# Arithmetic and other operations on numbers¶

x + y

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 built-in). 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 built-in). 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 built-in). This means that it can be extended by the user.

Example

In> 2-3
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 built-in). 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 element-wise.

Hint

Division is implemented in the standard math library (as opposed to being built-in). 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 built-in). 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. If Div(x,y) returns a and Mod(x,y) equals b, 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. If Div(x,y) returns a and Mod(x,y) equals b, 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 and m or of all elements of list.

Lcm(n, m)
Lcm(list)

least common multiple

This command returns the least common multiple of n and m or of all elements of list.

Example

In> Lcm(60,24)
Out> 120
In> Lcm({3,5,7,9})
Out> 315

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 sign-extended 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 non-decimal 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 non-decimal 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 expression expr, using prec 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 of 1.41421). Application of the N() operator will make yacas calculate floating point representations of functions whenever possible. In addition, the variable Pi is bound to the value of $$\pi$$ calculated at the current precision.

Note

N() is a macro. Its argument expr 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;

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};

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 by rest. 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;


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;

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 $$x$$. Half-integers (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 and y is returned. If one uses the second form, the smallest of the entries in list 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 and y is returned. If one uses the second form, the largest of the entries in list 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. If expr 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;

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 returns 1. If expr 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;

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 the N() 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 is False 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} returns True), or the quotient of two numbers, or an infinity (i.e. {IsInfinity} returns True). 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;

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 is False 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} returns True), or the quotient of two numbers, or an infinity (i.e. {IsInfinity} returns True). 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

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 is False 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} returns True), or the quotient of two numbers, or an infinity (i.e. {IsInfinity} returns True). 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

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 is False 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} returns True), or the quotient of two numbers, or an infinity (i.e. {IsInfinity} returns True). 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

IsZero(n)

test whether argument is zero

Param n

number to test

IsZero(n) evaluates to True if n is zero. In case n is not a number, the function returns False.

Example

In> IsZero(3.25)
Out> False;
In> IsZero(0)
Out> True;
In> IsZero(x)
Out> False;

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;