# Tagged Questions

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 ...

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### How to find the base of this expression?

I need to determine the radix of the numbers for make this expression true 15 = 8 How I can do that?
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### Converting repeating decimal in base b to a fraction the same base

The repeating decimal .36666... in base 8 can be written in a fraction in base 8. I understand simple patterns such as 1/9 in base 10 is .1111.... so 1/7 in base 8 is .1111. But I'm not too sure how ...
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### Decimal Place and crossing the boder

I always was a little confused by this notion but never thought to investigate it. In school and as I grew older people in this world (mathematics) would just say " that's the way it is " as in other ...
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### Distinct digits in a combination of 6 digits

How many 6-digit numbers contain exactly 4 different digits? My approach is: For any 3 digis same and the remaining 3 different(aaabcd) 4*9*8*7*6 For any 2 duplicate digits(aabb) and the remaining ...
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### The last eight digits of the binary development of $27^{1986}$

Find the last eight digits of the binary development of $27^{1986}$. We define $v_p(x)$ such that if $v_p(x) = n$, then $p^n \mid x$ but $p^{n+1} \nmid x$. Now we see that if $n \geq 2$ is an ...
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### How many 6-digit numbers contain exactly 4 different digits? [duplicate]

my solution is----> NUMBER can be 777210 this i denote by aaabcd ------ this can be done in ---> 10*1*1*9*8*7*[6!/3!] {1 for a thrice} NUMBER can be 772210 this i ...
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### Converting from base $x$ to base $y$

I'm trying to convert from base $x$ to base $y$, but am having trouble understanding why the following method only works when converting to base $10$. Take for instance the number $2132$ (base $4$). ...
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### the Greatest Number Among $3^{50} ,4^{40} ,5^{30}$ and $6^{20}$ [closed]

how to find the Greatest Number Among $3^{50} ,4^{40} ,5^{30}$ and $6^{20}$ please give a short cut method
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### Is there tree-based numeral system? [closed]

Preferrably such that ((()) ()), (() (())), (((()))), (() () ()) all were different numbers and any tree can be a number.
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### Numbers that have constant digits value independent of base

I was wondering why for example $\dfrac{1111_b \cdot 111_b}{11_b \cdot 1_b} = 11211_b$. Is there a good explanation for this and is there a name for products like this which have constant digits value ...
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### No square has a decimal expansion ending in 79

Show that no square number has a decimal ending in 79. More generally, find all possible two-digit endings for squares. Let any digit number ending at 79 be represented as $$a_nx^n+.....+7x+9$$ Plug ...
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### Relationship between decimal length and Fibonacci number

There are 6 single digit Fibonacci numbers. For all other number of digits in the decimal system, there are either 4 or 5 Fibonacci numbers. For example, between 10000 and 99999 there are 5: 10946 ...
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### Numbers divisible by $11$ [duplicate]

A number is divisible by $11$, when the difference between the sum of the digits in the odd positions counting from the left (the first, third, ....) and the sum of the remaining digits is either 0 or ...
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### Convert the following between octal, decimal and hexadecimal

(a) Convert $61502$ from base $8$ to decimal. (b) Convert $EB7C5$ from base $16$ to octal. My answer: a) $6\times8^4+1\times8^3+5\times8^2+\times8^1+2\times8^0=25410$ b) not sure: converting $E=14$,...
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### Roman Numbers - Conversion to decimal number

I have read that if a smaller number is to the left of a larger number means that the smaller number has to be subtracted from the larger number. Ok I can understand quickly for below Roman Numbers : ...
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### How can I tell if a number in base 5 is divisible by 3?

I know of the sum of digits divisible by 3 method, but it seems to not be working for base 5. How can I check if number in base 5 is divisible by 3 without ...
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### Find the rightmost digit of: $1^n+2^n+3^n+4^n+5^n+6^n+7^n+8^n+9^n$

Find the rightmost digit of: $1^n+2^n+3^n+4^n+5^n+6^n+7^n+8^n+9^n(n$ arbitrary positive integer) First of all I checked a few cases for small $n$'s and in all cases the rightmost digit was $5$, so ...
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### Is it an overflow or not?

The addition of 4 bit, 2's complement binary numbers 1101 and 0100 is $$\begin{array} \\\hphantom{+}1101\\ + 0100\\ \hline \\ 1 \ 0001 \end{array}$$ there occurs a carry out above, but this will ...
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### Convert floating point number to short int [closed]

This is the transfer function of a FIR filter $$H(z) = -0.125 + 0.25z - 0.125z^2$$ How to convert the coefficients to short int? Is it correct if I just multiply ...
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### Quaternary numeral system: fractions

I have a question related to the expression of a real number in base 4. Consider the table here: it is clear to me how all columns of the table are obtained except the fourth one: how do they get the ...
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### Why do they call it base 10?

Now, I know intuitively why it's called base 10: because there's 10 numbers. But see here's the thing, if we're working with numbers 0-9 (and of course we are), we use up our numerical artillery at 9....
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### How number system works in cases of hex, bin, dec and oct?

I have this question or a confusion from college time. When we convert a decimal digit to binary, we divide the decimal digit with the base of binary number system. Why there is no similar method ...
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### How to calculate after decimal point of 99.21(Base 10) to (Base 8)?

As I know by 0.21 x8 and -4 for the answer to convert to base 8. However, I could not calculate and keep on expanding the value to millions.
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### Expressing fractions in different bases. [closed]

In particular, a fraction that, in a certain base, will be recurring - such as $\frac12$ in base 3. (I used a base changing calculator to find it out) I have tried using repeated multiplation as ...
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### Why $82000$? Numbers that can be written from base $2$ to base $5$ using only the digits $0$ and $1$

This is really very curious. Many links on http://oeis.org/A146025 about this but -- why? I mean, this is not some abstract mathematical notation but rather something inherent in, I dunno, the ...
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### Convert this into fractional number step by step?

3.41287548754875... Convert the above number to a rational number? I was reviewing some pre calculus on my own but couldn't figure this out.
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### Alternative decimal number representation

In class we had an overview of different representations of numbers. Some examples were: decimal, roman numerals, binary, Redundant binary representation, church numerals... Then the teacher gave an ...
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### Mayan Number System Explained.

I have recently been studying the Mayans and have encountered their number system. A dot represents 1 A line represents 5 A shell represents 0 The base of the number system is 20 During my ...
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### Is PA the first axiomatization of arithmetic to be discovered? [closed]

Is Peano Arithmetic the first axiomatization of arithmetic to be discovered?
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### How many ordered pairs (x, y) exist which satisfy the following equation? Both x and y are whole numbers. [closed]

$x \cdot\ y=2^2\cdot\ 3^4 \cdot\ 5^7 \cdot(x+y)$ I have tried rearranging the equation to different formats but not getting anywhere.
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### Converting from decimal to any number base.

I know that this can a silly question. But I can not find the answer. When converting number bases (from decimal to any other number bases), we divide the decimal number by that base, and write down ...
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### Can every real number be written as unique infinite decimals?

A slightly variation of this question: Can every real number be represented by a (possibly infinite) decimal? Here I restrict my discussion into infinite decimals (without ending with infinite zeros ...
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### What's the real life purpose of Knuth arrows?

I recently read about Knuth's Arrows. Didn't even know those operations existed. My questions is: Do they have real-life applications? Most of the times a mathematical development follows a real-life ...
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### converting fraction into binary by division

So i have a question regarding converting from farctions into binary decimals. if i have $\frac{3}{17},\frac{2}{9},\frac{1}{7}$ How can i convert those fraction into binary decimals using divison. ...
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### What does leading $0s$ in a number in scientific notation mean?

Source: Computer Organization and Design: The Hardware/software Interface, David A. Patterson,John L. Hennessy It doesn't seem as the author is using leading $0$ like leading $1$ in a matrix. What ...
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### Are there names for any of these four classes of numbers related to divisors and totatives?

Are there names for any of these four classes of numbers related to divisors and totatives? A [insert name here] of $n$ is a positive integer $\leq n$ that isn't a divisor of $n$ and that can be ...
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### a*b = a/b = b/a (what's this symmetry called?)

I was playing around with numbers the other day, and I found an interesting symmetry, that I would like to know if it has any specific name assigned to it. Let's assume the notation n:a to refer to ...
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### Delectable numbers in other bases

So there's a pretty useless number called the delectable number. A delectable number has nine digits, using the numbers 1-9 once in each digit. The first digit of a delectable number must be divisible ...
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### How to prove that every power of 6 ends in 6?

Yesterday I had the traditional math matriculation exam, and in it there was a question "In what digit does the number $2016^{2016}$ end in?" After the test The Matriculation Examination Board ...
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### Let $x \in \mathbb{R}$. Then $x^2 < x^3$ if and only if $x > 1$ [closed]

I am having a problem with proving this statement. Any help would be appreciated! Thank you
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### Base ten representation proof

I want to prove that the pattern repeats but I can't figure out how.. Any help would be deeply appreciated!
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### Given a base, a mantissa size and a range for the exponent, what's the largest number?

I was introduced to floating-point numbers week ago or so, and although I think I understood the basics, I'm still not sure how to apply them. A floating-point number is usually represented with the ...
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### Could we “invent” a number $h$ such that $h = {{1}\over{0}}$, similarly to the way we “invented” $i=\sqrt{-1}$? [duplicate]

$\sqrt{-1}$ was completely undefined in the world before complex numbers. So we came up with $i$. $1\over0$ is completely undefined in today's world; is there a reason we haven't come up with a new ...
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### Inequalities (Natural Numbers)

Suppose that $1 < x$ and $z < x^{z}$ is true where $x, z \in \mathbb{N}$. Prove that $z + 1 < x^{z+1}$. I have tried to use every inequality but have not been able to find the proof. This ...
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### Defective coin weighing problem and ternary representation when number of coins is not a power of 3

We are given $N$ coins and a set of scales. We are told that there is a defective coin and we know whether it is lighter or heavier than the others. Our goal is to identify it in as few weighings as ...
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### The last two digits of $13^{1010}$.

$13^{1010}$ $13^{\phi(100)} \equiv 1 \mod 100$ $13^{40} \equiv 1 \mod 100$ $(13^{40})^{25} \equiv 1^{25} \mod 100$ $13^{1000} \equiv 1 \mod 100$ $13^{1010} \equiv 13^{10} \mod 100$ That's all I ...
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### rows of pascals triangle as powers of 11 in different numeral systems

It's not too difficult to see that (and understand why) in a base n system the ciphers of $(11_n)^k$ are equivalent to the k-th (0-indexed) row of pascals triangle until one of the numbers becomes ...
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### What's the technical term for “place-value”?

When talking about positional notation, is there a technical term for "place-value" (as in, "the place-value of the 9 in 792 is 10), or is that it? Somehow, "place-value" sounds informal, but I don't ...