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.
980 questions
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Why is the expansion of numbers in other bases always in base $10$?
Examples:
$4705_{10}=4 \times 10^3 + 7\times 10^2+0\times10^1+5\times10^0$
$2453_{7}=2\times7^3+4\times7^2+5\times7^1+3\times7^0$
But $2\times7^3+4\times7^2+5\times7^1+3\times7^0=920_{10}$
I know $...
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4
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What sets of digits can be used to represent all of the real numbers in a ternary numeral system?
For a typical ternary (base 3) numeral system, only the digits $0$, $1$, and $2$ can be used to write numerals. For instance, the number $29_{10}$ is represented as $1002_{3}$ in base 3.
However, ...
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Same digit patterns when applying exponents to factors of 10. True beyond base-10?
Is the relationship between the factors of 10 (e.g. 2 & 5) unique with base-10?
When using powers and inverses, you get the same digits (ignoring the decimal point). Would this similarity always ...
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Can we define division by 0 in an extended algebraic structure? [duplicate]
Body:
In the usual field of real or complex numbers, division by 0 is undefined because no x satisfies 0x=1.
However, various extensions (projective arithmetic, wheels, non-standard analysis, or ...
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What is the algorithm for a square root in balanced ternary?
I'm confused by wikipedia's explanation of balanced ternary's algorithm for the square root. They only show a formula I don't know how to generalize to more digits than 2, and the example given doesn'...
1
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0
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Multiplication Algorithm for a mixed radix base?
I was looking at the factorial number system, a mixed-radix numeral system where the place value of each digit is a factorial number. I figured out addition and subtraction are actually quite simple ...
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1
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Is there a compact base conversion for ternary numbers that avoids carryover operations?
Binary numbers (base 2) can be easily converted into more compact forms using hexadecimal (base 16) or octal (base 8).
Depending on the target base, it is done by first grouping binary digits into ...
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Concatenation of consecutive positive integers: Does a prime also exist in base $4$ (as in bases $2, 3, 5, 6, 7, 8, 9, \ldots$)?
I have verified that prime numbers of the form $123\ldots n$ (i.e., the base-$b$ integer obtained by concatenating $1,2,\ldots,n$ written in base $b$) exist for bases $b=2,3,5,6,7,8,9,\ldots$ (e.g., $(...
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Why no unique symbols for hex decimal values? [closed]
Why doesn't hexadecimals have unique symbols for all 0 thru 15 values?
Yes, there would be 16 more symbols to learn, but I immediately know if this is a hex.
Longterm I see less mistakes, faster/...
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Wrong way to solve a question containing number system radix conversion
Pardon my MathJax formatting skills, as this is my initial attempt.
The textbook question was:
If
$$\sqrt{41} = 5$$
find the radix of the number system.
Approach:
$$ \sqrt{(4.r^1 + 1.r^0)} = (5.r^0) $$...
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1
answer
51
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What is the (full) sequence of isogrammic numerals? [closed]
I read that 5000 is the largest isogrammic numeral, meaning that if you write it out, each letter appears only once. I’m looking for the entire sequence of numbers with such a property. I can’t find ...
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Can a composite number be equal the concatenation of the primes and exponents in its prime factorization in more than one base?
Someone called James Davis (see here) found that
$$
13532385396179 = 13\times 53^2\times 3853\times 96179,
$$
showing that a composite number can be equal the concatenation of the primes and exponents ...
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Why does the choice of variable for induction not matter? [duplicate]
Taken from Pedrick's A First Course in Analysis (1994) Exercise 11:
Prove that $ (n + k) \cdot m = n \cdot m + k \cdot m$
In other words, I have to prove right distributivity. The book doesn't say a ...
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I need help with a question in regards to "the conversion from base 4 to base 6 without going through base 10
QUESTION: Show how we could convert $123_4$ into an equivalent base-six numeral without first going through base-ten, but rather just by using only base-six notation. For example, use such results as $...
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Why is the decimal expansion in binary system done using repeated multiplication by 2? [duplicate]
I understand that converting fractional part of a decimal to binary involves this methods of repeating multiplication by 2:
For example, to convert $.375$(decimal) to binary:
$.375×2=0.75$ → integer ...
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