1.4 Numbers

FriCAS distinguishes very carefully between different kinds of numbers, how they are represented and what their properties are. Here are a sampling of some of these kinds of numbers and some things you can do with them.

Integer arithmetic is always exact.

11^13 * 13^11 * 17^7 - 19^5 * 23^3

\[25387751112538918594666224484237298\]

_{Type: PositiveInteger}

Integers can be represented in factored form.

factor 643238070748569023720594412551704344145570763243

\[11131311177195233292\]

_{Type: Factored Integer}

Results stay factored when you do arithmetic. Note that the 12 is automatically factored for you.

% * 12

_{Type: Factored Integer}

Integers can also be displayed to bases other than 10. This is an integer in base 11.

radix(25937424601,11)

\[10000000000\]

_{Type: RadixExpansion 11}

Roman numerals are also available for those special occasions. Roman numerals

roman(1992)

\[MCMXCII\]

_{Type: RomanNumeral}

Rational number arithmetic is also exact.

r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9

\[557392520\]

_{Type: Fraction Integer}

To factor fractions, you have to pmap factor onto the numerator and denominator.

map(factor,r)

\[139401233257\]

_{Type: Fraction Factored Integer}

SingleInteger refers to machine word-length integers.

In English, this expression means 11 as a small integer.

11@SingleInteger

\[11\]

_{Type: SingleInteger}

Machine double-precision floating-point numbers are also available for numeric and graphical applications.

123.21@DoubleFloat

\[123.21000000000001\]

_{Type: DoubleFloat}

The normal floating-point type in FriCAS, Float, is a software implementation of floating-point numbers in which the exponent and the mantissa may have any number of digits. The types Complex(Float) and Complex(DoubleFloat) are the corresponding software implementations of complex floating-point numbers.

This is a floating-point approximation to about twenty digits. floating point The :: is used here to change from one kind of object (here, a rational number) to another (a floating-point number).

r :: Float

\[22.118650793650793651\]

_{Type: Float}

Use digits to change the number of digits in the representation. This operation returns the previous value so you can reset it later.

digits(22)

\[20\]

_{Type: PositiveInteger}

To 22 digits of precision, the number eπ163.0 appears to be an integer.

exp(%pi * sqrt 163.0)

\[262537412640768744.0\]

_{Type: Float}

Increase the precision to forty digits and try again.

digits(40); exp(%pi * sqrt 163.0)

\[262537412640768743.9999999999992500725976\]

_{Type: Float}

Here are complex numbers with rational numbers as real and complex numbers imaginary parts.

(2/3 + %i)^3

\[-{{46} \over {27}}+{{1 \over 3} \ i}\]

_{Type: Complex Fraction Integer}

The standard operations on complex numbers are available.

conjugate %

\[-{{46} \over {27}} -{{1 \over 3} \ i}\]

_{Type: Complex Fraction Integer}

You can factor complex integers.

factor(89 - 23 * %i)

\[-{{\left( 1+i \right)} \ {{{\left( 2+i \right)}} ^ {2}} \ {{{\left( 3+{2 \ i} \right)}} ^ {2}}}\]

_{Type: Factored Complex Integer}

Complex numbers with floating point parts are also available.

exp(%pi/4.0 * %i)

\[{0.7071067811865475244}+{{0.7071067811865475244} \ i}\]

_{Type: Complex Float}

The real and imaginary parts can be symbolic.

complex(u,v)

\[u+{v \ i}\]

_{Type: Complex Polynomial Integer}

Of course, you can do complex arithmetic with these also.

% ^ 2

\[-{{v} ^ {2}}+{{u} ^ {2}}+{2 \ u \ v \ i}\]

_{Type: Complex Polynomial Integer}

Every rational number has an exact representation as a repeating decimal expansion

decimal(1/352)

\[0.{00284}{\overline {09}}\]

_{Type: DecimalExpansion}

A rational number can also be expressed as a continued fraction.

continuedFraction(6543/210)

\[ \begin{align}\begin{aligned}\def\zag#1#2{{{\left.{#1}\right|}\over{\left|{#2}\right.}}}\\{31}+ \zag{1}{6}+ \zag{1}{2}+ \zag{1}{1}+ \zag{1}{3}\end{aligned}\end{align} \]

_{Type: ContinuedFraction Integer}

Also, partial fractions can be used and can be displayed in a partial fraction compact format fraction:partial

partialFraction(1,factorial(10))

\[{{159} \over {{2} ^ {8}}} -{{23} \over {{3} ^ {4}}} -{{12} \over {{5} ^ {2}}}+{1 \over 7}\]

_{Type: PartialFraction Integer}

or expanded format.

padicFraction(%)

\[{1 \over 2}+{1 \over {{2} ^ {4}}}+{1 \over {{2} ^ {5}}}+{1 \over {{2} ^ {6}}}+{1 \over {{2} ^ {7}}}+{1 \over {{2} ^ {8}}} -{2 \over {{3} ^ {2}}} -{1 \over {{3} ^ {3}}} -{2 \over {{3} ^ {4}}} -{2 \over 5} -{2 \over {{5} ^ {2}}}+{1 \over 7}\]

_{Type: PartialFraction Integer}

Like integers, bases (radices) other than ten can be used for rational numbers. Here we use base eight.

radix(4/7, 8)

\[0.{\overline 4}\]

_{Type: RadixExpansion 8}

Of course, there are complex versions of these as well. FriCAS decides to make the result a complex rational number.

% + 2/3*%i

\[{4 \over 7}+{{2 \over 3} \ i}\]

_{Type: Complex Fraction Integer}

You can also use FriCAS to manipulate fractional powers.

(5 + sqrt 63 + sqrt 847)^(1/3)

\[\root {3} \of {{{{14} \ {\sqrt {7}}}+5}}\]

_{Type: AlgebraicNumber}

You can also compute with integers modulo a prime.

x : PrimeField 7 := 5

\[5\]

_{Type: PrimeField 7}

Arithmetic is then done modulo 7.

x^3

\[6\]

_{Type: PrimeField 7}

Since 7 is prime, you can invert nonzero values.

1/x

\[3\]

_{Type: PrimeField 7}

You can also compute modulo an integer that is not a prime.

y : IntegerMod 6 := 5

\[5\]

_{Type: IntegerMod 6}

All of the usual arithmetic operations are available.

y^3

\[5\]

_{Type: IntegerMod 6}

Inversion is not available if the modulus is not a prime number. Modular arithmetic and prime fields are discussed in Section ugxProblemFinitePrime .

1/y

There are 12 exposed and 13 unexposed library operations named /
   having 2 argument(s) but none was determined to be applicable.
   Use HyperDoc Browse, or issue
                             )display op /
   to learn more about the available operations. Perhaps
   package-calling the operation or using coercions on the arguments
   will allow you to apply the operation.

Cannot find a definition or applicable library operation named /
   with argument type(s)
                            PositiveInteger
                             IntegerMod 6

   Perhaps you should use "@" to indicate the required return type,
   or "$" to specify which version of the function you need.

This defines a to be an algebraic number, that is, a root of a polynomial equation.

a := rootOf(a^5 + a^3 + a^2 + 3,a)

\[a\]

_{Type: Expression Integer}

Computations with a are reduced according to the polynomial equation.

(a + 1)^10

\[-{{85} \ {{a} ^ {4}}} -{{264} \ {{a} ^ {3}}} -{{378} \ {{a} ^ {2}}} -{{458} \ a} -{287}\]

_{Type: Expression Integer}

Define b to be an algebraic number involving a.

b := rootOf(b^4 + a,b)

\[b\]

_{Type: Expression Integer}

Do some arithmetic.

2/(b - 1)

\[2 \over {b -1}\]

_{Type: Expression Integer}

To expand and simplify this, call ratDenom to rationalize the denominator.

ratDenom(%)

\[\scriptstyle{ {{\left( {{a} ^ {4}} -{{a} ^ {3}}+{2 \ {{a} ^ {2}}} -a+1 \right)} \ {{b} ^ {3}}}+{{\left( {{a} ^ {4}} -{{a} ^ {3}}+{2 \ {{a} ^ {2}}} -a+1 \right)} \ {{b} ^ {2}}}+{{\left( {{a} ^ {4}} -{{a} ^ {3}}+{2 \ {{a} ^ {2}}} -a+1 \right)} \ b}+{{a} ^ {4}} -{{a} ^ {3}}+{2 \ {{a} ^ {2}}} -a+1}\]

_{Type: Expression Integer}

If we do this, we should get b.

2/%+1

\[\scriptstyle{ {{{\left( {{a} ^ {4}} -{{a} ^ {3}}+{2 \ {{a} ^ {2}}} -a+1 \right)} \ {{b} ^ {3}}}+{{\left( {{a} ^ {4}} -{{a} ^ {3}}+{2 \ {{a} ^ {2}}} -a+1 \right)} \ {{b} ^ {2}}}+{{\left( {{a} ^ {4}} -{{a} ^ {3}}+{2 \ {{a} ^ {2}}} -a+1 \right)} \ b}+{{a} ^ {4}} -{{a} ^ {3}}+{2 \ {{a} ^ {2}}} -a+3} \over {{{\left( {{a} ^ {4}} -{{a} ^ {3}}+{2 \ {{a} ^ {2}}} -a+1 \right)} \ {{b} ^ {3}}}+{{\left( {{a} ^ {4}} -{{a} ^ {3}}+{2 \ {{a} ^ {2}}} -a+1 \right)} \ {{b} ^ {2}}}+{{\left( {{a} ^ {4}} -{{a} ^ {3}}+{2 \ {{a} ^ {2}}} -a+1 \right)} \ b}+{{a} ^ {4}} -{{a} ^ {3}}+{2 \ {{a} ^ {2}}} -a+1}}\]

_{Type: Expression Integer}

But we need to rationalize the denominator again.

ratDenom(%)

\[b\]

_{Type: Expression Integer}

Types Quaternion and Octonion are also available. Multiplication of quaternions is non-commutative, as expected.

q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)

\[-{8 \ i}+{{16} \ j} -{8 \ k}\]

_{Type: Quaternion Integer}