How to use
Keisan Online Calculator service aims to achieve the high precision of calculation by applying Arithmetic Operation System with variable number of digits developed by Casio.
Feature of Keisan Online Calculator Service
1. The digit number can be specified.| digit 6 | √5 =sqrt(5) =2.23607 |
| digit 14 | √5 =sqrt(5) =2.2360679774998 |
2. All operations are carried out in the decimal system.
| keisan | (0.9-0.3*3) =0 |
| Excel | (0.9-0.3*3) =1.11022E-16 |
3. The number of correct digits in the operation result is automatically judged and the precise operation result is returned.
| digit 6 | 1-0.999876 =1.24000E-4 |
| digit 10 | gamma(3.5) =3.323350970 |
4. Complex number, probability function, special function, and repeated calculation can be handled.
| [digit 6] | |||||||||||||||||
| Complex number | (1+i)^(1+i) =0.273957+0.583701i | ||||||||||||||||
| Error function | x=0.1 erf(x) =0.112463 | ||||||||||||||||
| Repeated calculation | x=2,1,5 sqrt(x) | ||||||||||||||||
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5. The library of expressions can be utilized.
| Ex) Area of triangle | s=(a+b+c)/2; a=3; b=4; c=5; sqrt(s*(s-a)*(s-b)*(s-c)) |
Reference.
| Patent: | USA Patent No.7716267(May 5, 2010) Japan Patent No.4483491(April 2, 2010) |
|
Description rule of expression 1. Numerical values
The numerical value is displayed with combination of
0,1,2,3,4,5,6,7,8,9, .(decimal point), E(exponent), i (imaginary number),+ and -
| Example | Explanation |
|---|---|
| 12340.56789 | 1.234056789E4 |
| E2 | 100 |
| 1.23E+4 | 12300 Note: + after E is the sign of exponent |
| 1.23E-4 | 0.000123 Note: - after E is the sign of exponent |
| i | 0+1i |
| E+2i | 0+100i |
| 1.23+4.56i | Note: In order to handle the expression as complex number, put it in the parenthesis |
| 1.23E3+4.56E2i | 1230+456i |
| 1.23E-3-4.56E+2i | 0.00123-456i |
2. Variables
The variable begins with an English letter and is composed of letters, words and/or numbers.
However, it is not possible to use the same variable identifier as special sign and reserved word.
| Valid | a, sinx, a3, interest_rate and the like |
|---|---|
| Invalid | reserved word such as e, i, inf, sin |
| #abc, 3abc |
3. Functions
The function consists of the reserved word, and the one or more of arguments which are surrounded by the parenthesis.
if there are more than two arguments, they have to be divided by comma.
The numerical value, variable, function and expression can be described as the argument.
| Ex) sin(x) normalcd(x,μ,2) normalcd(x+y,ln(12),σ) |
| The parenthesis is not required for constant. Ex) pi inf e |
4. Operators
The operators (+, -, *, /, ^, ! and = ) execute operations between each element of numerical value, variable and function.
| Example | Explanation |
|---|---|
| Element 1 + Element 2 | Add Element 1 to Element 2. |
| Element 1 - Element 2 | Subtract Element 2 from Element 1. |
| Element 1 * Element 2 | Multiply Element 1 by Element 2. (* it is possible also to abbreviate) |
| Element 1 / Element 2 | Divide Element 1 by Element 2. |
| Element 1 ^ Element 2 | Calculate Element 1 to the power of Element 2. |
| Element 1 ! | Factorial of Element 1. (Note that the factorial operation is performed only for one element.) |
| Element 1 = Element 2 | Substitute the operation result of Element 2 into Element 1. Therefore Element 1 must be variable. |
5. Expressions
The expression is composed of numerical value, variable and function, each element of which is connected by the operator. Further, the expressions can be connected by the operator as one element.
The operational order between each element follows "the operational order of expressions".
| Ex) element1 + element2 - element3 * element4 / element5 ^ element6 ! + ･･･ |
6. Miscellaneous
Minus "-", excluding the exponential part of expression, is regarded as a binary operator.
Minus "-" at the beginning of the expression or immediately after the open parenthesis or immediately after the semicolon, is considered as that the beginning of zero is abbreviated.
| Ex) | -a+b | → | (0-a)+b |
| -a^b | → | 0-(a^b) | |
| a^-b | → | Error | |
| a^(-b) | → | a^(0-b) | |
| a*b;-a*b | → | a*b;0-(a*b) | |
| a*-b | → | Error | |
| In order to make expression easy to see, insert the space between words. | |||
Order of operations in expression
| 1. The expression is operated in the higher priority order of the operator. |
| 2. The operation is performed from the left side if there is more than one operator of the same priority. (The power operator is performed from the right side.) |
| 3. The operation inside the parenthesis is performed with high priority. |
| 4. Further, the parenthesis can be described inside the parenthesis. In that case, the priority is given to the inside parenthesis. |
Type and priority of operator
| Priority | Operator | Meaning |
|---|---|---|
| high ↑ ↓ low |
Function | Intrinsic function |
| !!, !, ^ | Double factorial, Factorial and Power | |
| *, / | Multiplication and Division | |
| +, - | Addition and Subtraction | |
| = | Substitution |
Mistakable description example of expression.
1. The order of operations without parenthesis..
| 2+3*4 | → | 2+(3*4) | The multiplication is calculated first than the addition. |
| 3*4^5 | → | 3*(4^5) | The power is calculated first than the multiplication. |
| sin(2)^3 | → | (sin(2))^3 | The function is calculated first than the power. |
| 1/2i | → | 1/(2i) | The imaginary number sign takes precedence. |
| 3!!! | → | (3!!)! | The double factorial takes precedence from the factorial. |
2. The expression without multiplication sign.
| 3a | ○ | It is calculated as 3*a. |
| a3 | × | a3 is handled as variable identifier. |
| a(b+c) | × | It is not calculated as a* (b+c). "a" is handled as user defined function. |
| (a+b)(b+c) | ○ | It is calculated as (a+b)*(b+c). |
Numerical values, operators and constants
| Functional name | Reserved word | ||
|---|---|---|---|
| Numerical values | |||
| Number | 1234567890 | 0 ~9 | |
| Decimal point | 0.123 | . | |
| Sign | -123, +456 | +,- | |
| Exponent | 1.23E-45 | E | |
| Imaginary number | 2+3i | i | |
| Operators | |||
| Addition | x+y | + | |
| Subtraction | x-y | - | |
| Multiplication | x*y | * | |
| Division | x/y | / | |
| Power | x^y | ^ | |
| Factorial | n! | ! | |
| Double factorial | n!! | !! | |
| Substitution | a=123 | = | |
| Multiple expressions | |||
| Expression separator | a=2+3; b=4/7 | ; | |
| Constants | |||
| Napier's constant | e= 2.71828 | e | |
| Infinity | ∞ | inf | |
| Pi | π = 3.14159... | pi | |
| Euler's constant | γ= 0.5772... | euler | |
| Logical value | true = 1 false = 0 |
true false |
|
Elementary functions
| Functional name | Reserved word | ||
|---|---|---|---|
| Logarithm and Exponential functions | |||
| Square root | √ | sqrt(x) | |
| Common logarithm | log | log(x) | |
| Natural logarithm | ln | ln(x) | |
| Exponential | ex | e^x | |
| Power | xy | x^y | |
| Trigonometric functions | |||
| Sine | sin | sin(x) | |
| Cosine | cos | cos(x) | |
| Tangent | tan | tan(x) | |
| Invese sine | sin-1 | asin(x) | |
| Invese cosine | cos-1 | acos(x) | |
| Invese tangent | tan-1 | atan(x) | |
| Hyperbolic functions | |||
| Hyperbolic sine | sinh | sinh(x) | |
| Hyperbolic cosine | cosh | cosh(x) | |
| Hyperbolic tangent | tanh | tanh(x) | |
| Invese hyperbolic sine | sinh-1 | asinh(x) | |
| Invese hyperbolic cosine | cosh-1 | acosh(x) | |
| Invese hyperbolic tangent | tanh-1 | atanh(x) | |
| Combination and Permutation | |||
| Combination | nCr | combination(n,r) | |
| Permutation | nPr | permutation(n,r) | |
| Factorial | x! | x! | |
| Double factorial | x!! | x!! | |
| Numerical functions (real number) | |||
| Floor function | int(4.2) =4 int(-4.2) =-5 |
int(x) | |
| Ceiling function | ceiling(4.2) =5 ceiling(-4.2) =-4 |
ceiling(x) | |
| Integer part | IP(4.2) =4 IP(-4.2) =-4 |
IP(x) | |
| Fractional part | FP(4.2) =0.2 FP(-4.2) =-0.2 |
FP(x) | |
| Remainder on division of x by y | mod(9,5) =4 mod(-9,5) =1 |
mod(x, y) | |
| y*FP(x/y) | remainder(9,5)=4 remainder(-9,5)=-4 |
remainder(x, y) | |
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Rounding x to n decimal place(s)
|
round(2.35,1)=2.4 round(-2.35,1)=-2.4 |
round(x, n) | |
| Round down x to n decimal place(s) | rounddown(2.34,1) =2.3 rounddown(-2.34,1) =-2.4 |
rounddown(x, n) | |
| Round up x to n decimal place(s) | roundup(2.34,1) =2.4 roundup(-2.34,1) =-2.3 |
roundup(x, n) | |
| IP(x*10n)/10n | truncate(2.34,1) =2.3 truncate(-2.34,1) =-2.3 |
truncate(x, n) | |
| Random # (decimal) | random decimal number in [0,1] | random_f() | |
| Random # (decimal) | random decimal number in [a,b] | random_f(a,b) | |
| Random # (integer) | random integer number in [a,b] | random_i(a,b) | |
| Numerical functions (complex number) | |||
| Sign | sign(z) =1,0,-1 | sign(z) | |
| Complex conjugate number | x+iy | conjugate(z) | |
| Real part | x+iy | Re(z) | |
| Imaginary part | x+iy | lm(z) | |
| Absolute value | |z| =|reiθ| =r | abs(z) | |
| Complex argument | reiθ| argument(z) |
| |
| Cartesian coordinate conversion | reiθ | cartesian(z) | |
| Cartesian coordinate conversion | (r,θ) | cartesian(r,θ) | |
| Polar coordinate conversion | x+iy | polar(z) | |
| Polar coordinate conversion | (x, y) | polar(x,y) | |
Probability functions Continuous distribution functions
| Functional name | Reserved word | ||||||
|---|---|---|---|---|---|---|---|
| Q: Upper cumulative distribution f: Probability density x: Percentage point λ: Non centrality |
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| Standard normal distribution | Q: normalcd(x) f: normalpd(x) x: normalicd(Q) |
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| Normal distribution | Q: normalcd(x,μ,σ) f: normalpd(x,μ,σ) x: normalicd(Q,μ,σ) | ||||||
| Logarithmic normal distribution | Q: lognormalcd(x,μ,σ) f: lognormalpd(x,μ,σ) x: lognormalicd(Q,μ,σ) | ||||||
| Chi-square distribution | Q: chi2cd(x,ν) f: chi2pd(x,ν) x: chi2icd(Q,ν) |
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| t-distribution of student | Q: tcd(x,ν) f: tpd(x,ν) x: ticd(Q,ν) |
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| F-distribution | Q: fcd(x,ν1,ν2) f: fpd(x,ν1,ν2) x: ficd(Q,ν1,ν2) |
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| Noncentral chi-square distribution | Q: ncchi2cd(x,ν,λ) f: ncchi2pd(x,ν,λ) x: ncchi2icd(Q,ν,λ) λ: ncchi2il(P,x,ν) |
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| Noncentral t-distribution | Q: nctcd(x,ν,λ) f: nctpd(x,ν,λ) x: ncticd(Q,ν,λ) λ: nctil(Q,x,ν) |
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| Noncentral F-distribution | Q: ncfcd(x,ν1,ν2,λ) f: ncfpd(x,ν1,ν2,λ) x: ncficd(Q,ν1,ν2,λ) λ: ncfil(Q,x,ν1,ν2) |
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| Uniform distribution | Q: uniformcd(x,a,b) f: uniformpd(x,a,b) x: uniformicd(Q,a,b) |
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| Exponential distribution | Q: exponentialcd(x,b) f: exponentialpd(x,b) x: exponentialicd(Q,b) |
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| Weibull distribution | Q: weibullcd(x,a,b) f: weibullpd(x,a,b) x: weibullicd(Q,a,b) |
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| Gamma distribution | Q: gammacd(x,a,b) f: gammapd(x,a,b) x: gammaicd(Q,a,b) |
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| Beta distribution | Q: betacd(x,a,b) f: betapd(x,a,b) x: betaicd(Q,a,b) |
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| Laplace distribution | Q: laplacecd(x,a,b) f: laplacepd(x,a,b) x: laplaceicd(Q,a,b) |
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| Cauchy distribution | Q: cauchycd(x,a,b) f: cauchypd(x,a,b) x: cauchyicd(Q,a,b) |
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| Pareto distribution | Q: logisticcd(x,α,β) f: paretopd(x,xm,α) x: paretoicd(Q,xm,α) |
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| Generalized Pareto distribution | Q: gparetocd(x,μ,σ,ξ) f: gparetopd(x,μ,σ,ξ) x: gparetoicd(Q,μ,σ,ξ) |
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Discrete distribution functions
| Functional name | Reserved word |
|---|---|
| Q: Upper cumulative distribution f: Probability density x: Percentage point λ: Non centrality |
|
| Poisson distribution | Q: poissoncd(x,λ) f: poissonpd(x,λ) x: poissonicd(Q,λ) λ: poissonil(Q,x) |
| Binomial distribution | Q: binomialcd(x,n,p) f: binomialpd(x,n,p) x: binomialicd(Q,n,p) |
| Negative binomial distribution | Q: negbinomialcd(x,k,p) f: negbinomialpd(x,k,p) x: negbinomialicd(Q,k,p) |
| Geometric distribution | Q: geometriccd(x,p) f: geometricpd(x,p) x: geometricicd(Q,p) |
| Hypergeometric distribution | Q: hypgeometriccd(x,n,M,N) f: hypgeometricpd(x,n,M,N) |
| < Note > For the reserved word which uses lower cumulative probability P, add "lower" to the above-mentioned each reserved word end. Ex) P=poissoncdlower (x,λ), x=poissonicdlower (P,λ), λ=poissonillower (P, x) |
|
Bessel functions Bessel functions
| Functional name | Reserved word | |
|---|---|---|
| f (x) f '(x) f-1 (0), s=1,2... f '-1 (0), s=1,2... |
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| Bessel function of the 1st kind | Jν(x) | besselj(ν,x) besseljdf(ν,x) besseljzeros(ν,s) besseljdfzeros(ν,s) |
| Bessel function of the 2nd kind | Yν(x) | bessely(ν,x) besselydf(ν,x) besselyzeros(ν,s) besselydfseros(ν,s) |
| Modified bessel function of the 1st kind | Iν(x) | besseli(ν,x) besselidf(ν,x) |
| Modified bessel function of the 2nd kind | Kν(x) | besselk(ν,x) besselkdf(ν,x) |
| Hankel function of the 1st kind | H(1)ν(x) | hankelH1(ν,x) hankelH1df(ν,x) |
| Hankel function of the 2nd kind | H(2)ν(x) | hankelH2(ν,x) hankelH2df(ν,x) |
Spherical Bessel functions
| Functional name | Reserved word | |
|---|---|---|
| f (x) f '(x) f-1 (0), s=1,2... f '-1 (0), s=1,2... |
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| Spherical bessel function of the 1st kind | jν(x) | sphericalbesselj(ν,x) sphericalbesseljdf(ν,x) sphericalbesseljzeros(ν,s) sphericalbesseljdfzeros(ν,s) |
| Spherical bessel function of the 2nd kind | yν(x) | sphericalbessely(ν,x) sphericalbesselydf(ν,x) sphericalbesselyzeros(ν,s) sphericalbesselydfzeros(ν,s) |
| Modified spherical bessel function of the 1st kind | iν(x) | sphericalbesseli(ν,x) sphericalbesselidf(ν,x) |
| Modified spherical bessel function of the 2nd kind | kν(x) | sphericalbesselk(ν,x) sphericalbesselkdf(ν,x) |
| Spherical hankel function of the 1st kind | h(1)ν(x) | sphericalhankelH1(ν,x) sphericalhankelH1df(ν,x) |
| Spherical hankel function of the 2nd kind | h(2)ν(x) | sphericalhankelH2(ν,x) sphericalhankelH2df(ν,x) |
Airy functions
| Functional name | Reserved word | |
|---|---|---|
| f (x) f '(x) f-1 (0), s=1,2... f '-1 (0), s=1,2... |
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| Airy function | Ai(x) | airyai(x) airyaidf(x) airyaizeros(s) airyaidfzeros(s) |
| Airy function | Bi(x) | airybi(x) airybidf(x) airybizeros(s) airybidfzeros(s) |
Kelvin functions
| Functional name | Reserved word | |
|---|---|---|
| f (x) f '(x) |
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| Kelvin function of the 1st kind | berν(x) | ber(ν,x) berdf(ν,x) |
| Kelvin function of the 1st kind | beiν(x) | bei(ν,x) beidf(ν,x) |
| Kelvin function of the 2nd kind | kerν(x) | ker(ν,x) kerdf(ν,x) |
| Kelvin function of the 2nd kind | keiν(x) | kei(ν,x) keidf(ν,x) |
| Kelvin function of the 3rd kind | herν(x) | her(ν,x) herdf(ν,x) |
| Kelvin function of the 3rd kind | heiν(x) | hei(ν,x) heidf(ν,x) |
Special functions Gamma functions
| Functional name | Reserved word | |
|---|---|---|
| Gamma function | Γ(a) | gamma(a) |
| Lower incomplete gamma function |
γ(a,x) | igamma1(a,x) |
| Upper incomplete gamma function |
Γ(a,x) | igamma2(a,x) gamma(a,x) |
| Lower regularized incomplete gamma function |
γ(a,x)/Γ(a) | gammap(a,x) |
| Upper regularized incomplete gamma function |
Γ(a,x)/Γ(a) | gammaq(a,x) |
| Logarithm of gamma function | lnΓ(a) | lngamma(a) |
| Digamma function | Ψ(a) | polygamma(a) |
| Polygamma function | Ψ(n)(a) | polygamma(n,a) |
| Double factorial | x!! | x!! |
| Pochhammer function | (x)n | pochhammer(x,n) |
Beta functions
| Functional name | Reserved word | |
|---|---|---|
| Beta function | B(a,b) | beta(a,b) |
| Regularized incomplete beta function |
Ix(a,b) | y: ibeta(x,a,b) x: ibetaix(y,a,b) |
| Regularized incomplete beta function |
I1-x(a,b) | y: ibetac(x,a,b) x: ibetacix(y,a,b) |
Error functions
| Functional name | Reserved word | |
|---|---|---|
| Error function | erf(x) | y: erf(x) x: erfix(y) |
| Complementary error function | erfc(x) | y: erfc(x) x: erfcix(y) |
| Imaginary error function | erfi(x) | y: erfi(x) |
Elementary function integrals and Fresnel integrals
| Functional name | Reserved word | |
|---|---|---|
| Exponential integral | Ei(x) | Ei(x) |
| Exponential integral | En(x) | Ei(n,x) |
| Logarithmic integral | li(x) | li(x) |
| Sine integral | Si(x) | Si(x) |
| Cosine integral | Ci(x) | Ci(x) |
| Hyperbolic sine integral | Shi(x) | Shi(x) |
| Hyperbolic cosine integral | Chi(x) | Chi(x) |
| Fresnel sine integral | S(x) | fresnelS(x) |
| Fresnel cosine integral | C(x) | fresnelC(x) |
Elliptic integrals and Elliptic functions
| Functional name | Reserved word | |
|---|---|---|
| Complete elliptic integral of the 1st kind | K(k) | ellipticK(k) |
| Complete elliptic integral of the 2nd kind | E(k) | ellipticE(k) |
| Complete elliptic integral of the 3rd kind | Π(n,k) | ellipticPi(n,k) |
| Incomplete elliptic integral of the 1st kind | F(x,k) | ellipticF(x,k) |
| Incomplete elliptic integral of the 2nd kind | E(x,k) | ellipticE(x,k) |
| Incomplete elliptic integral of the 3rd kind | Π(x,n,k) | ellipticPi(x,n,k) |
| Jacobi elliptic function sn | sn(u) | jacobiSN(u,k) |
| Jacobi elliptic function cn | cn(u) | jacobiCN(u,k) |
| Jacobi elliptic function dn | dn(u) | jacobiDN(u,k) |
| Jacobi amplitude function | am(u) | jacobiAM(u,k) |
Orthogonal polynomials
| Functional name | Reserved word | |
|---|---|---|
| Hermite polynomial | Hn(x) | hermiteH(n,x) |
| Chebyshev polynomial of the 1st kind | Tn(x) | chebyshevT(n,x) |
| Chebyshev polynomial of the 2nd kind | Un(x) | chebyshevU(n,x) |
| Legendre polynomial | Pn(x) | legendreP(n,x) |
| Associated Legendre polynomial | Pnm(x) | legendreP(n,,x) |
| Laguerre polynomial | Ln(x) | laguerreL(n,x) |
| Associated Laguerre polynomial | Lnα(x) | laguerreL(n,α,x) |
| Gegenbauer polynomial | Cnλ(x) | gegenbauerC(n,λ,x) |
| Jacobi polynomial | Pnα,β(x) | jacobiP(n,α,β,x) |
| Spherical harmonics | Ynm(θ,φ) | sphericalharmonicY(n, m,θ,φ) |
Other functions
| Functional name | Reserved word | |
|---|---|---|
| Confluent hypergeometric function of the 1st kind | 1F1(a;b;x) | F11(a,b,x) |
| Confluent hypergeometric function of the 2nd kind | U(a;b;x) | FU11(a,b,x) |
| Gauss hypergeometric function | 2F1(a,b;c;x) | F21(a,b,c,x) |
| Bernoulli number | Bn | bernoulli(n) |
| Riemann zeta function | ζ(x) | zeta(x) |
Programming
This programming language is a "C-language-like" simple one and specialized for numerical computation.
| Reserved word | Quick instruction |
|---|---|
| if(expression) {statement;...} elseif(expression) {statement;...} else{statement;...} | If the conditional expression of "if" is true, the statement is executed. If false, the statement is executed by the conditional expression of "elseif". The if statement ends after the execution of one of statements. |
| while(expression) {statement;...} | If the conditional expression is true, the statement is executed. The statement is repeatedly executed and ends when the conditional expression becomes false. |
| do{statement;...} while(expression); | After the statement is unconditionally executed, if the conditional expression is true, the statement is again executed. If false, the statement ends. |
| for(initialize; check; update) {statement;...} | At first the variable is initialized, and if the conditional expression
is true, the statement is executed. After the execution of the statement, the variable is incremented, the conditional expression is again evaluated, and then the same procedure described above is repeatedly executed. When the conditional expression becomes false, the statement ends. |
| break | It escapes from a loop unconditionally. |
| continue | It returns to the beginning of a loop unconditionally. |
Display controls and Comment
| Reserved word | Quick instruction |
|---|---|
| print(expression1,expression2,...) | The value of expression is outputted. |
| println(expression1,expression2,...) | After outputting the value of expression, a new line is started. |
| /*comment*/ | It participates in neither expression nor control. |
Logical operators and Logical values
If a conditional expression is truth, the value is true. If not, the value is false.
| Classification | Reserved word | Quick instruction |
|---|---|---|
| Logical operators |
not | Priority order: not > and > or |
| and | ||
| or | ||
| == | The left expression is equal to the right expression. = | |
| <> | The left expression is not equal to the right expression. != | |
| =<, <= | The left expression is less than or equal to the right expression. <= | |
| =>, >= | The left expression is greater than or equal to the right expression. >= | |
| < | The left expression is less than the right expression. < | |
| > | The left expression is greater than the right expression. > | |
|
Logical values
|
true | true=1 (!=0) |
| false | false=0 |
Array
| Reserved word | Quick instruction | Examples |
|---|---|---|
| numeric Name[Length]; numeric Name[Length][Length]; numeric Name[Length][Length][Length]; .... |
Declare an array | numeric test1[4]; numeric test2[4][4]; numeric test3[4][4][4]; |
| numeric Name[] = {Value1,Value2,....}; numeric Name[][] = {{....},{....}}; numeric Name[][][] = {{....},{....},{....}}; .... |
Declare and initialize an array | numeric test1[] = {1,2,3}; numeric test2[][] = {{1,2,3},{4,5,6}}; |
| kei_length(a) | Get a length of an array | numeric test[4]; n = kei_length(test); |
| function FunctionName(numeric k[]) {...} |
Passing an array as a function parameter | numeric test[]={0,1,2,3}; func1(test); function func1(numeric k[]) { print(k[0]); ..... } |
