#1023

1023 is interesting because it shows up everywhere from floating point arithmetic to error-correcting codes to satellites!

1023 is a sum of 5 consecutive primes (193, 197, 199, 211, and 223)

It is a sub-perfect power (a perfect power minus 1) that can be expressed in two different ways, namely 4^5 - 1 and 2^10 - 1. The latter form makes this a Mersenne number.

1023 is a palindromic number in base 2 and base 4. In base 2, it is a string of 1's, and in base 4 it is a string of 3's.

Polycubes are the 3-dimensional analogue of planar polyominoes (like Tetris pieces). Below is a picture of the tetracubes (4 blocks that are connected by one or more faces). If you consider the number of ways you can arrange 7 blocks to be distinct (polycubes with 7 blocks), then there are 1023 distinct combinations.

It is a Stirling number of the second kind, specifically S(11, 2). This means that if you have 11 objects, and you want to group them into 2 unlabeled nonempty subsets, then there are 1023 ways to do it.

In computer science, floating point numbers are often represented using something known as the IEEE-754 double-precision floating point standard. This is the standard used for representing doubles in a ton of different programming languages. It uses 1 sign bit, 11 exponent bits, and 52 fraction (mantissa) bits. To represent a number, you consider the 52 fraction bits to be 0.1101101011101... (52 binary bits beyond the decimal point), and then you multiply by the exponent bit (specifically, multiply by 2 ^ (exponent number - 1023)). For example, 3/4 can be represented as 0 01111111110 1000000000000000000000000000000000000000000000000000. The 0 in the sign bit means this is positive. The 01111111110 represents 1022 in binary, and with the bias, this is equal to 1022 - 1023 = -1. Then the mantissa is 1.1000... (there's a hidden 1 in front for more precision). So this number is 2^-1 * 1.1 = 0.11, which in binary is 1/2 + 1/4 = 3/4. The reason 1023 appears as the bias is because with 11 exponent bits, there are 2^11 = 2048 possible exponent values. IEEE-753 chooses the bias to be 2^{11-1} - 1 = 1023 which centers the representable exponents roughly around 0. So this makes the smallest normal exponent 1 - 1023 = -1022, and the largest normal exponent to be 2046 - 1023 = 1023 (Values 0 and 2047 in the exponent field are reserved for subnormals, infinity, and NaN). And as a quick note, IEEE-754 uses a bias instead of two's complement because comparing floating point numbers becomes easier in hardware. Positive exponents sort correctly as unsigned integers, and the exponent field can be compared lexicographically. It also avoids needing a separate sign bit for the exponent.

A Reed-Muller code is a type of error-correcting code where you turn a polynomial into a long pattern of bits, and the redundancy lets you recover from errors (it's one of the ways a CD can still play music even if it's scratched a little bit). A Boolean polynomial uses variables that are only 0 or 1. For example, f(x, y, z) = x + yz where arithmetic is mod 2, multiplication is AND and addition is XOR is an example of a boolean polynomial since yz = 1 only if both are 1 and x + yz means XOR. You then evaluate it on every possible input. x | y | z | f(x, y, z) 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 This gives the bit string 00011110. That bit string is the codeword. This helps with errors since not every 8-bit string is allowed. Only strings that come from evaluating low-degree polynomials are valid codewords. So if noise flips a few bits during transmission, the receiver can ask which valid polynomial-generated codeword is closest and recover the original message. Generally, Reed-Muller codes are written RM(r,m) where r = maximum degree allowed and m = number of variables. For example, RM(1,3) means you can have a max degree of 1 with 3 variables. So allowed polynomials are of the form a + bx + cy + dz. There is 1 constant terms and 3 linear terms, which gives 4 coefficients and thus a code dimension of 4. The code length is 2^3 = 8 because we evaluate on all 8 inputs. Note that the dimensions of Reed-Muller codes comes from this formula: dim(RM(r,m)) = \Sum_{i=0}^r (m, i) (add up the binomials from m choose 1 to m choose r). 1023 also appears in Reed-Muller codes. For the code RM(9,10), the dimension is \Sum_{i=1}^9 (10, i) = 1023 while the code length is 1024. So RM(9,10) encodes 1023 its of information into 1024 codeworks with enough redundancy for error correction.

This number also appears in technology. A GPS (Global Positioning System) satellite continuously broadcasts its current position, the current time, and a special pseudorandom bit pattern (the PRN code). Your phone receives signals from multiple satellites and computes distance = c * time delay (where c is speed of light and time delay is how long the signal took to arrive). With distances to several satellites, your phone triangulates your location. GPS time is represented as a week number plus the seconds within that week. Because GPS started on January 6th, 1980, you can store time as week 2417, second 123456 of that week instead of storing a full date like May 24th, 2026. GPS works on a 10-digit binary counter that runs for 1024 weeks, 0 to 1023, at which point an integer overflow causes its internal value to roll over to 0 again. 1024 weeks is around 19.7 years, which is why GPS rollovers happened in 1999, 2019, and will happen again around 2038.

But there's another place where 1023 is found with GPS. Each satellite sends a radio signal that looks roughly like a very accurate carrier wave, multiplied by a rapidly changing binary code, and then navigation data. That rapidly changing binary code is the famous C/A code which you can think of as an "identifier" for a given satellite. The C/A code exists because if every satellite transmitted just a plain sine wave, you would have satellites that interfere with each other which would making timing ambiguous. So each satellite transmits a unique pseudorandom sequence like 11011010011100110011101010111000111100 . . . These sequences look random, but they are actually deterministic and repeat every 1023 bits ("chips"). Your receiver (phone) generates the same code internally and slides it back and forth until it matches. The amount of shift tells you the travel time and which satellite gave the transmission. The civilian GPS C/A code is generated using a 10-stage linear feedback shift register (LFSR), which is a simple way to generate a long sequence of bits that looks random but is actually completely deterministic. It's basically a row of bits, a shift operation, and some XOR logic feeding back into the front. An n-stage maximal LFSR produces a sequence length of 2^n - 1 before repeating. So for n = 10, we have 2^10 - 1 = 1023. Thus the GPS C/A code has length 1023 chips. This also means that a GPS's radio signals at certain levels operate with a period of 1023, and the frequencies used are multiples of 1023 kHz. This makes it so that one full 1023-chip code period lasts exactly 1 millisecond. Since 1023 chips / ms = 1.023 * 10^6 chips / s, we get the famous 1.023 MHz chip clock. This makes timing and ranging calculations extremely convenient. Hence the system clock architecture was built around this base frequency.

PT: 1023: Princess~, or princesstilde /end PT

DEFINITION ⦂⠀A term for when one identifies as a princess, or connect to princesses in a loved, attached, and loving way.

PT: Definition: A term for when one identifies as a princess, or connect to princesses in a loved, attached, and loving way. /end PT

ADDITIONAL ⦂⠀Coined on the 7th of April, 2026.

PT: Additional: Coined on the 7th of April, 2026. /end PT

TAGGING ⦂⠀@c1rcus-of-silliness @dearestchild @goregender @radiomogai @theeself