RR -- the class of all real numbers

Description

A real number is entered as a sequence of decimal digits with a point. It is stored internally as an arbitrary precision floating point number, using the MPFR library.

i1 : 3.14159

o1 = 3.14159

o1 : RR (of precision 53)

The precision is measured in bits, is visible in the ring displayed on the second of each pair of output lines, and can be recovered using precision.

i2 : precision 3.14159

o2 = 53

For real numbers, the functions class and ring yield different results. That allows numbers of various precisions to be used without creating a new ring for each precision.

i3 : class 3.1

o3 = RR

o3 : InexactFieldFamily

i4 : ring 3.1

o4 = RR
       53

o4 : RealField

The precision can be specified on input by appending the letter p and a positive number.

i5 : 3p300

o5 = 3

o5 : RR (of precision 300)

An optional exponent (for the power of ten to multiply by) can be specified on input by appending the letter e and a number.

i6 : 3e3

o6 = 3000

o6 : RR (of precision 53)

i7 : -3e-3

o7 = -.003

o7 : RR (of precision 53)

i8 : -3p111e-3

o8 = -.003

o8 : RR (of precision 111)

Numbers that appear alone on an output line are displayed with all their meaningful digits. (Specifying 100 bits of precision yields about 30 decimal digits of precision.)

i9 : 1/3.

o9 = .3333333333333333

o9 : RR (of precision 53)

i10 : 1/3p100

o10 = .3333333333333333333333333333335

o10 : RR (of precision 100)

i11 : 100 * log(10,2)

o11 = 30.10299956639811

o11 : RR (of precision 53)

Numbers displayed inside more complicated objects are printed with the number of digits specified by printingPrecision.

i12 : printingPrecision

o12 = 6

i13 : {1/3.,1/3p100}

o13 = {.333333, .333333}

o13 : List

The notion of equality tested by == amounts to equality of the internal binary digits.

i14 : .5p100 == .5p30

o14 = true

i15 : .2p100 == .2p30

o15 = false

The notion of (strict) equality tested by === also takes the precision into account.

i16 : .5p100 === .5p30

o16 = false

i17 : .2p100 === .2p30

o17 = false

Perhaps surprisingly, the IEEE floating point standard also specifies that every number, including 0, has a sign bit, and strict equality testing takes it into account, as it must do, because some arithmetic and transcendental functions take it into account.

i18 : 0.

o18 = 0

o18 : RR (of precision 53)

i19 : -0.

o19 = -0

o19 : RR (of precision 53)

i20 : 1/0.

o20 = infinity

o20 : RR (of precision 53)

i21 : 1/-0.

o21 = -infinity

o21 : RR (of precision 53)

i22 : log 0

o22 = -infinity

o22 : RR (of precision 53)

i23 : csc (0.)

o23 = infinity

o23 : RR (of precision 53)

i24 : csc (-0.)

o24 = -infinity

o24 : RR (of precision 53)

Use toExternalString to produce something that, when encountered as input, will reproduce exactly what you had before.

i25 : x = {1/3.,1/3p100}

o25 = {.333333, .333333}

o25 : List

i26 : x == {.333333, .333333}

o26 = false

i27 : y = toExternalString x

o27 = {.33333333333333331p53,.33333333333333333333333333333346p100}

i28 : x === value y

o28 = true

Transcendental constants and functions are available to high precision, with numeric.

i29 : numeric pi

o29 = 3.141592653589793

o29 : RR (of precision 53)

i30 : numeric_200 pi

o30 = 3.141592653589793238462643383279502884197169399375105820974944

o30 : RR (of precision 200)

i31 : Gamma oo

o31 = 2.288037795340032417959588909060233922889688153356222441199382

o31 : RR (of precision 200)

Functions and methods returning a real number:

QQ * RR -- see * -- a binary operator, usually used for multiplication
RR * QQ -- see * -- a binary operator, usually used for multiplication
RR * RR -- see * -- a binary operator, usually used for multiplication
RR * ZZ -- see * -- a binary operator, usually used for multiplication
ZZ * RR -- see * -- a binary operator, usually used for multiplication
+ RR -- see + -- a unary or binary operator, usually used for addition
QQ + RR -- see + -- a unary or binary operator, usually used for addition
RR + QQ -- see + -- a unary or binary operator, usually used for addition
RR + RR -- see + -- a unary or binary operator, usually used for addition
RR + ZZ -- see + -- a unary or binary operator, usually used for addition
ZZ + RR -- see + -- a unary or binary operator, usually used for addition
- RR -- see - -- a unary or binary operator, usually used for negation or subtraction
QQ - RR -- see - -- a unary or binary operator, usually used for negation or subtraction
RR - QQ -- see - -- a unary or binary operator, usually used for negation or subtraction
RR - RR -- see - -- a unary or binary operator, usually used for negation or subtraction
RR - ZZ -- see - -- a unary or binary operator, usually used for negation or subtraction
ZZ - RR -- see - -- a unary or binary operator, usually used for negation or subtraction
QQ / RR -- see / -- a binary operator, usually used for division
RR / QQ -- see / -- a binary operator, usually used for division
RR / RR -- see / -- a binary operator, usually used for division
RR / ZZ -- see / -- a binary operator, usually used for division
ZZ / RR -- see / -- a binary operator, usually used for division
acos(RR) -- see acos -- arccosine
agm(RR,RR) -- see agm -- arithmetic-geometric mean
asin(RR) -- see asin -- arcsine
atan(RR) -- see atan -- compute the arctangent of a number
atan2(RR,RR) -- see atan2 -- compute an angle of a certain triangle
Beta(RR,RR) -- see Beta -- Beta function
cos(RR) -- see cos -- compute the cosine
cosh(RR) -- see cosh -- compute the hyperbolic cosine
cot(RR) -- see cot -- cotangent
coth(RR) -- see coth -- hyperbolic cotangent
csc(RR) -- see csc -- cosecant
csch(RR) -- see csch -- hyperbolic cosecant
Digamma(RR) -- see Digamma -- Digamma function
eint(RR) -- see eint -- exponential integral
erf(RR) -- see erf -- error function
erfc(RR) -- see erfc -- complementary error function
exp(RR) -- see exp -- exponential function
expm1(RR) -- see expm1 -- exponential minus 1
Gamma(RR) -- see Gamma -- Gamma function
Gamma(RR,RR) -- see Gamma -- Gamma function
inverseErf(RR) -- see inverseErf -- inverse error function
inverseRegularizedBeta(RR,RR,RR) -- see inverseRegularizedBeta -- inverse of the regularized beta function
inverseRegularizedGamma(RR,RR) -- see inverseRegularizedGamma -- inverse of the upper regularized gamma function
log(RR) -- see log -- logarithm function
log1p(RR) -- see log1p -- logarithm of 1+x
random(RR) -- see random(ZZ,ZZ) -- get a random integer or real number
regularizedBeta(RR,RR,RR) -- see regularizedBeta -- regularized beta function
regularizedGamma(RR,RR) -- see regularizedGamma -- upper regularized gamma function
sec(RR) -- see sec -- secant
sech(RR) -- see sech -- hyperbolic secant
sin(RR) -- see sin -- compute the sine
sinh(RR) -- see sinh -- compute the hyperbolic sine
tan(RR) -- see tan -- compute the tangent
tanh(RR) -- see tanh -- compute the hyperbolic tangent
zeta(RR) -- see zeta -- Riemann zeta function

Methods that use a real number:

RR ! -- see ! -- factorial
CC % RR -- see % -- a binary operator, usually used for remainder and reduction
RR % QQ -- see % -- a binary operator, usually used for remainder and reduction
RR % RR -- see % -- a binary operator, usually used for remainder and reduction
RR % ZZ -- see % -- a binary operator, usually used for remainder and reduction
CC * RR -- see * -- a binary operator, usually used for multiplication
RR * CC -- see * -- a binary operator, usually used for multiplication
RR * RRi -- see * -- a binary operator, usually used for multiplication
RRi * RR -- see * -- a binary operator, usually used for multiplication
CC + RR -- see + -- a unary or binary operator, usually used for addition
RR + CC -- see + -- a unary or binary operator, usually used for addition
RR + RRi -- see + -- a unary or binary operator, usually used for addition
RRi + RR -- see + -- a unary or binary operator, usually used for addition
CC - RR -- see - -- a unary or binary operator, usually used for negation or subtraction
RR - CC -- see - -- a unary or binary operator, usually used for negation or subtraction
RR - RRi -- see - -- a unary or binary operator, usually used for negation or subtraction
RRi - RR -- see - -- a unary or binary operator, usually used for negation or subtraction
CC / RR -- see / -- a binary operator, usually used for division
RR / CC -- see / -- a binary operator, usually used for division
RR / RRi -- see / -- a binary operator, usually used for division
RRi / RR -- see / -- a binary operator, usually used for division
CC // RR -- see // -- a binary operator, usually used for quotient
InfiniteNumber // RR -- see // -- a binary operator, usually used for quotient
RR // QQ -- see // -- a binary operator, usually used for quotient
RR // RR -- see // -- a binary operator, usually used for quotient
RR // ZZ -- see // -- a binary operator, usually used for quotient
abs(RR) -- see abs -- absolute value function
agm(CC,RR) -- see agm -- arithmetic-geometric mean
agm(RR,CC) -- see agm -- arithmetic-geometric mean
atan2(RR,RRi) -- see atan2 -- compute an angle of a certain triangle
atan2(RRi,RR) -- see atan2 -- compute an angle of a certain triangle
Beta(CC,RR) -- see Beta -- Beta function
Beta(RR,CC) -- see Beta -- Beta function
Beta(RR,RRi) -- see Beta -- Beta function
Beta(RRi,RR) -- see Beta -- Beta function
CC == RR (missing documentation)
clean(RR,Matrix) -- see clean -- Set to zero elements that are approximately zero
clean(RR,MutableMatrix) -- see clean -- Set to zero elements that are approximately zero
clean(RR,Number) -- see clean -- Set to zero elements that are approximately zero
clean(RR,RingElement) -- see clean -- Set to zero elements that are approximately zero
format(RR) -- see format -- format a string or a real number
Gamma(CC,RR) -- see Gamma -- Gamma function
Gamma(RR,CC) -- see Gamma -- Gamma function
Gamma(RR,RRi) -- see Gamma -- Gamma function
Gamma(RRi,RR) -- see Gamma -- Gamma function
InfiniteNumber == RR (missing documentation)
InfiniteNumber ? RR (missing documentation)
interval(QQ,RR) -- see interval -- construct an interval
interval(RR) -- see interval -- construct an interval
interval(RR,QQ) -- see interval -- construct an interval
interval(RR,RR) -- see interval -- construct an interval
interval(RR,ZZ) -- see interval -- construct an interval
interval(ZZ,RR) -- see interval -- construct an interval
isc(RR) -- see isc -- ISC lookup
isMember(RR,RRi) -- see isMember(QQ,RRi) -- membership test in an interval
isReal(RR) -- see isReal -- whether a number is real
RR << ZZ -- see left shift
lift(RR,type of QQ) -- see lift -- lift to another ring
lift(RR,type of ZZ) -- see lift -- lift to another ring
lngamma(RR) -- see lngamma -- logarithm of the Gamma function
log(RR,CC) -- see log -- logarithm function
log(RR,RR) -- see log -- logarithm function
log(RR,RRi) -- see log -- logarithm function
log(RRi,RR) -- see log -- logarithm function
MutableMatrix * RR (missing documentation)
norm(RR,Matrix) -- see norm
norm(RR,MutableMatrix) -- see norm
norm(RR,Number) -- see norm
norm(RR,RingElement) -- see norm
numeric(RR) -- see numeric -- convert to floating point
promote(RR,type of QQ) -- see promote -- promote to another ring
QQ == RR (missing documentation)
random(RR,RR) -- see random(ZZ,ZZ) -- get a random integer or real number
randomMutableMatrix(ZZ,ZZ,RR,ZZ) -- a random mutable matrix of integers
regularizedBeta(CC,CC,RR) -- see regularizedBeta -- regularized beta function
regularizedBeta(CC,RR,CC) -- see regularizedBeta -- regularized beta function
regularizedBeta(CC,RR,RR) -- see regularizedBeta -- regularized beta function
regularizedBeta(RR,CC,CC) -- see regularizedBeta -- regularized beta function
regularizedBeta(RR,CC,RR) -- see regularizedBeta -- regularized beta function
regularizedBeta(RR,RR,CC) -- see regularizedBeta -- regularized beta function
regularizedBeta(RR,RR,RRi) -- see regularizedBeta -- regularized beta function
regularizedBeta(RR,RRi,RR) -- see regularizedBeta -- regularized beta function
regularizedBeta(RR,RRi,RRi) -- see regularizedBeta -- regularized beta function
regularizedBeta(RRi,RR,RR) -- see regularizedBeta -- regularized beta function
regularizedBeta(RRi,RR,RRi) -- see regularizedBeta -- regularized beta function
regularizedBeta(RRi,RRi,RR) -- see regularizedBeta -- regularized beta function
regularizedGamma(CC,RR) -- see regularizedGamma -- upper regularized gamma function
regularizedGamma(RR,CC) -- see regularizedGamma -- upper regularized gamma function
regularizedGamma(RR,RRi) -- see regularizedGamma -- upper regularized gamma function
regularizedGamma(RRi,RR) -- see regularizedGamma -- upper regularized gamma function
RR >> ZZ -- see right shift
ring(RR) -- see ring -- get the associated ring of an object
round(RR) -- see round -- round a number
round(ZZ,RR) -- see round -- round a number
RR * MutableMatrix (missing documentation)
RR == CC (missing documentation)
RR == InfiniteNumber (missing documentation)
RR == QQ (missing documentation)
RR == RR (missing documentation)
RR == RRi (missing documentation)
RR == ZZ (missing documentation)
RR ? InfiniteNumber (missing documentation)
RRi == RR (missing documentation)
size2(RR) -- see size2 -- number of binary digits to the left of the point
sqrt(RR) -- see sqrt -- square root function
toCC(RR) -- see toCC -- convert to high-precision complex number
toCC(RR,RR) -- see toCC -- convert to high-precision complex number
toCC(ZZ,QQ,RR) -- see toCC -- convert to high-precision complex number
toCC(ZZ,RR) -- see toCC -- convert to high-precision complex number
toCC(ZZ,RR,QQ) -- see toCC -- convert to high-precision complex number
toCC(ZZ,RR,RR) -- see toCC -- convert to high-precision complex number
toCC(ZZ,RR,ZZ) -- see toCC -- convert to high-precision complex number
toCC(ZZ,ZZ,RR) -- see toCC -- convert to high-precision complex number
ZZ == RR (missing documentation)

For the programmer

The object RR is an inexact field family, with ancestor classes InexactNumber < Number < Thing.

RR -- the class of all real numbers

Description

See also

Functions and methods returning a real number:

Methods that use a real number:

For the programmer