Rational numbers

Rational numbers
Tag	`0NCJ5WXN6Z`
URI	`m2://0NCJ5WXN6Z`
Aliases	`rational-number`, `rationals`, `set-of-rational-numbers`
Kind	object
Writing	W1 stated
MSC	11A99
Same as	Wikidata Q1244890 Wikipedia Rational number

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Description

A rational number is a number of the form $a / b$ , where $a$ and $b$ are integers and $b \neq 0$ . We write $\mathbb{Q}$ for the set of rational numbers. Two fractions $a / b$ and $c / d$ are the same rational number when $a d = b c$ . With addition and multiplication, $\mathbb{Q}$ is a field, and with its usual order it is an ordered field. The set $\mathbb{Q}$ contains $\mathbb{Z}$ , because each integer $a$ is the fraction $a / 1$ .

Remarks

Each rational number has exactly one representation in lowest terms $a / b$ with $b > 0$ . The set $\mathbb{Q}$ is countable. Not every real number is rational: $\sqrt{2}$ is irrational.

Explanations

L1 human

The rational numbers are the fractions, such as $1/2$ , $-3/4$ and $7 = 7/1$ . Many fractions name the same number: $1/2$ , $2/4$ and $3/6$ are all equal. In $\mathbb{Q}$ , you can divide by any number except $0$ .

What this node needs for W2

section Invariants: a table of the main invariants (0 of 1)
edge instance_of out (0 of 1)

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What depends on this

196 nodes use this node, directly or through other nodes.

Variance 007QED3E0Y
Prior probability 00NBBSXRQS
Absolute value 00S7C68D51
Chernoff bound 014PEBGDYF
Chapman–Kolmogorov equation 01APZ8GDQH
Bias of an estimator 01BD2TFFP4
Identically distributed random variables 01X8YSDTBW
Law of the unconscious statistician 01ZWX875AT
Density of the normal distribution 027JF1FBB0
Exponential distribution 028Z8WC5Y9
Covariance matrix 02CBHV4ZZQ
Chebyshev's inequality 02HZ1BK3MW
Characteristic function of a sum of independent random variables 02JGYG0T7M
Stationary distribution 02KYGSCTCN
Bernoulli process 02XPXR1FBK
Order on the reals 0337CYB0VD
Markov chain 033TJBZDG0
Statistic 036756NFD6
Law of total variance 03BY5XTNA4
Variance of the normal distribution 03EESZDB9A
Triangle inequality 03HF5PMY5T
Cauchy–Schwarz inequality for expectations 03PGHKAJG3
Monotonicity of probability 0464Z0JP2E
Every real number has a decimal expansion 047KEVEZ9Q
Correlation coefficient 04CTY6XYWN
Variance under translation 04CYN5G6RY
Density of the irrationals 04J6G4H78J
Almost sure convergence 04P7D63W8P
Dedekind cut 04QNZZ4HHY
Law of total expectation 04RRB6HM00
Probability density function 04YG690N5V
Expectation of a product of independent random variables 05AW1G0WE1
Likelihood function 05FSTXM92W
Markov's inequality 05PWWHYBX5
Standard score 05RZ4YZG4E
Poisson point process 05TG0SE4HT
Gamma distribution 06395GBNWD
Repeating decimals are rational 065S5B9HF1
De Moivre–Laplace theorem 065X9S8CM3
Buffon's needle problem 06X5PF6YFP
Kolmogorov's zero–one law 073Q8Z94KG
Independent random variables are uncorrelated 077GN9HDR6
Convolution of probability distributions 07CHJ5YX28
Normal distribution 07RXBVTVZB
Irrational number 080T2TXVQV
Mean of the Poisson distribution 086HEVXR9Z
Statistical hypothesis test 087P03NEND
Independence means the joint law is the product of the marginals 08QCA2F1NX
Median 08W2CW8PKW
Geometric probability 0924BCD257
Intermediate value theorem for connected spaces 098C0KZZAP
Dirac measure 099SXAG014
Method of moments 09J1H07VAH
Mean of the normal distribution 09PR1V6RWR
The rationals form an ordered field 09RFKMBQ63
Independent sigma-algebras 09T8ZHST1V
The reals are Archimedean 0A3GWZ86AY
Independent and identically distributed random variables 0AJN7WHBJZ
Classical definition of probability 0AKNPYQ04R
Pairwise independent events that are not mutually independent 0AR4B3K5QD
Ceiling function 0ASM4EPMB2
Marginal distribution 0AXP2Z5N9H
Moment-generating function of the normal distribution 0B1G4E21G7
Quantile function 0B3Q3XDM5X
Connected subsets of the real line are intervals 0B9ZQFMQNV
Weak law of large numbers 0BFDHF2H7N
Multiplication rule 0BS3RMX8HM
Borel sigma-algebra 0BVHZXWKCD
Addition rule for two events 0BZQ6ZX584
Random walk 0C08YMV95H
Random variable with a density 0C21RZG5V2
Uncorrelated random variables that are not independent 0C6J8BQXVR
Reverse triangle inequality 0C9S13WKMF
Kurtosis 0CBQVVNBMP
Martingale 0CDVK9RQBH
Variance of a sum 0D86GJEZ39
Mean of the geometric distribution 0DB0R4N5XX
Expected value 0DCT0QZVQ1
Strong law of large numbers 0DFWHQ839T
Covariance 0DJ8W3HP2D
Random variable 0DKRR22AEQ
Irrationality of the square root of 2 0DZGSDHD92
Base rate fallacy 0E44JMG6NV
Law of total probability 0E5G5FMT2R
Independence of two random variables 0E9502NT92
Neyman–Pearson lemma 0EE19CFEEF
Convergence in probability 0EFDDXTVD6
Boole's inequality 0EGSEE9J9R
P-value 0EHYZFZ4PK
Chi-squared test 0EN0V3NY20
Convergence in probability implies convergence in distribution 0EPTDZPJ83
Central limit theorem 0EPZC4Z1FC
68–95–99.7 rule 0EYQ03XCR3
Transition matrix 0F1KBVY161
Conditional variance 0F1SN93KXE
Conditional probability 0FJB06AD2P
Lévy's continuity theorem 0FPWF7TTMB
Convergence of random variables 0FQ968YKA1
Maximum likelihood estimation 0FREPFKNS7
Multivariate normal distribution 0G558GJQMW
Mutual independence of events 0GCVEV6BJA
Absolute value is multiplicative 0GMV7GP8W0
Skewness 0GQ3H7VW3V
Galton–Watson process 0H631CZP5K
Markov property 0HB8SXHPFA
Student's t-test 0HYP606RYF
Discrete random variable 0J6Z6H5VQJ
Bernoulli distribution 0J97P5KZY0
Convergence in distribution 0JDAQHWA9N
The cumulative distribution function determines the distribution 0JEVZAFDY6
Sample variance 0JN9P4A944
Bayesian inference 0JWRJX2RMP
Confidence interval 0K4NMMJBG2
Posterior probability 0KJ7ZS8AJA
Variance under scaling 0KVQ9D86BM
Discrete uniform distribution 0KYP951QWH
Coupon collector's problem 0MXCP7MXWJ
Cumulative distribution function 0MXMS3AE13
Characteristic function 0MYDNRT7PP
The rationals form a field 0N2WPCC00T
Pairwise independence 0N57D1XY6T
Cauchy distribution 0NA6Q82YR7
Bayes' theorem 0NM6E9VW04
Variance as the mean of the square minus the square of the mean 0NMF95YK8W
Birthday problem 0NMWG2PHJ0
Moment-generating function 0NNRNGKMFM
Student's t-distribution 0NQ5798SB1
Characteristic function of the normal distribution 0NS0TDA4E2
Second Borel–Cantelli lemma 0NWXRAMEE7
Sum of independent normal random variables 0NXGPJWNFN
Independence of two events 0P0MZ4X682
Linearity of expectation 0P7PTC8DDE
Monty Hall problem 0PBWWQD9KY
Moment 0PH0VHTQS0
Beta distribution 0PP5ZAJPP7
Probability of the complement 0PQEBTYDZM
Expectation with respect to the binomial distribution 0PVCTCKD3C
Cumulant generating function 0Q2QC81VTW
Joint probability distribution 0QAR0XDZH4
Probability distribution 0QB913WM5E
Stochastic process 0QH5ENKZM1
Sample mean 0QNRBHWV35
The reals form an ordered field 0QSPNASE50
Type I and type II errors 0R41PVFA39
Almost surely 0R46A1XY60
Fraction in lowest terms 0R64A2CVN4
Characterization of the floor 0R9B0B5458
Generating function of a sum of independent random variables 0REMT01N36
Chi-squared distribution 0RFMMBHFTV
Limit of the cumulative distribution function at infinity 0RJ5TZH6G0
Hypergeometric distribution 0SPBPW3K61
Moments are derivatives of the moment-generating function at zero 0STK8KEY9G
Density of the rationals 0STX0WRDNG
Almost sure convergence implies convergence in probability 0SVN6X695K
Bertrand paradox 0T0JWJ3M47
First Borel–Cantelli lemma 0T6WS7TT77
The characteristic function determines the distribution 0T8BYAFEXX
Interval 0TK8Y3J5RW
Law of a random variable 0TST4MHA3A
Convergence in mean 0TTA57SEJG
Estimator 0TTNRP5PFQ
Variance of a sum of independent random variables 0V0778DCW9
Mean squared error 0V4VSCBXP3
Poisson distribution 0VEFSSKX18
Probability measure 0VGMEBVW1M
Binomial distribution 0VRWZDWW6B
Standard deviation 0VTN5RKNC5
Law of total probability for an event and its complement 0W62TKMW7V
Probability mass function 0WF4EJP91W
Gambler's ruin 0WFMKK7DFZ
Continuous uniform distribution 0WGDPYR30Q
There is a natural number n with 1/n below any positive epsilon 0WH916TJNY
Hoeffding's inequality 0WPDNXGSSN
Moment-generating function of a sum of independent random variables 0WRZ170YEH
Continuity of probability measures 0X8HBKCSYB
Independence of random variables 0XCE070V8B
Geometric distribution 0XCVV5Q500
Monotonicity of the cumulative distribution function 0XD22PBMWA
Consistent estimator 0XJVF81QF6
Poisson limit theorem 0XKTZRWTXX
Negative binomial distribution 0XN7KEVSSS
A root of an integer is an integer or irrational 0XWCKJZRYH
Tail sum formula for expectation 0Y6QNF1J49
Conditional expectation 0YAMJRK04E
Absorbing Markov chain 0YB93GWTKK
Sum of independent Poisson random variables 0YH1M56SBN
Irrationality of the square root of a prime 0YHFMXJJAZ
Conditional probability distribution 0YVDC5PYDW
Floor function 0YVRNEDSVG
Probability-generating function 0Z8PSSAVBA
Memorylessness 0ZAZZ34WQS
Conditional independence 0ZJ87ZZ8PK
Path-connected spaces are connected 0ZMT1GP4KM
Central moment 0ZT24EC90K
Law of large numbers 0ZVEAYVRC3
Probability space 0ZYBZ34RT2

Sources

Web https://en.wikipedia.org/wiki/Rational_number, lead section
Web https://en.wikipedia.org/wiki/Rational_number, Arithmetic

What links here

Used by

Repeating decimals are rational 065S5B9HF1
Irrational number 080T2TXVQV
The rationals form a field 0N2WPCC00T
Fraction in lowest terms 0R64A2CVN4
Density of the rationals 0STX0WRDNG