InstantCalc1.1
for instantaneous calculations, ©1996-2000

Special Symbols

ˇ.˝decimal point or thousands separator˜(depending on the number format settings)
ˇ,˝decimal point or thousands separator˜(depending on the number format settings)
ˇ;˝separator of argument lists

ˇ"˝first character of a comment:¸sin(π/3)  "and a comment…

Dyadic Operators (sorted with respect to its precedence)

ˇ=˚subtraction with lowest priority (specially useful in combination with the
ˇNewton solver to subtract the right side of an equation from the left one)
ˇ´˝number of decimal digits:   π´5Ì=  3.14159
ˇ`˝number of significant digits:Â   10*π`4˚=  31.42
ˇ?˚comparison operator (behaves like a subtraction)

ˇ+     -Óaddition and subtraction
ˇ*     /Ômultiplication and division
ˇ÷     :     %¯modulo, integral division, remainder
ˇ^¸power:Â2^3^2È=  (2^3)^2ˆ=  64Ì(evaluation from the left)

ˇ√˚root:Ì3√5Ò=  √(3;5)˚=  3√5
ˇ@˜2D distance:Èx @ yÈ=  √(x2 + y2)

Monadic Operators (always highest priority)

ˇ-˝negation:ˆ-x
ˇ√˚square root:Î√3ˆ=  sqrt(3)  =  2√3  = 1.732050807568877294
ˇ|˛absolute value:˙|y˘=  |y|;   |-4 =  |4  =  4
ˇ!˝factorial:¯!n¯=  n! if n ≤ 25 and integral, else gamma(n+1)

ˇ°˝radiant -> grad:˚°x¯=  x*180/πÛ(angle conversions, useful for
ˇ•˝grad -> radiant:˚•x¯=  x*π/180Û the trigonometric functions)

ˇç˚pt -> cm:˘çxˆ=  x*2.54/72Î(measurement conversions,
ˇ¶˚cm -> pt:˘¶xˆ=  x*72/2.54Î useful in graphics applications)

Decimal/Hexadecimal/Binary Conversion Operators

ˇ$˚precedes a hexadecimal number (only for integral values)
ˇ$20 + 5ˇ=  37

ˇ&˙converts the result to a hexadecimal number (non integral results are rounded
ˇbefore (see rint). "&" must be the first character of the expression.)
ˇ&10 + 20 + $1E¸=  $3C

ˇ#˚precedes a binary number (only for integral values)
ˇ#11100101 + 5˝=  234 

ˇ\˝converts the result to a binary number (non integral results are rounded
ˇbefore (see rint). "\" must be the first character of the expression.)
ˇ\10 + 20 + $1Eˇ=  #00111100

Functions With One Argument

ˇabsÒabsolute value:˙abs(x)Ê=  |x|
ˇnegÒnegation:ˆneg(x)Ê=  -x
ˇsignÔsign function:Âsign(x) ˛= -1 for x < 0; 0 for x = 0; 1 for x > 0
ˇsqrtsquare root:Îsqrt(x)Â=  √x
ˇsqrÛpower of 2:sqr(x)Ë=  x2
ˇcubeÏpower of 3:cube(x)˝=  x3
ˇgradÓradians -> grad:˘grad(x)ˇ=  x*180/πÛ(angle conversions, useful for
ˇradÛgrad -> radians:˘rad(x)Ë=  x*π/180Û the trigonometric functions)

ˇrounding functions; please note: binary representations of decimal numbers are always inexact.
ˇFor example, the binary representation of 0.5 is identical to 0.49999999999999999999, therefore:

ˇrintÛrint(0.5)¸=  rint(0.49999999999999999999)Ë= 0 but not 1
ˇroundÈround(0.5)Ú=  round(0.49999999999999999999)˙= 1 but not 0
ˇrint(4.25)˜=  round(4.25)  =  4
ˇrint(5.66)˜=  round(5.66)  =  6

ˇtruncÎneglect decimals:Òtrunc(π)¸=  3

ˇfactfactorial:¯fact(x)Â=  x! if x ≤ 25 and integral, else gamma(x+1)
ˇfactLnÊlog-factorial:ÈfactLn(x)˜=  ln(x!) if x ≤ 25 and integral else gammaLn(x+1)
ˇpermÏpermutations: ˛perm(n) ˚=  n! if n ≤ 25 and integral, else gamma(n+1)
ˇgammaˇgamma function
ˇgammaLnılog-gamma function

Exponential Functions

˝expÒbase eÂexp(x)Ê=  ex
ˇexpm1Âexpm1(x)ˆ=  ex - 1
ˇexp10Ábase 10¸exp10(x)¯=  10x
ˇexp2Ïbase 2Âexp2(x)˝=  2x

Logarithms

ˇln˘base eÂln(x)
ˇlnp1Ôlnp1(x) ˛=  ln(x + 1)
ˇlg˘base 10¸lg(100) ˛=  2;   lg(2) = 0.3010299956639811952
ˇlb˘base 2Âlb(64) Á=  6

Trigonometric Functions (arguments always in radians)

ˇsinÙsineÔsin(π/3)˝=  0.8660254037844386467
ˇcosÒcosineÂcos(•45)˙=  0.7071067811865475244
ˇtanÛtangent
ˇcotÛcotangent

Inverse Trigonometric Functions (results always in radians)

ˇasinÔinverse sineÎasin(0.5)¯=  0.5235987755982988731
ˇacosÏinverse cosine˝°acos(0.5)Ú=  60
ˇatanÓinverse tangent
ˇacotÓinverse cotangent

Hyperbolic Functions

ˇsinhÔhyperbolic sine
ˇcoshÏhyperbolic cosine
ˇtanhÓhyperbolic tangent
ˇcothÓhyperbolic cotangent

ˇasinhÍinverse hyperbolic sine
ˇacoshÁinverse hyperbolic cosine
ˇatanhÈinverse hyperbolic tangent
ˇacothÈinverse hyperbolic cotangent

Functions With Two And Three Arguments (separated by semicolon)

ˇrootroot(n; x)˜=  √(n;x)
ˇscalbÍscalb(x; n)Ò=  x*2n
ˇpy2DÎpy2D(x; y)Ú=  √(x2 + y2)
ˇpy3DÎpy3D(x; y; z)Ë=  √(x2 + y2 + z2)
ˇphaseÁphase(x; y)Ó=  phase angle between x and y (electrical definition)
ˇgaussÁgauss(x; s) Ï=  Gauß (normal) distribution around 0 with standard deviation s
ˇgaussLn˘gaussLn(x; s) ˛=  log normal distribution around 1 with standard deviation s
ˇggtÛggt(x; y)˙=  largest common divisor
ˇkgvÒkgv(x; y)¯=  smallest common multiple

ˇcombÍcomb(n; k)Ò=  combinations˘= n!/[k!(n-k)!]Êbinomial coefficient
ˇ    comb(49; 6)Ù= 13983816
ˇvarÛvar(n; k)˙=  variationsÎ= n!/(n-k)!

ˇcompnd¸compnd(r; n)È=  (1 + r)n¯but compound is more exact
ˇannuity˛annuity(r; n)Î=  [1 - (1+r)-n]/r˛but annuity is more exact

Functions With Variable Number Of Arguments
(up to 1000 arguments, separated by semicolon or carriage return)

ˇ∑˙sum function:Â∑(1;2;3;4;5;6;7;8;9)Á=  45
ˇ∏¯product function:Û∏(1;2;3;4;5;6;7;8;9)Â=  362880
ˇm˘arithmetic mean:ım(1;2;3;4;5;6;7;8;9)Ê=  5
ˇs˚standard deviation:Ís(1;2;3;4;5;6;7;8;9)Ë=  2.738612787525830567

ˇ1˚∏(ı1˚m(ˆ1˚s(¯1
ˇ2˚2˚2˚2
ˇ3˚3˚3˚3
ˇ4˚4˚4˚4
ˇ5  =  15 ˛5)  =  120¯5)  =  3 5)  =  1.581138830084189666

Function Without Argument

ˇrandÓequally distributed random numbers between -1 and 1 (linear congruential method)
ˇri+1Û=  (3250998893 * ri + 907633343) mod 4294967295
ˇrandÓ=   ri+1/2147483647;  (-1 ≤ rand ≤ 1)

Numerical Methods (numerical accuracy can be adjusted in the InstantCalc control panel)

ˇ∫…∂Romberg integration:ı∫sin(u)∂u; 0; π]  =  2
ˇ∫1/(1 + sqr(t))∂t;-1; 1]  = 1.5707963… (=  π/2)
ˇ∫gauss(t; 1)∂t; -1; 1]  =  0.6826…

”…“ is a place holder for the function which should be integrated with respect to the independent variable which must follow ”∂“ directly. The integration limits, separated by semicolon, and a square bracket complete the formula.

ˇ«…»ÌNewton solver:¸«sin(u)»u; 3]  =  3.14159…
ˇ«3*x^2 + 6*x = 45»x; 2]  =  3

”…“ is a place holder for the function which should be solved to zero with respect to the independent variable which must follow ”»“ directly. The first guess of the Newton series, separated by semicolon, and a square bracket complete the formula.

Encapsulation of both methods into each other is possible:

ˇ∫∫py2D(x; y)∂x; -1; 1]∂y; -1; 1]  =  3.0607…
ˇ«∫gauss(x; 1)∂x; -y; y] = 0,95»y; 1] = 1.95996…

Logical Functions (result of the last argument is compared to 0)

ˇEQÛEQ(1; 0; x ? y)˛=  1Úif x = y,ˇelseÔ0˚(equal)
ˇODÛOD(1; 0; √x)Î=  1Úif x ≥ 0,ˇelseÔ0˚(ordered; √x Œ R?)
ˇUNÙUN(1; 0; ln(x))=  1Úif x < 0,ˇelseÔ0˚(unordered; ln(x) Œ R?)
ˇNEÙNE(1; 0; x ? y)ˇ=  1Úif x ≠ y,ˇelseÔ0˚(not equal)
ˇGTÙGT(1; 0; x ? y)ˇ=  1Úif x > y,ˇelseÔ0˚(greater than)
ˇGEÛGE(1; 0; x ? y)˛=  1Úif x ≥ y,ˇelseÔ0˚(greater or equal)
ˇLTˆLT(1; 0; x ? y) ˇ=  1Úif x < y,ˇelseÔ0˚(less than)
ˇLEıLE(1; 0; x ? y)=  1Úif x ≤ y,ˇelseÔ0˚(less or equal)

ˇSEQÌSEQ(5; x)Ù=  xÚif x = 0,ˇelseÔ5
ˇSODÌSOD(5; √x)Ô= √xÔif x ≥ 0,ˇelseÔ5
ˇSUNÓSUN(5; √x)= √xÔif x < 0,ˇelseÔ5˚(√-1 = NaN; Not a Number)
ˇSNEÓSNE(5; x)ı=  xÚif x ≠ 0,ˇelseÔ5
ˇSGTÓSGT(5; x)ı=  xÚif x > 0,ˇelseÔ5
ˇSGEÌSGE(5; x)Ù=  xÚif x ≥ 0,ˇelseÔ5
ˇSLTSLT(5; x)˜=  xÚif x < 0,ˇelseÔ5
ˇSLEÔSLE(5; x)ˆ=  xÚif x ≤ 0,ˇelseÔ5

Constants

ˇtrue=  1
ˇfalseÏ=  0
ˇe˚=  2.718281828459045235
ˇπ,  piÌ=  3.141592653589793238
ˇln10Ô=  2.302585092994045684
ˇgoldÔ=  0.6180339887498948482È(golden ratio)
ˇ∞, inf, INFÙ=  1/0 (IEEE-inf)
ˇnan, NaN˜=  Not a Number
ˇmaxIÏ=  32767
ˇmaxLÍ=  2147483647
ˇminRÏ=  1.822599765941237301e-4951
ˇmaxRÈ=  1.189731495357231765e+4932
ˇprecÓ=  1.084202172485504434e-19Û(Macintosh numerical precision = smallest
ˇ positive number p for which:  1 + p > 1)

Atomic Weights Of The Elements In The Periodic Table (in g/mol)

examplesÙH*2+OÂ=  18.01528Ìmolecular weightıof water
ˇC*2+H*5+O+H˛=  46.06904Ìof ethanol
ˇH*2+S+O*4 Î=  98.07948Ìof sulfuric acid

ˇ250*Cr/(Cr+O*3)´4Ì=  129.9977Ìamount of chromium in 250 g chromic acid

periodic table

1.ıH˙=˚    1.00794ÛHydrogen
ˇHeı=˚    4.002602ÓHelium

2.ıLi˘=˚    6.941˝Lithium
ˇBeı=˚    9.012182ÓBeryllium
ˇB˙=˚  10.811¸Boron
ˇC˙=˚  12.011¸Carbon
ˇN˙=˚  14.00674ÚNitrogen
ˇO˘=˚  15.9994˜Oxygen
ˇF˚=˚  18.9984˜Fluorine
ˇNeı=˚  20.1797˜Neon

3.ıNaı=˚  22.98977ÚSodium
ˇMgÙ=˚  24.305¸Magnesium
ˇAl¯=˚  26.98154ÚAluminium
ˇSi¯=˚  28.0855˜Silicon
ˇP˙=˚  30.97376ÚPhosphorus
ˇS˙=˚  32.066¸Sulfur
ˇCl¯=˚  35.4527˜Chlorine
ˇAr˜=˚  39.948¸Argon

4.ıK˙=˚  39.0983˜Potassium
ˇCaı=˚  40.078¸Calcium
ˇScı=˚  44.95591ÚScandium
ˇTi˘=˚  47.88 ˇTitanium
ˇV˙=˚  50.9415˜Vanadium
ˇCr˜=˚  51.9961˜Chromium
ˇMnÙ=˚  54.93805ÚManganese
ˇFeˆ=˚  55.847¸Iron
ˇCoı=˚  58.9332˜Cobalt
ˇNi¯=˚  58.69 ˇNickel
ˇCuı=˚  63.546¸Copper
ˇZnˆ=˚  65.39 ˇZinc
ˇGaÙ=˚  69.723¸Gallium
ˇGeÙ=˚  72.61 ˇGermanium
ˇAsı=˚  74.92159ÚArsenic
ˇSeı=˚  78.96 ˇSelenium
ˇBr˜=˚  79.904¸Bromium
ˇKr˜=˚  83.8ÍKrypton

5.ıRbı=˚  85.4678˜Rubidium
ˇSr˜=˚  87.62 ˇStrontium
ˇY˙=˚  88.90585ÚYttrium
ˇZr¯=˚  91.224¸Zirconium
ˇNbı=˚  92.90638ÚNiobium
ˇMoÙ=˚  95.94 ˇMolybdenum
ˇTcˆ=˚  98.9063˜Technetium
ˇRuı=˚101.07˚Ruthenium
ˇRhı=˚102.9055ˆRhodium
ˇPdı=˚106.42˚Palladium
ˇAgı=˚107.8682ˆSilver
ˇCdı=˚112.411˚Cadmium
ˇIn¯=˚114.82˚Indium
ˇSnı=˚118.71˚Tin
ˇSbı=˚121.75˚Antimony
ˇTeˆ=˚127.6ÈTellurium
ˇI˝=˚126.9045ˆIodine
ˇXeı=˚131.29˚Xenon

6.ıCsı=˚132.9054ˆCesium
ˇBaı=˚137.327˚Barium
ˇLaˆ=˚138.9055ˆLanthanum
ˇCeı=˚140.115˚Cerium
ˇPr˜=˚140.9077ˆPraseodymium
ˇNdı=˚144.24˚Neodymium
ˇPmÛ=˚146.9151ˆPromethium
ˇSmÛ=˚150.36˚Samarium
ˇEuı=˚151.965˚Europium
ˇGdÙ=˚157.25˚Gadolinium
ˇTbˆ=˚158.9253ˆTerbium
ˇDyı=˚162.5ÈDysprosium
ˇHoı=˚164.9303ˆHolmium
ˇEr˜=˚167.26˚Erbium
ˇTmÙ=˚168.9342ˆThulium
ˇYbı=˚173.04˚Ytterbium
ˇLuˆ=˚174.967˚Lutetium
ˇHf˜=˚178.49˚Hafnium
ˇTaˆ=˚180.9479ˆTantalum
ˇW¯=˚183.85˚Tungsten
ˇReı=˚186.207˚Rhenium
ˇOsÙ=˚190.2ÈOsmium
ˇIr˙=˚192.22Iridium
ˇPt˜=˚195.08˚Platinum
ˇAuı=˚196.9665ˆGold
ˇHgı=˚200.59˚Mercury
ˇTl˘=˚204.3833ˆThallium
ˇPbı=˚207.2ÈLead
ˇBi¯=˚208.9804ˆBismuth
ˇPoı=˚208.9824ˆPolonium
ˇAt˜=˚209.9871ˆAstatine
ˇRnı=˚222.0176ˆRadon

7.ıFr¯=˚223.0197ˆFrancium
ˇRaı=˚226.0254ˆRadium
ˇAcı=˚227.0278ˆActinium
ˇThˆ=˚232.0381ˆThorium
ˇPaı=˚231.0359ˆProtactinium
ˇU˙=˚238.0289ˆUranium
ˇNpı=˚237.0482ˆNeptunium
ˇPuı=˚244.0642ˆPlutonium
ˇAmÛ=˚243.0614ˆAmericium
ˇCmÛ=˚247.0703ˆCurium
ˇBkı=˚247.0703ˆBerkelium
ˇCf˜=˚251.0796ˆCalifornium
ˇEsı=˚252.0829ˆEinsteinium
ˇFmÙ=˚257.0951ˆFermium
ˇMdÙ=˚258.0986ˆMendelevium
ˇNoı=˚259.1009ˆNobelium
ˇLr¯=˚260.1053ˆLawrencium
ˇRf˜=˚261ÒRutherfordium
ˇHaı=˚262ÒHahnium