Arithmetic and other operations on numbers¶

x
+
y¶ addition
Addition can work on integers, rational numbers, complex numbers, vectors, matrices and lists.
Hint
Addition is implemented in the standard math library (as opposed to being builtin). This means that it can be extended by the user.
 Example
In> 2+3 Out> 5


x¶ negation
Negation can work on integers, rational numbers, complex numbers, vectors, matrices and lists.
Hint
Negation is implemented in the standard math library (as opposed to being builtin). This means that it can be extended by the user.
 Example
In>  3 Out> 3

x

y subtraction
Subtraction can work on integers, rational numbers, complex numbers, vectors, matrices and lists.
Hint
Subtraction is implemented in the standard math library (as opposed to being builtin). This means that it can be extended by the user.
 Example
In> 23 Out> 1

x
*
y¶ multiplication
Multiplication can work on integers, rational numbers, complex numbers, vectors, matrices and lists.
Note
In the case of matrices, multiplication is defined in terms of standard matrix product.
Hint
Multiplication is implemented in the standard math library (as opposed to being builtin). This means that it can be extended by the user.
 Example
In> 2*3 Out> 6

x
/
y¶ division
Division can work on integers, rational numbers, complex numbers, vectors, matrices and lists.
Note
For matrices division is elementwise.
Hint
Division is implemented in the standard math library (as opposed to being builtin). This means that it can be extended by the user.
 Example
In> 6/2 Out> 3

x
^
y¶ exponentiation
Exponentiation can work on integers, rational numbers, complex numbers, vectors, matrices and lists.
Note
In the case of matrices, exponentiation is defined in terms of standard matrix product.
Hint
Exponentiation is implemented in the standard math library (as opposed to being builtin). This means that it can be extended by the user.
 Example
In> 2^3 Out> 8

Div
(x, y)¶ determine divisor
Div()
performs integer division. IfDiv(x,y)
returnsa
andMod(x,y)
equalsb
, then these numbers satisfy \(x =ay + b\) and \(0 \leq b < y\). Example
In> Div(5,3) Out> 1

Mod
(x, y)¶ determine remainder
Mod()
returns the division remainder. IfDiv(x,y)
returnsa
andMod(x,y)
equalsb
, then these numbers satisfy \(x =ay + b\) and \(0 \leq b < y\). Example
In> Div(5,3) Out> 1 In> Mod(5,3) Out> 2

Gcd
(n, m)¶ 
Gcd
(list) greatest common divisor
This function returns the greatest common divisor of
n
andm
or of all elements oflist
.See also

Lcm
(n, m)¶ 
Lcm
(list) least common multiple
This command returns the least common multiple of
n
andm
or of all elements oflist
. Example
In> Lcm(60,24) Out> 120 In> Lcm({3,5,7,9}) Out> 315
See also

n
<<
m¶ 
n
>>
m¶ binary shift operators
These operators shift integers to the left or to the right. They are similar to the C shift operators. These are signextended shifts, so they act as multiplication or division by powers of 2.
 Example
In> 1 << 10 Out> 1024 In> 1024 >> 10 Out> 1

FromBase
(base, "string")¶ conversion of a number from nondecimal base to decimal base
 Param base
integer, base to convert to/from
 Param number
integer, number to write out in a different base
 Param “string”
string representing a number in a different base
In Yacas, all numbers are written in decimal notation (base 10). The two functions {FromBase}, {ToBase} convert numbers between base 10 and a different base. Numbers in nondecimal notation are represented by strings. {FromBase} converts an integer, written as a string in base {base}, to base 10. {ToBase} converts {number}, written in base 10, to base {base}.

N
(expression)¶ try determine numerical approximation of expression
 Param expression
expression to evaluate
 Param precision
integer, precision to use
The function
N()
instructs yacas to try to coerce an expression in to a numerical approximation to the expressionexpr
, usingprec
digits precision if the second calling sequence is used, and the default precision otherwise. This overrides the normal behaviour, in which expressions are kept in symbolic form (eg.Sqrt(2)
instead of1.41421
). Application of theN()
operator will make yacas calculate floating point representations of functions whenever possible. In addition, the variablePi
is bound to the value of \(\pi\) calculated at the current precision.Note
N()
is a macro. Its argumentexpr
will only be evaluated after switching to numeric mode. Example
In> 1/2 Out> 1/2; In> N(1/2) Out> 0.5; In> Sin(1) Out> Sin(1); In> N(Sin(1),10) Out> 0.8414709848; In> Pi Out> Pi; In> N(Pi,20) Out> 3.14159265358979323846;
See also

Rationalize
(expr)¶ convert floating point numbers to fractions
 Param expr
an expression containing real numbers
This command converts every real number in the expression “expr” into a rational number. This is useful when a calculation needs to be done on floating point numbers and the algorithm is unstable. Converting the floating point numbers to rational numbers will force calculations to be done with infinite precision (by using rational numbers as representations). It does this by finding the smallest integer \(n\) such that multiplying the number with \(10^n\) is an integer. Then it divides by \(10^n\) again, depending on the internal gcd calculation to reduce the resulting division of integers.
 Example
In> {1.2,3.123,4.5} Out> {1.2,3.123,4.5}; In> Rationalize(%) Out> {6/5,3123/1000,9/2};
See also

ContFrac
(x[, depth=6])¶ continued fraction expansion
 Param x
number or polynomial to expand in continued fractions
 Param depth
positive integer, maximum required depth
This command returns the continued fraction expansion of
x
, which should be either a floating point number or a polynomial. The remainder is denoted byrest
. This is especially useful for polynomials, since series expansions that converge slowly will typically converge a lot faster if calculated using a continued fraction expansion. Example
In> PrettyForm(ContFrac(N(Pi))) 1  + 3 1  + 7 1  + 15 1  + 1 1  + 292 rest + 1 Out> True; In> PrettyForm(ContFrac(x^2+x+1, 3)) x  + 1 x 1   x  + 1 rest + 1 Out> True;
See also

Decimal
(frac)¶ decimal representation of a rational
 Param frac
a rational number
This function returns the infinite decimal representation of a rational number {frac}. It returns a list, with the first element being the number before the decimal point and the last element the sequence of digits that will repeat forever. All the intermediate list elements are the initial digits before the period sets in.
 Example
In> Decimal(1/22) Out> {0,0,{4,5}}; In> N(1/22,30) Out> 0.045454545454545454545454545454;
See also

Floor
(x)¶ round a number downwards
 Param x
a number
This function returns \(\left \lfloor{x}\right \rfloor\), the largest integer smaller than or equal to
x
. Example
In> Floor(1.1) Out> 1; In> Floor(1.1) Out> 2;

Ceil
(x)¶ round a number upwards
 Param x
a number
This function returns \(\left \lceil{x}\right \rceil\), the smallest integer larger than or equal to
x
. Example
In> Ceil(1.1) Out> 2; In> Ceil(1.1) Out> 1;

Round
(x)¶ round a number to the nearest integer
 Param x
a number
This function returns the integer closest to \(x\). Halfintegers (i.e. numbers of the form \(n + 0.5\), with \(n\) an integer) are rounded upwards.
 Example
In> Round(1.49) Out> 1; In> Round(1.51) Out> 2; In> Round(1.49) Out> 1; In> Round(1.51) Out> 2;

Min
(x, y)¶ 
Min
(list) minimum of a number of values
This function returns the minimum value of its argument(s). If the first calling sequence is used, the smaller of
x
andy
is returned. If one uses the second form, the smallest of the entries inlist
is returned. In both cases, this function can only be used with numerical values and not with symbolic arguments. Example
In> Min(2,3) Out> 2 In> Min({5,8,4}) Out> 4 In> Min(Pi, Exp(1)) Out> Exp(1)

Max
(x, y)¶ 
Max
(list) maximum of a number of values
This function returns the maximum value of its argument(s). If the first calling sequence is used, the larger of
x
andy
is returned. If one uses the second form, the largest of the entries inlist
is returned. In both cases, this function can only be used with numerical values and not with symbolic arguments. Example
In> Max(2,3); Out> 3; In> Max({5,8,4}); Out> 8;

Numer
(expr)¶ numerator of an expression
This function determines the numerator of the rational expression
expr
and returns it. As a special case, if its argument is numeric but not rational, it returns this number. Ifexpr
is neither rational nor numeric, the function returns unevaluated. Example
In> Numer(2/7) Out> 2; In> Numer(a / x^2) Out> a; In> Numer(5) Out> 5;
See also

Denom
(expr)¶ denominator of an expression
This function determines the denominator of the rational expression
expr
and returns it. As a special case, if its argument is numeric but not rational, it returns1
. Ifexpr
is neither rational nor numeric, the function returns unevaluated. Example
In> Denom(2/7) Out> 7; In> Denom(a / x^2) Out> x^2; In> Denom(5) Out> 1;
See also

Pslq
(xlist[, precision=6])¶ search for integer relations between reals
 Param xlist
list of numbers
 Param precision
required number of digits precision of calculation
This function is an integer relation detection algorithm. This means that, given the numbers \(x_i\) in the list
xlist
, it tries to find integer coefficients \(a_i\) such that \(a_1*x_1+\ldots+a_n*x_n = 0\). The list of integer coefficients is returned. The numbers in “xlist” must evaluate to floating point numbers when theN()
operator is applied to them.

e1
<
e2¶ test for “less than”
 Param e1
expression to be compared
 Param e2
expression to be compared
The two expression are evaluated. If both results are numeric, they are compared. If the first expression is smaller than the second one, the result is
True
and it isFalse
otherwise. If either of the expression is not numeric, after evaluation, the expression is returned with evaluated arguments. The word “numeric” in the previous paragraph has the following meaning. An expression is numeric if it is either a number (i.e. {IsNumber} returnsTrue
), or the quotient of two numbers, or an infinity (i.e. {IsInfinity} returnsTrue
). Yacas will try to coerce the arguments passed to this comparison operator to a real value before making the comparison. Example
In> 2 < 5; Out> True; In> Cos(1) < 5; Out> True;
See also

e1
>
e2¶ test for “greater than”
 Param e1
expression to be compared
 Param e2
expression to be compared
The two expression are evaluated. If both results are numeric, they are compared. If the first expression is larger than the second one, the result is
True
and it isFalse
otherwise. If either of the expression is not numeric, after evaluation, the expression is returned with evaluated arguments. The word “numeric” in the previous paragraph has the following meaning. An expression is numeric if it is either a number (i.e. {IsNumber} returnsTrue
), or the quotient of two numbers, or an infinity (i.e. {IsInfinity} returnsTrue
). Yacas will try to coerce the arguments passed to this comparison operator to a real value before making the comparison. Example
In> 2 > 5; Out> False; In> Cos(1) > 5; Out> False
See also

e1
<=
e2¶ test for “less or equal”
 Param e1
expression to be compared
 Param e2
expression to be compared
The two expression are evaluated. If both results are numeric, they are compared. If the first expression is smaller than or equals the second one, the result is
True
and it isFalse
otherwise. If either of the expression is not numeric, after evaluation, the expression is returned with evaluated arguments. The word “numeric” in the previous paragraph has the following meaning. An expression is numeric if it is either a number (i.e. {IsNumber} returnsTrue
), or the quotient of two numbers, or an infinity (i.e. {IsInfinity} returnsTrue
). Yacas will try to coerce the arguments passed to this comparison operator to a real value before making the comparison. Example
In> 2 <= 5; Out> True; In> Cos(1) <= 5; Out> True
See also

e1
>=
e2¶ test for “greater or equal”
 Param e1
expression to be compared
 Param e2
expression to be compared
The two expression are evaluated. If both results are numeric, they are compared. If the first expression is larger than or equals the second one, the result is
True
and it isFalse
otherwise. If either of the expression is not numeric, after evaluation, the expression is returned with evaluated arguments. The word “numeric” in the previous paragraph has the following meaning. An expression is numeric if it is either a number (i.e. {IsNumber} returnsTrue
), or the quotient of two numbers, or an infinity (i.e. {IsInfinity} returnsTrue
). Yacas will try to coerce the arguments passed to this comparison operator to a real value before making the comparison. Example
In> 2 >= 5; Out> False; In> Cos(1) >= 5; Out> False
See also

IsZero
(n)¶ test whether argument is zero
 Param n
number to test
IsZero(n)
evaluates toTrue
ifn
is zero. In casen
is not a number, the function returnsFalse
. Example
In> IsZero(3.25) Out> False; In> IsZero(0) Out> True; In> IsZero(x) Out> False;
See also

IsRational
(expr)¶ test whether argument is a rational
 Param expr
expression to test
This commands tests whether the expression “expr” is a rational number, i.e. an integer or a fraction of integers.
 Example
In> IsRational(5) Out> False; In> IsRational(2/7) Out> True; In> IsRational(0.5) Out> False; In> IsRational(a/b) Out> False; In> IsRational(x + 1/x) Out> False;