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.

692 questions
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Is there an ordering on $\mathrm Z$ to make it dense?

It is well-known that the set of rational numbers $\mathrm Q$ is dense; that is, given any two rationals, say $r\ne s,$ then there exist infinitely many rationals $r_i$ such that $r<r_i<s$ for ...
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Composition of permutations in factoradic notation

The Lehmer code maps between permutations and natural numbers. My question: Is a "nice" algorithm known for composing permutations directly in the factoradic form? When the permutations commute, it ...
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Are numbers independent of the basis we use?

So I have recently read about Kaprekar's Constant (https://en.wikipedia.org/wiki/6174_(number)) and It made me wonder If this number is really "special"? It seems to me that the notion 6174 (and the ...
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Are the digits of $\log 2$ and $\log 3$, or $\sqrt{2}$ and $\sqrt{3}$, or $e$ and $\pi$, cross-correlated?

I try to find sequences of digits (in base $b$, with $b$ not necessarily an integer) that are not cross-correlated. While the digits in base $b$ of (say) $\pi$ and $e$ do not exhibit auto-correlations ...
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Is a Centillion bigger than a g Googol? [closed]

I heard that a googol is a 10 to the 100th power, in other words, 10^100, and a centillion is a 10 to the 303rd power, in other words, 10^303. Some people disagree though.
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Evaluate the number of different numbers one can write, having at one's disposal n digits of a q-ary system.

I am working on Zorich, Mathematical Analysis I and I am stuck on a problem in section 2.2. The problem asks me to evaluate the number of different numbers one can write with n digits of a q-ary ...
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Determine a property of $S_b(n)$, which is the sum of the digits of $n$ when $n$ is expressed in base $b$

The original question Let $S_b(n)$ be the sum of the digits of $n$ when $n$ is expressed in base $b$. was asked by Anson Chan on Jan. 24, 2019. Since it was deemed to not have enough context, it was ...
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Difficulty understanding how Babylonian reciprocals work

According to the reciprocal tables for Babylonian's base $60$ system, dividing by $2$ is like multiplying like $30$. Dividing by $3$ is like multiplying by $20$. Dividing by $4$ is like multiplying by ...
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How to prove that when we change numeral system same number becomes shorter

Let's start with an example: We have a binary number $1100_2$ In decimal it will be equal to $12_{10}$ In hexadecimal it will be equal to $C_{16}$ I'm trying to find a proof that for any positive ...
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Babylonian Operations Not Working Out So Well

I'm going to denote 1 as v and 10 as < to replace the Babylonian symbols my homework says to evaluate $$(vv)+(<vvvv)+(<vv+vv)+(<<v)+(vv)$$ Because Babylonians used powers of 60 for ...
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Confusion about normalised floating point number systems

I was assigned the following question: List all numbers that can be represented exactly in a normalised floating- point number system with base 10, two digits in the fraction, and an exponent 0 ≤ e ≤ ...
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Rational numbers in irrational bases

If you take the base-$b$ expansion of a rational number where $b$ is irrational, do you get a non-terminating sequence of digits (assuming you pick the right(?) digits)? More informally, do rational ...
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Number system conversions

I am an Electrical Engineering student but my question is related to number systems, more specificaly to conversions between octal, hexadecimal and binary systems. I know the rules of conversions but ...
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Why multiplication by numeral system base and raising to corresponding power converts any numeral system integer to decimal only

I am not very good in math so I apologize if my question is too simple and does not belong here... Why an integer in numeral system X can be converted to decimal by multiplying it's digits to X (...
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Big Integer Base Converter?

I need to convert some really big numbers from base-10 to base-8. All online converters I found cap the conversion at around 20 digits. Are there any downloadable programs with no digit limit? I'm ...
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A signed number in $2's$ compliment notation with $16-$bit register $P=DFA0$ What is the value of $P*4$? I have tried $P*4=0111111010000000$ but sign bit changed from $1$ to $0$. What is wrong ...
$\sqrt{(144)_r} =(12)_r$ I have tried $\sqrt{r^2+4r+4} =r+2$ From this, I am unable to find the value of $r$. Can anyone help me out to solve this problem?