Questions tagged [rational-numbers]
This challenge involves the manipulation of rational numbers, i.e. those which can be represented as a fraction of integers. Do not use this tag if rational numbers are just one of several admissible input/output formats, but rather if the use of exact rational arithmetic is required.
105
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Fractions nobody needs (because they can be reduced to a simpler form)
It happened in the 19th century. Georg was bored and started counting the rational numbers. Surprisingly, he discovered that there were no more of them than natural numbers. This insight made Georg ...
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answer
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Counterexample to Shapiro inequality
Input: A positive integer n such that n is even and greater than 12 or n is odd and greater than 23.
Output: A list of non-negative integers that violates Shapiro inequality.
More precisely, Let s be ...
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votes
16
answers
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Rabinowitz-Wagon \$\pi\$ formula
In 1995, Stanley Rabinowitz and Stan Wagon found an interesting algorithm to generate the digits of \$\pi\$ one by one without storing the previous results. The algorithm is called the spigot ...
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Factoriadic Fraction Addition
Objective
Given two rational numbers represented in fractional factoriadic as defined below, add them, and output the result in fractional factoriadic.
Fractional factoriadic
Fractional factoriadic is ...
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NaN is not equal to NaN
In many programming languages, the floating-point value NaN, or "not a number", in some programming languages generated by the expression ...
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10
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Egyptian fraction representations of 1 without prime denominators
Background
As noted in this question, for all positive integers \$n>2\$ there exists at least one Egyptian fraction representation (EFR) of \$n\$ distinct positive integers \$a_{1} < a_{2} < \...
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7
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Longest sequence of Egyptian fractions with n as denominator
Background
From Wikipedia: An Egyptian fraction is the sum of distinct unit fractions. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, ...
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Find the smallest integer multiple of a Decimal
The Challenge
Given a rational number, determine the smallest number which is a positive integer multiple of it. Eg.
...
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13
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Decimalize a Fraction
Preamble
A common pain-point when working with rational numbers and decimals is how infrequently one can represent their rational number as a clean, non-repeating decimal. Let's solve this by writing ...
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13
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Minkowski's ?(x) for rational x
Here is Minkowski's question mark function:
It is a strictly increasing and continuous function from the reals to themselves that, among other unusual properties, maps rational numbers to dyadic ...
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16
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15
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Find Index of Rational Number in Calkin-Wilf Sequence
Related
From Wikipedia:
In number theory, the Calkin–Wilf tree is a tree in which the vertices correspond one-to-one to the positive rational numbers. The tree is rooted at the number \$1\$, and any ...
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Whole Number Groups
Given a list of fractions, group them so that each group sums to a whole number. This should be done in such a way to maximize the number of non-empty groups.
You may assume a solution exists. Order ...
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10
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Enumerate the rationals
The cardinality of the set \$\mathbb Q\$ of rational numbers is known to be exactly the same as that of the set \$\mathbb Z\$ of integers. This means that it is possible to construct a bijection ...
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8
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Print ascending proper fractions using integers up to the given input
User inputs an integer. Print out proper fractions using all positive integers up to the user's input, in ascending order.
Rule 1: Eliminate equal fractions.
Rule 2: Fractions should be in their ...
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Next digit of rational number
Story:
The π was recently computed with accuracy to 100 trillions digits, but it is useless to us. We can't do accurate enough math, because rational numbers are too boring and so we don't know that ...
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18
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12
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In between fractions
Given two positive integer fractions \$x\$ and \$y\$ such that \$x < y\$, give the fraction \$z\$ with the smallest positive integer denominator such that it is between \$x\$ and \$y\$.
For example ...
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1
vote
1
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Best performance on x/(y+z) + y/(x+z) + z/(x+y) = N
Consider the equation $$\frac x {y+z} + \frac y {x+z} + \frac z {x+y} = n$$ for positive integers \$x, y, z\$ and \$n \ge 4\$. Your code will receive \$n\$ as an input, and output three integers \$x, ...
- 1,548
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17
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Infinite Candle Sequence
I have a cake shop that specialises in birthday cakes. The cakes that I sell must have candles placed in a circle. You would probably think I can just divide 360° by the number of candles, but the ...
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votes
7
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Factorials of primes decomposition
You have to decompose a positive integer/fraction as a product of powers of factorials of prime numbers.
For example
...
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20
answers
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Egyptian fraction representations of 1
An Egyptian fraction is a representation of a rational number using the sum of distinct unit fractions (a unit fraction is of the form \$ \frac 1 x \$ where \$ x \$ is a positive integer).
For all[1] ...
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31
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17
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Iterate your way to a fraction
I recently learned from a comment by MathOverflow user pregunton that it is possible to enumerate all rational numbers using iterated maps of the form \$f(x) = x+1\$ or \$\displaystyle g(x) = -\frac ...
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Harmonic divisor numbers
Consider the \$4\$ divisors of \$6\$: \$1, 2, 3, 6\$. We can calculate the harmonic mean of these numbers as
$$\frac 4 {\frac 1 1 + \frac 1 2 + \frac 1 3 + \frac 1 6} = \frac 4 {\frac {12} 6} = \frac ...
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17
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Print this sequence I just made up
To get this sequence I just made up, which will subsequently be referred to as TSIJMU, consider the harmonic series:
\$ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} ...\$
But what if you only add a term if ...
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20
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Exact generalised harmonic numbers
The generalised harmonic number of order \$m\$ of \$n\$ is
$$H_{n,m} = \sum_{k=1}^n \frac 1 {k^m}$$
For example, the harmonic numbers are \$H_{n,1}\$, and \$H_{\infty,2} = \frac {\pi^2} 6\$. These are ...
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17
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Wolstenholme numbers
The generalised harmonic number of order \$m\$ of \$n\$ is
$$H_{n,m} = \sum^n_{k=1} \frac 1 {k^m}$$
In this challenge, we'll be considering the generalised harmonic numbers of order \$2\$:
$$H_{n,2} = ...
- 50.2k
15
votes
21
answers
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Do I need a win streak?
You have played \$N\$ matches in some game where each match can only result in one of the two outcomes: win or loss. Currently, you have \$W\$ wins. You want to have a win percentage of \$P\$ or more, ...
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10
votes
2
answers
433
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Quote a rational number
Quote notation[1] is a way of expressing rational integers in a precise, finite manner, based on the concept of \$p\$-adic numbers. The notation is in the form of a string of digits (\$0123456789\$) ...
- 50.2k
13
votes
9
answers
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Truncate continued fractions
Related: Cleaning up decimal numbers
Background
A continued fraction is a way to represent a real number as a sequence of integers in the following sense:
$$
x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \...
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19
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Quoted rational numbers
Quote notation is a way of expressing rational numbers based on the concept of \$p\$-adic numbers, written in the form \$x'y\$.
The quote indicates that the number to it's left (\$x\$) is "...
- 50.2k
8
votes
15
answers
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Equalizing fractions
When I was in grade 3, we were taught how to solve a very simple math problem. It was equaling the denominators of two or more fractions.
Let's take two proper fractions:-
$$
\frac{1}{2},\frac{2}{3}
$$...
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11
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Calculate the probability of getting to the target first (exactly)
Consider the following probability puzzle.
We start with a string of bits all set to 0. At each step we choose a bit uniformly and independently at random and flip it. The value your code has to ...
user7467
26
votes
19
answers
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Convert a decimal to a fraction, approximately
Take the decimal number \$0.70710678\$. As a fraction, it'd be \$\frac{70710678}{100000000}\$, which simplifies to \$\frac{35355339}{50000000}\$. If you were to make the denominator \$1\$, the closest ...
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16
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15
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Repetend length in 1/n
This problem is based on non-terminating, repeating decimal points.
Let \$n\$ be any positive integer \$(n > 1 \text{ and } n < 10000)\$, say \$7\$. Then, \$1/n = 1/7 = 0.142857142857142857...\$
...
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Diophantine Approximation: find lowest possible denominator to approximate within given precision
Challenge
Given a number x and a precision e, find the lowest positive integer q such that <...
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5
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Cantor Function, Cruel
A ripoff of this challenge. Go upvote it!
Objective
Given a rational number amongst \$[0,1]\$, apply the Cantor function to it and output the rational number that's produced.
The Cantor function
The ...
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votes
13
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Parse a Unicode vulgar fraction
Objective
Given a string with single Unicode vulgar fraction, parse it to a rational number.
Valid inputs
A valid input is one of:
¼ U+00BC; one quarter
...
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3
votes
5
answers
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Find the binary period [duplicate]
We know that not all fractions have a terminating binary representation. However every fraction can be written as a leading portion followed by a repeating portion. For example \$1/3\$ starts with \$...
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39
votes
13
answers
3k
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12
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9
answers
1k
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Integer Logarithm
Objective
Take \$a \in ℤ_{>1}\$ and \$b \in ℤ_+\$ as inputs. Write a function \$f\$ such that:
$$
f(a,b) =
\left\{
\begin{array}{ll}
\log_ab & \quad \text{if} \space \log_ab \in ℚ \\...
- 6,299
29
votes
11
answers
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Rational Number RNG
The Objective
Build a random number generator whose range is \$\left[0,1\right) \cap \mathbb{Q} .\$
This is, build a random number generator that can produce any value that's:
at least \$0 \,;\$
...
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11
votes
8
answers
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Approximate the perfect fifth
Starting at 1-TET, give equal temperaments that have better and better approximation of the perfect fifth(just ratio 3/2). (OEIS sequence A060528)
The formal description of the sequence, copied from ...
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21
votes
71
answers
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Round towards zero
This is a simple task. Given a positive or negative real number, round it to the next whole integer closer to zero.
The challenge
Take input through any reasonable form (stdin, function, etc.) of one ...
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6
votes
2
answers
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Surreal Numbers
Surreal Numbers are one way of describing numbers using sets. In this challenge you will determine the value of a surreal number.
Intro
A surreal number consists of two sets: a left and right. The ...
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votes
7
answers
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Satisfying Rounding
Satisfying Rounding
You know when you're in science class, and asked to round to 2 sig figs, but your answer is 5.2501...? You should round to ...
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votes
7
answers
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Ryley's Theorem
S. Ryley proved following theorem in 1825:
Every rational number can be expressed as a sum of three rational cubes.
Challenge
Given some rational number \$r \in \mathbb Q \$ find three rational ...
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36
votes
46
answers
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Half, Half Half, and, Half
Consider the following number sequence:
\$ 0, \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8}, \frac{1}{16}, \frac{3}{16}, \frac{5}{16}, \frac{7}{16}, \...
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Convert a percentage to a "simple" ratio
You run a political website, and have determined that people have a better intuitive understanding when the chance of winning or losing an election is expressed as a ratio ("5 in 7") than when it is ...
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25
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Mixed Fraction Equality
In elementary school, children learn about proper fractions, where the numerator is less than the denominator, and thus the value of the fraction is less than one. Later, they are taught about ...
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Exact Partial Sum of Harmonic Series
Challenge
Given a positive integer N, output the sum of the first N reciprocals as an exact fraction, which is represented as a ...
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votes
9
answers
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Number the positive rationals
The positive rational numbers can be shown to be numerable with the following process:
Zero has the ordinal 0
Arrange the other numbers in a grid so that row a, column b contains a/b
Plot a diagonal ...
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