Standard functions

8.3.2 Standard functions

The max command finds the maximum of a sequence of real numbers.

max takes seq, a sequence (or list) of real numbers.
max(seq) returns the largest number in the sequence seq.

Example

max(0,1,2,-1,-2)

The min command finds the minimum of a sequence of real numbers.

min takes seq, a sequence (or list) of real numbers.
min(seq) returns the smallest number in the sequence seq.

Example

min(0,1,2,-1,-2)

−2

The abs command finds the absolute value of a real or complex number.

abs takes x, a real or complex number.
abs(x) returns the absolute value of x.

Examples

abs(-5)

abs(3+4*i)

The sign command finds the sign of a real number (+1 if it is positive, 0 if it is zero, and -1 if it is negative).

sign takes x, a real number.
sign(x) returns the sign of x.

Examples

sign(-4)

−1

sign(0)

The floor or iPart) command finds the floor of a real number; namely, the largest integer less than or equal to the number.

floor takes x, a real number.
floor(x) returns the floor of x.

Examples

floor(4.1)

floor(-4.1)

−5

The round command rounds a number to the nearest integer, rounding up in the case of a half-integer.

round takes x, a real number.
round(x) returns the nearest integer to x.

Examples

round(3.4)

round(-3.4)

−3

round(3.5)

The ceil or ceiling command finds the ceiling of a real number; namely, the smallest integer greater than or equal to the number.

ceil takes x, a real number.
ceil(x) returns the ceiling of x.

Examples

ceiling(4.1)

ceiling(-4.1)

−4

The frac or fPart command finds the fractional part of a number; informally, the part of the number to the right of the decimal point with the appropriate plus or minus sign. For a positive real number x, the fractional part is x minus the floor of x; for a negative real number x, the fractional part is x minus the ceiling of x.

frac takes x, a real number.
frac(x) returns the fractional part of x.

Examples

frac(3.24)

0.24

frac(-3.24)

−0.24

The trunc command truncates a real number; namely, it removes the fractional part. The truncated number added to the fractional part will equal the original number.

trunc takes x, a real number.
trunc(x) returns the truncated value of x.

Examples

trunc(3.24)

trunc(-3.24)

−3

The id command is the identity function.

id takes one argument or a sequence of:
seq, whose elements can be any type.
id(seq) returns seq.

Example

id(a,1,"abc",[1,2,3])

a,1,“abc”,

⎡
⎣

1,2,3

⎤
⎦

The sq command squares its argument.

sq takes x, any object that can be multiplied by itself.
sq(x) returns x².

Examples

sq(5)

sq(x+y)

⎛
⎝

x+y

⎞
⎠

sq([[1,2],[3,4]])

⎡
⎢
⎣

7	10
15	22

⎤
⎥
⎦

(This is a matrix product; see Section 14.2).

sq([1,2,3])

(This is the dot product of [1,2,3] with itself.)

The sqrt command finds the square root of its argument.

sqrt takes x, any object for which the 1/2 power makes sense.
sqrt(x) returns x^1/2.

Examples

sqrt(9)

sqrt((x+y)^2)

⎪
⎪

x+y

⎪
⎪

simplify(sqrt([[2,3],[3,5]]))

⎡
⎢
⎣

1	1
1	2

⎤
⎥
⎦

The surd command finds roots of quantities.

surd takes two arguments: x and n, numbers.
surd(x,n) returns the nth root of x; i.e., x^1/n.

Example

surd(15.625,3)

2.5

The exp command computes the exponential function.

exp takes x, a number.
exp(x) returns e^x.

Example

exp(1.0)

2.71828182846

The log or ln command computes the natural logarithm.

log takes x, a number.
log(x) returns the natural logarithm of x.

Example

log(2.0)

0.69314718056

The log10 command computes the common logarithm.

log10 takes x, a number.
log10(x) returns the base-10 logarithm of x.

Example

log10(1000)

The log2 command computes the binary logarithm.

log2 takes x, a number.
log2(x) returns the base-2 logarithm of x.

Example

log2(1024)

The logb computes the logarithm to a specified base.

logb takes two arguments: x and b, non-zero numbers.
logb(x,b) returns the base-b logarithm of x.

Example

logb(10.0,2)

3.32192809489

Trigonometric functions.

The sin command is the sine function.
The cos command is the cosine function.
The tan command is the tangent function, tanx=sinx/cosx.
The cot command is the cotangent function, cotx=cosx/sinx.
The sec command is the secant function, secx=1/cosx.
The csc command is the cosecant function, cscx=1/sinx.
These commands take one argument: x, a number.
The number x will by default represent an angle measured in radians, but you can set Xcas to use degrees (see Section 2.5.3) by setting the variable angle_radian to 0; resetting it to 1 will change the angle measure to radians again.
sin(x) returns the sine of x.

Examples

Input with angle_radian equal to 1:

sin(pi/4)

√

Input with angle_radian equal to 0:

sin(30)

cos(x) returns the cosine of x. For example, input with angle_radian equal to 1:

cos(pi/6)

√

Input with angle_radian equal to 0:

cos(90)

tan(x) returns the tangent of x. For example, input with angle_radian equal to 1:

tan(pi/4)

Input with angle_radian equal to 0:

tan(60)

√

cot(x) returns the cotangent of x. For example, input with angle_radian equal to 1:

cot(pi/6)

√

Input with angle_radian equal to 0:

cot(45)

sec(x) returns the secant of x. For example, input with angle_radian equal to 1:

sec(pi/3)

Input with angle_radian equal to 0:

sec(30)

√

csc(x) returns the cosecant of x. For example, input with angle_radian equal to 1:

csc(pi/4)

√

Input with angle_radian equal to 0:

csc(30)

Inverse trigonometric functions.

The asin, acos, atan, acot, asec, acsc commands are the inverse trigonometric functions. The latter are defined by:

asec(x)=acos(1/x),
acsc(x)=asin(1/x),
acot(x)=atan(1/x).

arcsin, arccos and arctan are synonyms for asin, acos and atan, respectively.

These functions take one argument: x, a number.
They return a number which can represent an angle; by default, the angles will be in radians, but you can set Xcas to use degrees (see Section 2.5.3) by setting the variable angle_radian to 0; resetting it to 1 will change the angle measure to radians again.

Examples

Input with angle_radian equal to 1:

asin(1/2)

Input with angle_radian equal to 0:

asin(1)

acos(x) returns the arccosine of x. For example: Input with angle_radian equal to 1:

acos(sqrt(3)/2)

Input with angle_radian equal to 0:

acos(-1/2)

120

atan(x) returns the arctangent of x. For example, input with angle_radian equal to 1:

atan(sqrt(3))

Input with angle_radian equal to 1:

atan(1)

acot(x) returns the arccotangent of x. For example, input with angle_radian equal to 1:

acot(sqrt(3))

Input with angle_radian equal to 0:

acot(1/sqrt(3))

asec(x) returns the arcsecant of x. For example, input with angle_radian equal to 1:

asec(1)

Input with angle_radian equal to 0:

asec(sqrt(2))

acsc(x) returns the arccosecant of x. For example, input with angle_radian equal to 1:

acsc(1)

Input with angle_radian equal to 0:

acsc(2)

Hyperbolic functions.

The sinh, cosh, and tanh commands compute the hyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions.

These functions take one argument: x, a number.
sinh(x) returns the hyperbolic sine of x.

Examples

sinh(1.0)

1.17520119364

cosh(x) returns the hyperbolic cosine of x. For example:

cosh(0)

tanh(x) returns the hyperbolic tangent of x. For example:

tanh(-1.0)

−0.761594155956

Inverse hyperbolic functions.

The asinh, acosh, and atanh commands compute the inverse hyperbolic functions.

arcsinh, arccosh and arctanh are synonyms for asinh, acosh and atanh, respectively.

These functions take one argument: x, a number.
asinh(x), acosh(x), and atanh(x) return the inverse hyperbolic sine, cosine, and tangent of x, respectively.

Examples

asinh(2)

⎛
⎜
⎝

√

⎞
⎟
⎠

acosh(x) returns the inverse hyperbolic cosine of x. For example:

acosh(1)

atanh(x) returns the inverse hyperbolic tangent of x. For example:

atanh(1/2)

⎛
⎝

⎞
⎠