of "quinquadecillion".
There's no row for 0.
Zeroes simply have no corresponding prefix; the prefix for 103 would simply be "trecentillion". (Not to be confused with the one for 300, "trescentillion". Note the S.)