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.

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how this number system works the examples provided with this thread?

A B 0,0 1 0,1 2 -1,1 3 -1,0 4 0,-1 5 1,-1 6 1,0 7 1,1 8 0,2 9 -1,2 10 -2,2 11 -2,1 12 -2,0 13 -1,-1 14 0,-2 15 1,-2 16 2,-2 17 2,-1 18 2,0 19 2,1 20 1,2 21 0,3 22 -1,3 23 for ...
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How to deal with irrational values with a bijective number system?

Having done some research on bijective numeration - that is, a number system in which every non-negative integer can be represented in exactly one way using a finite string using a finite set of ...
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Question regarding r's complement

This question was asked in a test in my undergrad course A computer, using 3-digit radix complement arithmetic with an unknown radix r, gives the following results expressed in radix complement form: ...
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Numbers in $-A - (-B)$ form for $ 2's$ Complement [closed]

I know for $2's$ complement representation, you are supposed to perform an operation like $A - B$ into addition but perform $2's$ complement on $B$. So $0101 - 1001 = 0101 + 0111$ Now, if I had a ...
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Can this binary number be represented as a fraction $a/b$? [closed]

Can the binary number $0.1010\ldots$ be represented as a fraction $a/b$ with $a$ and $b$ as integer numbers?
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2answers
41 views

How to find $N-60$ when it is based on the sum of its prime divisors?

The problem is as follows: A certain number coincides with the sum of the prime positive divisors of the number of zeros in which the product of the first $784$ integers ends positive, when expressed ...
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2answers
40 views

Nos Systems Ten

How do I find $4$ digit $10$'s complement of $-456$ (withbase $10$)? I am able to do $4$ digit $10$'s complement of a positive number (base 10) using the formula $base^n - number$ where $n$ is the ...
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1answer
97 views

Problem regarding a MST proof of 0.99999…=1

I came across an answer of this (which is the highest voted, and also awarded bounties worth 50 reputations). To quote, this is what the answer said:- "Suppose this was not the case, i.e. $0.9999....
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45 views

Trouble in the proof of Theorem 1.1. in the book Structure of the Real Number System

The Question THEOREM 1.1 There is exactly one natural number which is not the successor of any natural number. PROOF: Let $N_S$ be the range of $S$. By Axiom $A_2$, there exists a natural number $u$ ...
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A problem in arithmetic

Okay, so I recently began learning the number system, i.e., the place value system, viz., the decimal system. I understand how the 1st place from the right represents the number of ones, the second ...
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2answers
46 views

Significant figures: Is the answer on this question correct?

A google form question So I got this question in a test and options 2, 3 and 4 were wrong, in my view. So I chose a random option and I got it wrong. Upon further enquiry, the teacher explained that ...
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45 views

How to find $\overline{abc}_{7}-\overline{cba}_{7}=\overline{xy4}_{7}$?

The problem is as follows: Let: $\overline{abc}_{7}-\overline{cba}_{7}=\overline{xy4}_{7}$ Assuming $a=b-c$ Find: $cba_{7}$ in terms of decimal system. How exactly should I do this?. Can someone ...
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The Soup Problem: how to asymptotically fairly split a geometric series and a constant one using a single pattern?

Literally every time I'm serving some soup I'm thinking of this little mathematical problem I devised. Imagine you have a very large (= infinite, for the purposes of the actual problem) bowl of soup ...
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Can all naturals can be expressed as sums of prime powers of $\{3,5,7,13\}$?

Define the set $$S=\left\{(p,k) \mid p\in\{3,5,7,13\} \land k \in\mathbb Z^{0+}\right\}.$$ The question is whether every natural number $n$ can be expressed as the sum of some subset $s\subseteq S$ of ...
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38 views

How to tell if a base for a number system is consistent (a number has only one representation)?

Let's say I have a number system with an array defining the bases at each position. For example, [1,1,1] would be a binary number system (each digit can be 0 or 1) in which any number from 0 to 3 can ...
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73 views

in which base is the number 221 the factor of 1215

I basically tried to find a number $n$ which would be the base and then divide both the numbers $221$ and $1215$ so as to get a combo which has same remainder But I am not able to proceed in doing so. ...
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How we can write 36 in base 37?

i've been imagining that in the base 16 the number 10 is A and 11 is B so in the base 37, 35 will be Z so how we can write 36 ?
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31 views

Fourier transform and spinning wheels with numbers

I was wondering if there is an analogy that can be built with spinning wheels with number markings on them and the Fourier transform. For example if the markings are in $[0,10)$. Then fast frequency ...
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1answer
58 views

Are there any properties that distinguish the decimal numeral system?

Since early childhood we are used to work with the decimal numbers. Living in the world, where everything is written is terms of decimals people feel most convenient, when performing the calcuations ...
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2answers
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Base Number Addition (base 6)

So, I'm studying base number addition (AoPS book) and I'm kind of stumped. Any help would be thankful. So the question was: Perform the following base number addition: $ 1_6 + 5_6 = ? $ So initially,...
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Find all positive integer solutions of the equation

Find all positive integer solutions of the equation $1650_x = 1131_3$$_y$ where $x$ and $3y$ are the bases of the numbers on the two sides. I have this example in my book but have never come across an ...
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25 views

Why is the range of reciprocal function equal to $\mathbb R\smallsetminus\{0\}$?

Why is the range of reciprocal function $0$ and not the entire real number set? My reasoning is that, because the domain of the reciprocal function is $\mathbb R\smallsetminus\{0\}$ to avoid the ...
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2answers
45 views

Is there any notation for the undecimal system? [closed]

Decimal is 0 1;2;3;4;5;6;7;8;9 10;11;... But undecimal 0 1;2;3;4;5;6;7;8;9;? ?0;?1 TWas this system used by an ancient civilisation ? (I don't know what to tag)
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Positional notation: proof of relations for different base but same digits

I have the following situation: Be $n_1, n_2, b_1, b_2 \in \mathbb{N}$ with the notation of $n_1$ in base $b_1$ and $n_2$ in $b_2$ and the digits $z_{l-1}z_{l-2}...z_0$ Furthermore, be $b_1 > b_2$ ...
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Complement calculations - I don't get the expected results?

I've just started to study complement calculations and I don't understand why I don't get the correct result. I need to do $47 - 63$ in complement, using $8$ bits. $47$ is $00101111$ and $-63$ is $...
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66 views

Sequence that avoids the equation $a + b + c = 3 \cdot d$

Let us define $G$ as the sequence of non-negative integer numbers such that the terms of indexes 0, 1, 2 are the numbers 0, 1, 2 and every next term is defined as the smallest integer number larger ...
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1answer
26 views

How to find the base number of of two number systems when there's a difference in the quanity of numbers?

The problem is as follows: In two numbering systems which are of consecutive bases, there are about $120$ three-digit numerals more than the other. Find the smallest of these bases. The alternatives ...
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29 views

How to Most Succinctly Distinguish Base-n Numbers From Actual Numbers?

When reading mathematical texts I encounter phrases such as "the decimal number 2," the "binary number 10," etc., so much that I begin to imagine there really is such a thing as a &...
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1answer
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How many 5-digit numbers have distinct digits and are divisible by 3? [duplicate]

Problem: How many 5-digit numbers have distinct digits and are divisible by 3? My direction: I have tried to find all numbers with 5 digits that are divisible by 3 (30000 numbers) and then I will ...
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2answers
60 views

how to get the vector solution with a system of 3 variables and 4 equations

I'm doing an exercise of solid mechanics in which I got these 4 equations Own vector associated with eI = 7,391 (i) System of equations: $1.39116499156263*nx + 0.5*ny + 3.0*nz = 0$ $0.5*nx - 7....
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48 views

What is a formula that checks if a number is negative?

I am working on a project involving different number systems. Right now I am trying to come up with a non-recursive formula about converting a decimal number into the binary equivalent. Right now I ...
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1answer
46 views

How to find the nth binary number?

I was wondering if there is mathematical formula or a way to get the nth binary number where n is given. Obviously, this is a very easy problem for a programmer and I can construct a program (using ...
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5answers
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Need help converting 79 to base 5

I am struggling with the topic of converting a number to a base 5.... Tried looking some stuff up but still confused The one example I have on my homework is 79 to base 5
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1answer
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How to find the values which would equate the division between two numbers of swapped digits without knowing anything further? [duplicate]

I'm having difficulty in assess this problem in a method without relying too much in too many trial and error. I have noticed there is a similar puzzle but in such it includes the value of one of the ...
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1answer
35 views

How to find the number of marbles which someone has when it is a multiple of an unknown number?

The problem is as follows: Mark has $\overline{ab}$ marbles and Ferdinand has $\overline{cde}$ marbles. Find the total of marbles which both have knowing that the product of the number of marbles ...
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168 views

Number systems and Fourier Transforms

I was wondering if there is some connection between positional notation for number systems and the Fourier transform. Here is what led to this proposition. Say we look at the first few integers in ...
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1answer
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Multiplying number that is split into billions and remainder

I am working with 32 bit integers which have a maximum value of 2,147,483,647, but am wanting to support higher numbers by having two variables (...
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1answer
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A question from floating point number representation.

The numbers in a floating-point system are defined by a base B, a mantissa length t, and an exponent range [L, U]. A nonzero floating-point number x has the form x = +/-(o.b1b2.....bt)B^e ---1 then ...
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What's the explicit formula {$a_n$} for the Stanley Sequence beginning $a_0=0$, and $a_1=1$.

A Stanley sequence is a sequence of non-negative integers. In this case, the initial term is 0, and the next is 1. Every following term is the smallest integer bigger than the last term such that no ...
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GCD computation in Modular Residue Number System

Is there any known algorithm for computing the GCD of two numbers in the modular representation (i.e, residues modulo pair-wise coprime integers) that does not require the computation of the actual ...
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108 views

Can we reach infinity if we remove repeating digits from $a_k=m\cdot a_{k-1}$?

Inspired by Does this sequence reach infinity?, I propose the following generalization: $F_b(n)$ replaces consecutive repeating digits of $n$ in base $b$, with just one digit. Find smallest $m\in\...
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Can I solve for a radix?

If I know that $$100=(5\text{A})_{18}=(6\text{A})_{15}$$ Can I solve for the (presumably non-integer) radix $r$ that satisfies the following? $$100=(7\text{A})_r$$
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printing out a base 2^32 number in base 10

I wasn't really sure weather I should post this in here or in Stack Overflow, but I am creating a big_int class and have a vector of unsigned integers, each of them being 32 bit; this vector ...
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1answer
79 views

On the construction of the number systems

Recently, I read something about how set theory is used to built the number systems (starting with $\mathbb{N}$ and building the sets $\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}.$) I would like to ...
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Can there be more than 7 consecutive repdigit primes (extended from 1)?

Introduction The number $33333331$ is a prime (or $1$), regardless of how many $3$s are removed. That is, $31,331,3331,33331,333331,3333331,33333331$ are all prime numbers. I was wondering if we can ...
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2answers
79 views

Base $7$ is backwards base $16$

Find the sum of all positive integers such that their expression in base $7$ digits is the reverse of their expression in base $16$ digits. Express your answer in base $10$. I tried expressing the ...
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118 views

Is there a numeral system for real numbers that is always unique, but still has the usual convenient properties?

For each integer $b\ge 2,$ we know that representations of real numbers are usually unique in the base-$b$ positional notation. The only time that uniqueness fails is if the form ends in a tail of $0$'...
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2answers
30 views

What is the correct expression about Units digit? (revised)

I want to express it accurately. For example, consider $1872_{(9)}$. I would like to refer $2$. There are many expression such as 'units of digit of $1872_{(9)}$ is 2', 'unit digit of $1872_{(9)}$ is ...
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38 views

Number representation without equal adjacent digits

I want to represent numbers in a way where no adjacent digits are the same—so that no digit occurs two times in a row. Thus, representing a number as $1818$ would be fine, whereas representing as $...
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3answers
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what is the purpose of number system conversions e.g decimal to base 5?

I'm learning Number system conversion youtube. so far I know there are decimal, binary, octal, and Hex numbers. There is a purpose behind converting decimal to binary But what is the purpose of ...

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