Learn · Advanced incremental
Idle games are the rare games that run out of numbers. A balance can pass a billion in its first hour, and some games are built to go far past 10308. How the game stores it decides where it stops, and what it quietly loses on the way. Here are six ways to hold a number, and where each one gives out.
Every value exact, down to the last coin, up to a hard ceiling.
Also called int, Int32
A whole number in 32 bits: exact, fast, and finished at 2,147,483,647. One more and it wraps around to −2,147,483,648, unless the code checks first.
Also called long, Int64
Twice the bits: every whole number up to about 9.2 quintillion, exactly. Plenty for a game that grows by adding, but anything that multiplies gets there fast; the eight-tier chain on Generator chains does in about 15 minutes.
A handful of significant digits and a power of two to scale them, so the range is enormous but the number of digits is fixed.
Also called single precision, Single
About seven significant digits, scaled by a power of two, up to 3.4 × 10³⁸. Past 16,777,216 not every whole number fits, so adding one coin can do nothing at all.
Also called double precision, Number in JavaScript
About sixteen significant digits, up to 1.8 × 10³⁰⁸. It's every number in JavaScript, and the Infinity of Antimatter Dimensions, which sits just under 2¹⁰²⁴, where doubles run out.
Libraries that store the exponent as a number of its own, for games whose numbers outgrow the hardware's.
Also called break_infinity, BigDouble
Keep the leading digits as a double and the power of ten as a number of its own, and the ceiling jumps to 10 to the 9 quadrillion. The digits kept are still a double's sixteen.
Also called break_eternity, tetration
Store a sign, a magnitude, and a layer: how many times 10 is raised to the power of it. Numbers become towers like the 10^10^10^(1.2 × 10³⁴) above, up to a tower 1.8 × 10³⁰⁸ tens tall.
digits in n = ⌊log10 n⌋ + 1
So 2,147,483,647 has 10 digits, the largest double has 309, and break_infinity's ceiling has nine quadrillion. Idle games are easiest to think about in logs: an income that doubles every minute adds about 0.3 to its log each minute, so it gains a digit every three and a third minutes.
value = significand × 2exponent
A float keeps 24 binary digits of significand (24 × log10 2 ≈ 7.2 decimal digits) and a double 53 (≈ 16). Between neighbouring values there's a gap of 2exponent − 23 for a float. Below 16,777,216 = 224 the gap is at most 1, so every whole number fits; from there it's 2, so 16,777,216 + 1 lands halfway between two values and rounds to the even one, back where it started.
(a × 10m) × (b × 10n) = (a × b) × 10m + n · (a × 10m)k = ak × 10m × k
Store the exponent as its own number and multiplying is adding exponents, and powers are multiplying them, so overflow waits until the exponent itself runs out: break_infinity stops it at 9 × 1015. Adding is the hard part: the exponents have to be lined up first, and if they differ by more than about 16 the smaller number vanishes, the same limit a double has. More range never means more digits.
10↑↑3 = 101010
Past even that, break_eternity counts how many times 10 is raised to a power, its layer, and keeps a double for the top of the tower. That's tetration, the step after exponentiation, as exponentiation is the step after multiplying.
Add one coin every second for an hour, to a balance held in a float and in a double. Pick a starting balance.
Balance 1,000,000 + 1 coin a second, for an hour
Both keep every coin: the gaps between the values they can hold are no bigger than a coin.
Balance 16,777,000 + 1 coin a second, for an hour
The float keeps 216 coins, until the balance reaches 16,777,216 (2²⁴), where its steps grow to 2 coins and every +1 rounds back. The double keeps it all.
Balance 100,000,000 + 1 coin a second, for an hour
The float can only step 8 coins at a time here, so each +1 rounds back and the whole hour is lost. The double keeps it all.
Balance 1,000,000,000,000,000 + 1 coin a second, for an hour
The float can only step 67,108,864 coins at a time here, so each +1 rounds back and the whole hour is lost. The double keeps it all.
Balance 9,007,199,254,740,000 + 1 coin a second, for an hour
The double keeps 992 coins, until the balance reaches 9,007,199,254,740,992 (2⁵³), where its steps grow to 2 coins: the float's wall again, 29 bits further on.
Balance 10,000,000,000,000,000 + 1 coin a second, for an hour
Now the double's steps are 2 coins, so it loses the hour too: more range wouldn't help, only more digits.
Nothing crashes and nothing warns: the small income simply stops counting. In a real game the culprit is usually a tiny per-frame amount added to a big balance sixty times a second, or a weak early generator still paying into a late-game total. Adding up the small amounts on their own first, and paying them in once they've grown, loses far less.
A number type is a vault with a fixed size, and each kind fails in
its own way when it fills. Integers wrap round, which is why a
checked add (checked in C#) or a clamp belongs on every
balance. Floats overflow to Infinity, then turn the maths into
Infinity and NaN. And long before either, floating point starts
rounding small amounts away, as above.
Antimatter Dimensions turned the vault's door into the game: reaching the largest double opens its first prestige layer, called Infinity, and past it the game's numbers run on break_infinity.js.
The same values in four common notations, to three significant figures (the logarithm to two decimal places). Antimatter Dimensions lets players pick from these and more.
| Notation | 1,234 | 12,345,678 | 1.234 × 1015 | 1.234 × 1040 | 1.797 × 10308 |
|---|---|---|---|---|---|
| Standard | 1.23 K | 12.3 M | 1.23 Qa | 12.3 DDc | – |
| Scientific | 1.23e3 | 1.23e7 | 1.23e15 | 1.23e40 | 1.80e308 |
| Engineering | 1.23e3 | 12.3e6 | 1.23e15 | 12.3e39 | 180e306 |
| Logarithm | e3.09 | e7.09 | e15.09 | e40.09 | e308.25 |
| Type | Largest | Digits kept | Size | Good for |
|---|---|---|---|---|
| 32-bit integer | 2,147,483,647 | Every whole number | 4 bytes | Counts that stay small: levels, items, kills |
| 64-bit integer | 9,223,372,036,854,775,807 | Every whole number | 8 bytes | Money in a game that adds rather than multiplies |
| Float | 3.4 × 10³⁸ | About 7 | 4 bytes | Positions, short timers, and anything that needs only a few digits |
| Double | 1.8 × 10³⁰⁸ | About 16 | 8 bytes | Nearly every idle game's money, from day one |
| Mantissa and exponent | 10^(9 × 10¹⁵) | About 16 | 16 bytes (two numbers) | Games that mean to pass 10³⁰⁸ |
| Towers of exponents | 10↑↑(1.8 × 10³⁰⁸) | About 16, at the top of the tower | Three numbers: sign, layer, magnitude | Games about growth itself, with layers on layers of resets |