Questions tagged [number-systems]

Representations of numeric values in decimal, binary, octal, hexadecimal, and other bases; one's-complement and two's-complement signed numbers; scientific notation; floating-point numbers in digital computers; history of number systems; nonstandard number systems; algorithms for arithmetic within specific number systems or for conversions between number systems.

Number systems provide systematic ways to write numeric values such as the (base-ten) numbers $289$ or $2.125$. Some questions with this tag involve algorithms for converting base-ten numbers to or from another number system; conversions between other number systems; algorithms to perform arithmetic (addition, subtraction, multiplication, etc.) within a specific number system without converting the operands to base ten; symbols for writing numbers in systems other than base ten; ancient number systems (such as Roman numerals) and the historical development of number systems; and specialized or unusual number systems.

A base-$b$ number system represents an integer as a sequence of digits, each of which is an integer such that $0 \leq d < b$. Ordinary decimal numbers are written in base ten; other well-known bases include binary (base $2$), octal (base $8$), and hexadecimal (base sixteen). Optionally, the base or radix, $b$, may be appended as a subscript. The value of such a numeric representation is

$${\left(d_m d_{m-1} \cdots d_2 d_1 d_0\right)}_b = d_m b^m + d_{m-1} b^{m-1} + \cdots + d_2 b^2 + d_1 b^1 + d_0 b^0.$$

For example, $21_{16} = 33_{10} = 41_8 = 100001_2$, representing the same value as hexadecimal, decimal, octal, and binary numbers, respectively. The factors $b^0$, $b^1$, $b^2$, and so forth are the place values of the digits. A base-$b$ number with a fractional part is written by appending a decimal point and digits with place values $b^{-1}$, $b^{-2}$, $b^{-3}$, and so forth; for example, $$101.011_2 = 1\cdot2^2 + 0\cdot2^1 + 1\cdot2^0 + 0\cdot2^{-1} + 1\cdot2^{-2} + 1\cdot2^{-3} = 4 + 1 + \frac14 + \frac18 = 5.375_{10}.$$

In a mixed-radix number system, such as the factorial number system, the ratio between the places value of two digits depends on their distances from the decimal point. A number system can have a negative radix, for example the negabinary number system, which has the radix $-2$.

Digital computing has raised interest in various other number systems. In an $n$-digit $b$'s-complement base-$b$ representation, the integer $-x$ is represented by $b^n - x$, whereas in a $(b-1)$'s complement representation, $-x$ is represented by $(b^n - 1) - x$. Computers often use two's-complement (or sometimes one's-complement) binary numbers.

Very large or small numbers can be written in scientific notation, for example $1.234 \times 10^9$. Floating-point numbers in digital computers, typically using the IEEE 754 standard, serve a similar purpose.

More esoteric number systems of interest in computer science include:

  • Balanced base-$b$ number systems, which use both positive and negative digit values. The balanced ternary (base $3$) system with digit values $\{-1,0,1\}$ is an example of this kind of number system.
  • Redundant base-$b$ systems, which allow more than $n$ values of each digit. There may be many ways to represent a given number in such a number system.
  • Residue number systems, in which each digit position is assigned a fixed modulus and the digit in that position is the remainder when the number's value is divided by that modulus.

Other possible numbering systems include the Fibonacci base system and systems using a non-integer radix such as the $\phi$ number system.

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Possible values of $n$ such that $n!$ ends with $30$ zeroes

If $n!$ ends with $30$ zeroes, how many values of $n$ are possible? I know that $10$ will occur when pairs of $2$ and $5$ will occur and since $2$ occurs more times than $5$ does in $n!$, the number of trailing zeroes will be the highest power of…
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