Trigintaduonion

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ca.uwaterloo.alumni.dwharder.Numbers
Class Trigintaduonion

java.lang.Object
  ca.uwaterloo.alumni.dwharder.Numbers.Trigintaduonion

All Implemented Interfaces:: java.io.Serializable

public final class Trigintaduonion
extends java.lang.Object
implements java.io.Serializable
extends java.lang.Object
implements java.io.Serializable

A class implementing trigintaduonions using thirty-two doubles.

This package allows the user to construct, manipulate, and model with trigintaduonions. A trigintaduonion is of the form a₀ + ia₁ + ja₂ + ka₃ + e'a₄ + i'a₅ +...+ k⁽⁷⁾a₃₁ where each a_n is a real numbers. The symbols i, j, k, e', i', j', k', e'', i'', j'', k'', e''', i''', j''', k''', e⁽⁴⁾, i⁽⁴⁾, j⁽⁴⁾, k⁽⁴⁾, e⁽⁵⁾, i⁽⁵⁾, j⁽⁵⁾, k⁽⁵⁾, e⁽⁶⁾, i⁽⁶⁾, j⁽⁶⁾, k⁽⁶⁾, e⁽⁷⁾, i⁽⁷⁾, j⁽⁷⁾, k⁽⁷⁾ commute with real numbers and obey the multiplication rules i² = j² = k² = e'² = i'² =...= k⁽⁷⁾² = -1, ij = k, ie' = i', je' = j', ke' = k', ie'' = i'', je'' = j'', ke'' = k'', ie''' = i''', je''' = j''', and ke''' = k'''.

Trigintaduonions are neither commutative, associative, nor alternative. The norm is not algebraic and there exist zero divisors. All these are a conseqence of trigintaduons contain the sedenions.

In the description of each non-static method, this trigintaduonion is represented by s = a₀ + ia₁ + ja₂ + ka₃ + e'a₄ + i'a₅ +...+ k⁽⁷⁾a₃₁. a₀ and ia₁ + ja₂ + ka₃ + e'a₄ + i'a₅ +...+ k⁽⁷⁾a₃₁ are the real and imaginary parts of s, respectively. The real numbers a_n are the coefficients of the trigintaduonion s. The components a₀, ia₁, ..., k⁽⁷⁾a₃₁ will be referred to as the 0th, 1st, ..., and 17th terms of the trigintaduonion s, respectively.

Any other trigintaduonion is represented by t = b₀ + ib₁ + jb₂ + kb₃ + e'b₄ + i'b₅ +...+ k⁽⁷⁾b₃₁. The variables x and n are used to represent real numbers and integers, respectively.

In reference to methods which must deal with branch cuts, the symbol u refers to any unit imaginary trigintaduonion (real(u) = 0 and |u| = 1).

Instances of this class are immutable.

This class returns correct answers on branch cuts. Additionally, this class tries to give reasonable treatment to ∞ and NaN. For example, ONE.multiply(Double.POSITIVE_INFINITY) returns ∞ + i0 + ... + k'''0. Most implementations would return ∞ + iNaN + ... + k'''NaN because they would simply multiply each coefficient by the multiplier. This is probably not the desired result.

This class is based on the methods found in the Java classes Double, Math, and java.math.BigDecimal.

There is very little interaction between between the classes in this package. If there is a significant request for such behaviour, it will be added (seemlessly) in the future.

Since:: JDK 5.0
Version:: 1.0beta 22-Mar-2005
Author:: Douglas Wilhelm Harder, Department of Electrical and Computer Engineering, University of Waterloo
See Also:: Double, Math, Serialized Form

Field Summary
`static Trigintaduonion`	`E1` The constant e'.
`static Trigintaduonion`	`E2` The constant e''.
`static Trigintaduonion`	`E3` The constant e'''.
`static Trigintaduonion`	`E4` The constant e⁽⁴⁾.
`static Trigintaduonion`	`E5` The constant e⁽⁵⁾.
`static Trigintaduonion`	`E6` The constant e⁽⁶⁾.
`static Trigintaduonion`	`E7` The constant e⁽⁷⁾.
`static Trigintaduonion`	`I` The constant i.
`static Trigintaduonion`	`I1` The constant i'.
`static Trigintaduonion`	`I2` The constant i''.
`static Trigintaduonion`	`I3` The constant i'''.
`static Trigintaduonion`	`I4` The constant i⁽⁴⁾.
`static Trigintaduonion`	`I5` The constant i⁽⁵⁾.
`static Trigintaduonion`	`I6` The constant i⁽⁶⁾.
`static Trigintaduonion`	`I7` The constant i⁽⁷⁾.
`static Trigintaduonion`	`J` The constant j.
`static Trigintaduonion`	`J1` The constant j'.
`static Trigintaduonion`	`J2` The constant j''.
`static Trigintaduonion`	`J3` The constant j'''.
`static Trigintaduonion`	`J4` The constant j⁽⁴⁾.
`static Trigintaduonion`	`J5` The constant j⁽⁵⁾.
`static Trigintaduonion`	`J6` The constant j⁽⁶⁾.
`static Trigintaduonion`	`J7` The constant j⁽⁷⁾.
`static Trigintaduonion`	`K` The constant k.
`static Trigintaduonion`	`K1` The constant k'.
`static Trigintaduonion`	`K2` The constant k''.
`static Trigintaduonion`	`K3` The constant k'''.
`static Trigintaduonion`	`K4` The constant k⁽⁴⁾.
`static Trigintaduonion`	`K5` The constant k⁽⁵⁾.
`static Trigintaduonion`	`K6` The constant k⁽⁶⁾.
`static Trigintaduonion`	`K7` The constant k⁽⁷⁾.
`static Trigintaduonion`	`NaN` The constant NaN + iNaN + jNaN +...+ k⁽⁷⁾NaN.
`static Trigintaduonion`	`ONE` The constant 1.
`static Trigintaduonion`	`ZERO` The constant 0.

Constructor Summary
`Trigintaduonion(double a0)` Generates the trigintaduonion a₀ + i0 + j0 +...+ k⁽⁷⁾0.
`Trigintaduonion(double[] v)` Generates a trigintaduonion based on the entries of the array v.
`Trigintaduonion(double x, double[] v)` Generates the trigintaduonion x + iv₀ + jv₁ +...+ k⁽⁷⁾v₃₀.
`Trigintaduonion(double a1, double a2, double a3, double a4, double a5, double a6, double a7, double a8, double a9, double a10, double a11, double a12, double a13, double a14, double a15, double a16, double a17, double a18, double a19, double a20, double a21, double a22, double a23, double a24, double a25, double a26, double a27, double a28, double a29, double a30, double a31)` Generates the imaginary trigintaduonion ia₁ + ja₂ +...+ k⁽⁷⁾a₃₁.
`Trigintaduonion(double a0, double a1, double a2, double a3, double a4, double a5, double a6, double a7, double a8, double a9, double a10, double a11, double a12, double a13, double a14, double a15, double a16, double a17, double a18, double a19, double a20, double a21, double a22, double a23, double a24, double a25, double a26, double a27, double a28, double a29, double a30, double a31)` Generates the trigintaduonion a₀ + ia₁ + ...+ k⁽⁷⁾a₃₁.

Method Summary
`double`	`abs()` Returns the absolute value of this trigintaduonion s.
`double`	`abs2()` Returns the square of the absolute value of this trigintaduonion s.
`double`	`abs2Imag()` Returns the square of the absolute value of the imaginary part of this trigintaduonion s.
`double`	`absImag()` Returns the absolute value of the imaginary part of this trigintaduonion s.
`Trigintaduonion`	`acos()` Returns the principal inverse cosine of this trigintaduonion s.
`Trigintaduonion`	`acos(Trigintaduonion u)` Returns the principal inverse cosine of this trigintaduonion s.
`Trigintaduonion`	`acosh()` Returns the principal inverse hyperbolic cosine of this trigintaduonion s.
`Trigintaduonion`	`acosh(Trigintaduonion u)` Returns the principal inverse hyperbolic cosine of this trigintaduonion s.
`Trigintaduonion`	`acot()` Returns the principal inverse cotangent of this trigintaduonion s.
`Trigintaduonion`	`acot(Trigintaduonion u)` Returns the principal inverse cotangent of this trigintaduonion s.
`Trigintaduonion`	`acoth()` Returns the principal inverse hyperbolic cotangent of this trigintaduonion s.
`Trigintaduonion`	`acoth(Trigintaduonion u)` Returns the principal inverse hyperbolic cotangent of this trigintaduonion s.
`Trigintaduonion`	`acsc()` Returns the principal inverse cosecant of this trigintaduonion s.
`Trigintaduonion`	`acsc(Trigintaduonion u)` Returns the principal inverse cosecant of this trigintaduonion s.
`Trigintaduonion`	`acsch()` Returns the principal inverse hyperbolic cosecant of this trigintaduonion s.
`Trigintaduonion`	`acsch(Trigintaduonion u)` Returns the principal inverse hyperbolic cosecant of this trigintaduonion s.
`Trigintaduonion`	`add(double x)` Returns the sum of this trigintaduonion s and the real number x.
`Trigintaduonion`	`add(int n, double x)` Returns this sedenian s with x added onto its nth term.
`Trigintaduonion`	`add(Trigintaduonion t)` Returns the sum of this trigintaduonion s and the trigintaduonion t = b₀ + ib₁ + jb₂ +...+ k⁽⁷⁾b₃₁.
`double`	`argument()` Returns the argument of this trigintaduonion s.
`Trigintaduonion`	`asec()` Returns the principal inverse secant of this trigintaduonion s.
`Trigintaduonion`	`asec(Trigintaduonion u)` Returns the principal inverse secant of this trigintaduonion s.
`Trigintaduonion`	`asech()` Returns the principal inverse hyperbolic secant of this trigintaduonion s.
`Trigintaduonion`	`asech(Trigintaduonion u)` Returns the principal inverse hyperbolic secant of this trigintaduonion s.
`Trigintaduonion`	`asin()` Returns the principal inverse sine of this trigintaduonion s.
`Trigintaduonion`	`asin(Trigintaduonion u)` Returns the principal inverse sine of this trigintaduonion s.
`Trigintaduonion`	`asinh()` Returns the principal inverse hyperbolic sine of this trigintaduonion s.
`Trigintaduonion`	`asinh(Trigintaduonion u)` Returns the principal inverse hyperbolic sine of this trigintaduonion s.
`Trigintaduonion`	`atan()` Returns the principal inverse tangent of this trigintaduonion s.
`Trigintaduonion`	`atan(Trigintaduonion u)` Returns the principal inverse tangent of this trigintaduonion s.
`Trigintaduonion`	`atanh()` Returns the principal inverse hyperbolic tangent of this trigintaduonion s.
`Trigintaduonion`	`atanh(Trigintaduonion u)` Returns the principal inverse hyperbolic tangent of this trigintaduonion s.
`Trigintaduonion`	`ceil()` Returns the ceiling of each of the coefficients of this trigintaduonion s.
`double`	`coefficient(int n)` Returns the coefficient of the nth term of this trigintaduonion s.
`Trigintaduonion`	`conjugate()` Returns the conjugate of this trigintaduonion s.
`Trigintaduonion`	`cos()` Returns the cosine of this trigintaduonion s.
`Trigintaduonion`	`cosh()` Returns the hyperbolic cosine of this trigintaduonion s.
`Trigintaduonion`	`cot()` Returns the cotangent of this trigintaduonion s.
`Trigintaduonion`	`coth()` Returns the hyperbolic cotangent of this trigintaduonion s.
`Trigintaduonion`	`csc()` Returns the cosecant of this trigintaduonion s.
`Trigintaduonion`	`csch()` Returns the hyperbolic cosecant of this trigintaduonion s.
`Trigintaduonion`	`divide(Trigintaduonion t)` Returns the quotient of this trigintaduonion s over the trigintaduonion t = b₀ + ib₁ + jb₂ +...+ k⁽⁷⁾b₃₁.
`boolean`	`equals(java.lang.Object t)` Returns `true` if this trigintaduonion s equals the trigintaduonion t = b₀ + ib₁ + jb₂ +...+ k⁽⁷⁾b₃₁.
`boolean`	`equals(java.lang.Object t, double eps)` Returns `true` if this trigintaduonion s equals the trigintaduonion t = b₀ + ib₁ + jb₂ +...+ k⁽⁷⁾b₃₁.
`Trigintaduonion`	`exp()` Returns Euler's number (e) raised to the power of this trigintaduonion s.
`Trigintaduonion`	`floor()` Returns the floor of each of the coefficients of this trigintaduonion s.
`int`	`hashCode()` Returns a hash code for the this trigintaduonion s based on the coefficients.
`Trigintaduonion`	`horner(double[] v)` Evaluate the polynomial with real coefficients, given by the array of doubles v, at this trigintaduonion s.
`Trigintaduonion`	`imag()` Returns the imaginary part of this trigintaduonion s.
`static java.lang.String[]`	`imaginaryUnits(java.lang.String[] units)` Set the strings to be used for the thirty-one imaginary units.
`boolean`	`isImaginary()` Returns `true` if the real part of this trigintaduonion s is zero and `false` otherwise.
`boolean`	`isImaginary(int n)` Returns `true` if the nth imaginary coefficient of this octonian s is the only non-zero coefficient and `false` otherwise.
`boolean`	`isInfinite()` Returns `true` if any coefficient of this trigintaduonion s is infinite and `false` otherwise.
`boolean`	`isNaN()` Returns `true` if any coefficient of this trigintaduonion s is not-a-number (NaN) and `false` otherwise.
`boolean`	`isReal()` Returns `true` if all the coefficients of the imaginary part of this trigintaduonion s are zero, otherwise `false`.
`boolean`	`isZero()` Returns `true` if all coefficients of this trigintaduonion s are zero, otherwise `false`.
`Trigintaduonion`	`log()` Returns the natural logarithm of this trigintaduonion s.
`Trigintaduonion`	`log(Trigintaduonion u)` Returns the natural logarithm of this trigintaduonion s.
`Trigintaduonion`	`log10()` Returns the logarithm base 10 of this trigintaduonion s.
`Trigintaduonion`	`log10(Trigintaduonion u)` Returns the natural logarithm of this trigintaduonion s.
`Trigintaduonion`	`multiply(double x)` Returns the product of this trigintaduonion s and the real number x.
`Trigintaduonion`	`multiply(Trigintaduonion t)` Returns the product of this trigintaduonion s and the trigintaduonion t = b₀ + ib₁ + jb₂ +...+ k⁽⁷⁾b₃₁.
`Trigintaduonion`	`negate()` Returns the negative of this trigintaduonion s.
`Trigintaduonion`	`pow(double x)` Returns this trigintaduonion s raised to the real number x.
`Trigintaduonion`	`pow(int n)` Returns this trigintaduonion s raised to the integer n.
`Trigintaduonion`	`pow(Trigintaduonion t)` Returns this trigintaduonion s raised to the trigintaduonion t.
`Trigintaduonion`	`project(boolean[] v)` Project this trigintaduonion s onto the subspace defined by the boolean vector v.
`Trigintaduonion`	`project(int n)` Project this trigintaduonion s onto the nth term.
`static Trigintaduonion`	`random()` A factory method for constructing an trigintaduonion with random real and imaginary parts.
`Trigintaduonion`	`random(boolean[] v)` A factory method to create a random trigintaduonion defined by the boolean vector v.
`static Trigintaduonion`	`random(int n)` A factory method for constructing a complex number with random nth coefficient.
`static Trigintaduonion`	`randomImaginary()` A factory method for constructing a random imaginary trigintaduonion.
`static Trigintaduonion`	`randomReal()` A factory method for constructing an trigintaduonion with random real part.
`double`	`real()` Returns the real part of this trigintaduonion s.
`Trigintaduonion`	`rint()` Returns each coefficient of the trigintaduonion s rounded to the nearest double equal to an integer.
`Trigintaduonion`	`sec()` Returns the secant of this trigintaduonion s.
`Trigintaduonion`	`sech()` Returns the hyperbolic secant of this trigintaduonion s.
`Trigintaduonion`	`signum()` Returns a normalized form of this trigintaduonion s when it is non-zero and s otherwise.
`Trigintaduonion`	`sin()` Returns the sine of this trigintaduonion s.
`Trigintaduonion`	`sinh()` Returns the hyperbolic sine of this trigintaduonion s.
`Trigintaduonion`	`sqr()` Returns the square of this trigintaduonion s.
`Trigintaduonion`	`sqrt()` Returns the square root of this trigintaduonion s.
`Trigintaduonion`	`sqrt(Trigintaduonion u)` Returns the square root of this trigintaduonion s.
`Trigintaduonion`	`subtract(Trigintaduonion t)` Returns the difference between this trigintaduonion s and the trigintaduonion t = b₀ + ib₁ + jb₂ +...+ k⁽⁷⁾b₃₁.
`Trigintaduonion`	`tan()` Returns the tangent of this trigintaduonion s.
`Trigintaduonion`	`tanh()` Returns the hyperbolic tangent of this trigintaduonion s.
`double[]`	`toArray()` Converts this trigintaduonion s in to an array of four doubles.
`double[]`	`toArray(byte[] v)` Convert this trigintaduonion o to an array of doubles.
`double[]`	`toImagArray()` Converts the imaginary part of this trigintaduonion s in to an array of three doubles.
`java.lang.String`	`toString()` Returns a string representing this trigintaduonion s.

Methods inherited from class java.lang.Object
`getClass, notify, notifyAll, wait, wait, wait`

Field Detail

ZERO

public static final Trigintaduonion ZERO

The constant 0.

ONE

public static final Trigintaduonion ONE

The constant 1.

I

public static final Trigintaduonion I

The constant i.

J

public static final Trigintaduonion J

The constant j.

K

public static final Trigintaduonion K

The constant k.

E1

public static final Trigintaduonion E1

The constant e'.

I1

public static final Trigintaduonion I1

The constant i'.

J1

public static final Trigintaduonion J1

The constant j'.

K1

public static final Trigintaduonion K1

The constant k'.

E2

public static final Trigintaduonion E2

The constant e''.

I2

public static final Trigintaduonion I2

The constant i''.

J2

public static final Trigintaduonion J2

The constant j''.

K2

public static final Trigintaduonion K2

The constant k''.

E3

public static final Trigintaduonion E3

The constant e'''.

I3

public static final Trigintaduonion I3

The constant i'''.

J3

public static final Trigintaduonion J3

The constant j'''.

K3

public static final Trigintaduonion K3

The constant k'''.

E4

public static final Trigintaduonion E4

The constant e⁽⁴⁾.

I4

public static final Trigintaduonion I4

The constant i⁽⁴⁾.

J4

public static final Trigintaduonion J4

The constant j⁽⁴⁾.

K4

public static final Trigintaduonion K4

The constant k⁽⁴⁾.

E5

public static final Trigintaduonion E5

The constant e⁽⁵⁾.

I5

public static final Trigintaduonion I5

The constant i⁽⁵⁾.

J5

public static final Trigintaduonion J5

The constant j⁽⁵⁾.

K5

public static final Trigintaduonion K5

The constant k⁽⁵⁾.

E6

public static final Trigintaduonion E6

The constant e⁽⁶⁾.

I6

public static final Trigintaduonion I6

The constant i⁽⁶⁾.

J6

public static final Trigintaduonion J6

The constant j⁽⁶⁾.

K6

public static final Trigintaduonion K6

The constant k⁽⁶⁾.

E7

public static final Trigintaduonion E7

The constant e⁽⁷⁾.

I7

public static final Trigintaduonion I7

The constant i⁽⁷⁾.

J7

public static final Trigintaduonion J7

The constant j⁽⁷⁾.

K7

public static final Trigintaduonion K7

The constant k⁽⁷⁾.

NaN

public static final Trigintaduonion NaN

The constant NaN + iNaN + jNaN +...+ k⁽⁷⁾NaN.

Constructor Detail

Trigintaduonion

public Trigintaduonion(double a0,
                       double a1,
                       double a2,
                       double a3,
                       double a4,
                       double a5,
                       double a6,
                       double a7,
                       double a8,
                       double a9,
                       double a10,
                       double a11,
                       double a12,
                       double a13,
                       double a14,
                       double a15,
                       double a16,
                       double a17,
                       double a18,
                       double a19,
                       double a20,
                       double a21,
                       double a22,
                       double a23,
                       double a24,
                       double a25,
                       double a26,
                       double a27,
                       double a28,
                       double a29,
                       double a30,
                       double a31)

Generates the trigintaduonion a₀ + ia₁ + ...+ k⁽⁷⁾a₃₁.

Parameters:: a0 - the real part; a1 - the coefficient of i; a2 - the coefficient of j; a3 - the coefficient of k; a4 - the coefficient of e'; a31 - the coefficient of k⁽⁷⁾

Trigintaduonion

public Trigintaduonion(double a0)

Generates the trigintaduonion a₀ + i0 + j0 +...+ k⁽⁷⁾0.

Parameters:: a0 - the real part

Trigintaduonion

public Trigintaduonion(double a1,
                       double a2,
                       double a3,
                       double a4,
                       double a5,
                       double a6,
                       double a7,
                       double a8,
                       double a9,
                       double a10,
                       double a11,
                       double a12,
                       double a13,
                       double a14,
                       double a15,
                       double a16,
                       double a17,
                       double a18,
                       double a19,
                       double a20,
                       double a21,
                       double a22,
                       double a23,
                       double a24,
                       double a25,
                       double a26,
                       double a27,
                       double a28,
                       double a29,
                       double a30,
                       double a31)

Generates the imaginary trigintaduonion ia₁ + ja₂ +...+ k⁽⁷⁾a₃₁.

Parameters:: a1 - the coefficient of i; a2 - the coefficient of j; a3 - the coefficient of k; a31 - the coefficient of k⁽⁷⁾

Trigintaduonion

public Trigintaduonion(double[] v)

Generates a trigintaduonion based on the entries of the array v.

An array v of length 32 defines the trigintaduonion v₀ + iv₁ + jv₂ +...+ k⁽⁷⁾v₃₁, and

An array v of length 31 defines the trigintaduonion 0 + iv₁ + jv₂ +...+ k⁽⁷⁾v₃₀, and

Parameters:: v - array of length 31 or 32

Trigintaduonion

public Trigintaduonion(double x,
                       double[] v)

Generates the trigintaduonion x + iv₀ + jv₁ +...+ k⁽⁷⁾v₃₀.

Parameters:: x - the real part; v - array of length 31

Method Detail

random

public static final Trigintaduonion random()

A factory method for constructing an trigintaduonion with random real and imaginary parts.

Returns:: r₀ + ir₁ + jr₂ + kr₃, where each r_i is a random double.
See Also:: Math.random()

random

public static final Trigintaduonion random(int n)

A factory method for constructing a complex number with random nth coefficient.

Returns:: r, ir, jr, ..., or k⁽⁷⁾r where r is a random double
See Also:: Math.random()

random

public final Trigintaduonion random(boolean[] v)

A factory method to create a random trigintaduonion defined by the boolean vector v.

In the result, the coefficient of the nth term is a random double if v_n == true.

Parameters:: v - an array indicating a subspace
Returns:: a random trigintaduonion s
See Also:: Math.random()

randomReal

public static final Trigintaduonion randomReal()

A factory method for constructing an trigintaduonion with random real part.

Returns:: r where r is a random double
See Also:: Math.random()

randomImaginary

public static final Trigintaduonion randomImaginary()

A factory method for constructing a random imaginary trigintaduonion.

Returns:: 0 + ir₁ + jr₂ +...+ k⁽⁷⁾r₃₁, where each r_n is a random double
See Also:: Math.random()

equals

public final boolean equals(java.lang.Object t)

Returns true if this trigintaduonion s equals the trigintaduonion t = b₀ + ib₁ + jb₂ +...+ k⁽⁷⁾b₃₁.

The sedendions are equal if a_n = b_n for each n.

Note: NaN == NaN returns false but 0.0 == -0.0 returns true.

Overrides:: equals in class java.lang.Object

Returns:: true if the trigintaduonions are equal.

equals

public final boolean equals(java.lang.Object t,
                            double eps)

Returns true if this trigintaduonion s equals the trigintaduonion t = b₀ + ib₁ + jb₂ +...+ k⁽⁷⁾b₃₁.

Two nonzero trigintaduonions s and t are equal up to ε if |s - t|/min( |s|, |t| ) < ε.

If s = 0 or t = 0 then we check that |t| < ε or |s| < ε, respectively.

Parameters:: eps - the allowable relative difference: greater than or equal to 0
Returns:: true if the trigintaduonions are approximately equal.

hashCode

public final int hashCode()

Returns a hash code for the this trigintaduonion s based on the coefficients.

Overrides:: hashCode in class java.lang.Object

toString

public final java.lang.String toString()

Returns a string representing this trigintaduonion s.

If N₀ = abs(a), N₁ = abs(b), N₂ = abs(c), and N₃ = abs(d), then the return value is "[-]N₀ (+|-) iN₁ (+|-) jN₂ (+|-) kN₃".

Overrides:: toString in class java.lang.Object

Returns:: a string of the form "a + ib + jc + kd"

imaginaryUnits

public static final java.lang.String[] imaginaryUnits(java.lang.String[] units)

Set the strings to be used for the thirty-one imaginary units.

For example, double[] {"i", "j", "k", "e(1)", "i(1)", "j(1)", "k(1)", "e(2)", "i(2)", "j(2)", "k(2)", "e(3)", "i(3)", "j(3)", "k(3)", "e(4)", "i(4)", "j(4)", "k(4)", "e(5)", "i(5)", "j(5)", "k(5)", "e(6)", "i(6)", "j(6)", "k(6)"}.

Returns:: the previously used array of imaginary units

toArray

public final double[] toArray()

Converts this trigintaduonion s in to an array of four doubles.

Returns:: new double[] {a₀, a₁, a₂, ..., a₃₁}

toArray

public final double[] toArray(byte[] v)

Convert this trigintaduonion o to an array of doubles.

The coefficient placed into the nth entry of the array is determined by v_n.

Parameters:: v - an array of 0, 1, 2, ..., or 31, indicating which coefficient is to be placed into the nth entry of the array
Returns:: an array of doubles of coefficients as specified by v

toImagArray

public final double[] toImagArray()

Converts the imaginary part of this trigintaduonion s in to an array of three doubles.

Returns:: new double[] {a₁, a₂, ..., a₃₁}

isZero

public final boolean isZero()

Returns true if all coefficients of this trigintaduonion s are zero, otherwise false.

Returns:: true if s is zero

isReal

public final boolean isReal()

Returns true if all the coefficients of the imaginary part of this trigintaduonion s are zero, otherwise false.

Returns:: true if s is real

isImaginary

public final boolean isImaginary()

Returns true if the real part of this trigintaduonion s is zero and false otherwise.

Returns:: true if s is imaginary

isImaginary

public final boolean isImaginary(int n)

Returns true if the nth imaginary coefficient of this octonian s is the only non-zero coefficient and false otherwise.

Returns:: true if s is non-zero in only one component

isNaN

public final boolean isNaN()

Returns true if any coefficient of this trigintaduonion s is not-a-number (NaN) and false otherwise.

Returns:: true if any coefficient of s is NaN
See Also:: Double.isNaN(double)

isInfinite

public final boolean isInfinite()

Returns true if any coefficient of this trigintaduonion s is infinite and false otherwise.

Returns:: true if any coefficient of s is infinite
See Also:: Double.isInfinite(double)

abs

public final double abs()

Returns the absolute value of this trigintaduonion s.

abs(s) = |s| = sqrt(a₀² + a₁² + a₂² +...+ a₃₁²).

Special case:

If |a_n| = ∞ for any n then |s| = ∞

Returns:: |s|

abs2

public final double abs2()

Returns the square of the absolute value of this trigintaduonion s.

abs(s)² = |s|² = a₀² + a₁² + a₂² +...+ a₃₁².

Special case:

If |a_n| = ∞ for any n then |s|² = ∞

Returns:: |s|²

absImag

public final double absImag()

Returns the absolute value of the imaginary part of this trigintaduonion s.

abs(imag(s)) = |imag(s)| = sqrt(a₁² + a₂² +...+ a₃₁²).

Special case:

If |a_n| = ∞ for any n ≥ 1 then |imag(s)| = ∞

Returns:: |imag(s)|

abs2Imag

public final double abs2Imag()

Returns the square of the absolute value of the imaginary part of this trigintaduonion s.

abs(imag(s))² = |imag(s)|² = a₁² + a₂² +...+ a₃₁².

Special case:

If |a_n| = ∞ for any n ≥ 1 then |imag(s)|² = ∞

Returns:: |imag(s)|²

real

public final double real()

Returns the real part of this trigintaduonion s.

real(s) = a₀.

Returns:: a₀
See Also:: project(int)

imag

public final Trigintaduonion imag()

Returns the imaginary part of this trigintaduonion s.

imag(s) = ia₁ + ja₂ +...+ k⁽⁷⁾a₃₁.

Returns:: 0 + imag(s)

coefficient

public final double coefficient(int n)

Returns the coefficient of the nth term of this trigintaduonion s.

Parameters:: n - the number (0, 1, 2, ..., 31) of the coefficient to return
Returns:: the nth coefficient

project

public final Trigintaduonion project(int n)

Project this trigintaduonion s onto the nth term.

Parameters:: n - the number of term (0, 1, 2, ..., 31) to be projected onto
Returns:: a₀, ia₁, ja₂, ..., or k⁽⁷⁾a₃₁

project

public final Trigintaduonion project(boolean[] v)

Project this trigintaduonion s onto the subspace defined by the boolean vector v.

In the result, the coefficient of the nth term is set to 0.0 if v_n == false.

Parameters:: v - an array indicating the terms to project onto
Returns:: a perpendicular projection of s

conjugate

public final Trigintaduonion conjugate()

Returns the conjugate of this trigintaduonion s.

conjugate(s) = a₀ - ia₁ - ja₂ -...- k⁽⁷⁾a₃₁.

Returns:: a₀ - ia₁ - ja₂ -...- k⁽⁷⁾a₃₁

argument

public final double argument()

Returns the argument of this trigintaduonion s.

This is a value on the interval [0, &pi]. For non-zero s, if u = imag(s)/|imag(s)| then s = |s|e^{u argument(s)}.

argument(s) = atan2( |imag(s)|, a ).

Returns:: atan2( |imag(s)|, a )

signum

public final Trigintaduonion signum()

Returns a normalized form of this trigintaduonion s when it is non-zero and s otherwise.

Returns:: s/|s| for nonzero s

add

public final Trigintaduonion add(Trigintaduonion t)

Returns the sum of this trigintaduonion s and the trigintaduonion t = b₀ + ib₁ + jb₂ +...+ k⁽⁷⁾b₃₁.

s + t = (a₀ + b₀) + i(a₁ + b₁) + j(a₂ + b₂) +...+ k⁽⁷⁾(a₃₁ + b₃₁).

Returns:: s + t

add

public final Trigintaduonion add(int n,
                                 double x)

Returns this sedenian s with x added onto its nth term.

Parameters:: n - the number (0, 1, 2, ..., 31) of the onto which to add x
Returns:: s with x added onto the coefficient of the nth term

add

public final Trigintaduonion add(double x)

Returns the sum of this trigintaduonion s and the real number x.

s + x = (a₀ + x) + ia₁ + ja₂ +...+ k⁽⁷⁾a₃₁.

Returns:: s + x

subtract

public final Trigintaduonion subtract(Trigintaduonion t)

Returns the difference between this trigintaduonion s and the trigintaduonion t = b₀ + ib₁ + jb₂ +...+ k⁽⁷⁾b₃₁.

s - t = (a₀ - b₀) + i(a₁ - b₁) + j(a₂ - b₂) +...+ k⁽⁷⁾(a₃₁ - b₃₁).

Returns:: s - t

negate

public final Trigintaduonion negate()

Returns the negative of this trigintaduonion s.

-s = -a₀ - ia₁ - ja₂ -...- k'''a₃₁.

Returns:: -s

multiply

public final Trigintaduonion multiply(Trigintaduonion t)

Returns the product of this trigintaduonion s and the trigintaduonion t = b₀ + ib₁ + jb₂ +...+ k⁽⁷⁾b₃₁.

Multiplication is distributive with the following multiplication matrix for 1 and the imaginary units:

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j⁽⁶⁾

k⁽⁶⁾

e⁽⁷⁾

i⁽⁷⁾

j⁽⁷⁾

k⁽⁷⁾

e⁽⁴⁾

-i⁽⁴⁾

-j⁽⁴⁾

-k⁽⁴⁾

-e⁽⁵⁾

-i⁽⁵⁾

-j⁽⁵⁾

-k⁽⁵⁾

-e''

i''

j''

k''

e'''

i'''

j'''

k'''

-1

-i

-j

-k

-e'

-i'

-j'

-k'

i⁽⁶⁾

i⁽⁶⁾

-e⁽⁶⁾

k⁽⁶⁾

-j⁽⁶⁾

i⁽⁷⁾

-e⁽⁷⁾

-k⁽⁷⁾

j⁽⁷⁾

i⁽⁴⁾

e⁽⁴⁾

k⁽⁴⁾

-j⁽⁴⁾

i⁽⁵⁾

-e⁽⁵⁾

-k⁽⁵⁾

j⁽⁵⁾

-i''

-e''

k''

-j''

i'''

-e'''

-k'''

j'''

-1

-j

-e'

-k'

j⁽⁶⁾

j⁽⁶⁾

-k⁽⁶⁾

-e⁽⁶⁾

i⁽⁶⁾

j⁽⁷⁾

k⁽⁷⁾

-e⁽⁷⁾

-i⁽⁷⁾

j⁽⁴⁾

-k⁽⁴⁾

e⁽⁴⁾

i⁽⁴⁾

j⁽⁵⁾

k⁽⁵⁾

-e⁽⁵⁾

-i⁽⁵⁾

-j''

-k''

-e''

i''

j'''

k'''

-e'''

-i'''

-k

-1

-e'

-i'

k⁽⁶⁾

k⁽⁶⁾

j⁽⁶⁾

-i⁽⁶⁾

-e⁽⁶⁾

k⁽⁷⁾

-j⁽⁷⁾

i⁽⁷⁾

-e⁽⁷⁾

k⁽⁴⁾

j⁽⁴⁾

-i⁽⁴⁾

e⁽⁴⁾

k⁽⁵⁾

-j⁽⁵⁾

i⁽⁵⁾

-e⁽⁵⁾

-k''

j''

-i''

-e''

k'''

-j'''

i'''

-e'''

-i

-1

-j'

-e'

e⁽⁷⁾

e⁽⁷⁾

-i⁽⁷⁾

-j⁽⁷⁾

-k⁽⁷⁾

-e⁽⁶⁾

i⁽⁶⁾

j⁽⁶⁾

k⁽⁶⁾

e⁽⁵⁾

-i⁽⁵⁾

-j⁽⁵⁾

-k⁽⁵⁾

e⁽⁴⁾

i⁽⁴⁾

j⁽⁴⁾

k⁽⁴⁾

-e'''

-i'''

-j'''

-k'''

-e''

i''

j''

k''

-i'

-j'

-k'

-1

i⁽⁷⁾

i⁽⁷⁾

e⁽⁷⁾

-k⁽⁷⁾

j⁽⁷⁾

-i⁽⁶⁾

-e⁽⁶⁾

-k⁽⁶⁾

j⁽⁶⁾

i⁽⁵⁾

e⁽⁵⁾

-k⁽⁵⁾

j⁽⁵⁾

-i⁽⁴⁾

e⁽⁴⁾

-k⁽⁴⁾

j⁽⁴⁾

-i'''

e'''

-k'''

j'''

-i''

-e''

-k''

j''

-k'

-i

-1

-k

j⁽⁷⁾

j⁽⁷⁾

k⁽⁷⁾

e⁽⁷⁾

-i⁽⁷⁾

-j⁽⁶⁾

k⁽⁶⁾

-e⁽⁶⁾

-i⁽⁶⁾

j⁽⁵⁾

k⁽⁵⁾

e⁽⁵⁾

-i⁽⁵⁾

-j⁽⁴⁾

k⁽⁴⁾

e⁽⁴⁾

-i⁽⁴⁾

-j'''

k'''

e'''

-i'''

-j''

k''

-e''

-i''

-i'

-j

-1

-i

k⁽⁷⁾

k⁽⁷⁾

-j⁽⁷⁾

i⁽⁷⁾

e⁽⁷⁾

-k⁽⁶⁾

-j⁽⁶⁾

i⁽⁶⁾

-e⁽⁶⁾

k⁽⁵⁾

-j⁽⁵⁾

i⁽⁵⁾

e⁽⁵⁾

-k⁽⁴⁾

-j⁽⁴⁾

i⁽⁴⁾

e⁽⁴⁾

-k'''

-j'''

i'''

e'''

-k''

-j''

i''

-e''

-j'

-k

-j

-1

In general, st ≠ ts, s(tu) ≠ (st)u, and s(st) ≠ s²t.

Returns:: st

multiply

public final Trigintaduonion multiply(double x)

Returns the product of this trigintaduonion s and the real number x.

sx = xa₀ + ixa₁ + jxa₂ +...+ k'''xa₃₁

Returns:: sx

divide

public final Trigintaduonion divide(Trigintaduonion t)

Returns the quotient of this trigintaduonion s over the trigintaduonion t = b₀ + ib₁ + jb₂ +...+ k⁽⁷⁾b₃₁.

s/t = sconjugate(t)/|t|². In general, s/t ≠ (1/p)q.

Returns:: s/t

pow

public final Trigintaduonion pow(Trigintaduonion t)

Returns this trigintaduonion s raised to the trigintaduonion t.

Returns:: s^t

pow

public final Trigintaduonion pow(double x)

Returns this trigintaduonion s raised to the real number x.

Returns:: s^x

pow

public final Trigintaduonion pow(int n)

Returns this trigintaduonion s raised to the integer n.

Returns:: sⁿ

sqr

public final Trigintaduonion sqr()

Returns the square of this trigintaduonion s.

Returns:: s²

horner

public final Trigintaduonion horner(double[] v)

Evaluate the polynomial with real coefficients, given by the array of doubles v, at this trigintaduonion s.

If n = v.length - 1 then this method evaluates v₀ + v₁s + ... + v_nsⁿ using Horner's rule.

Returns:: v₀ + v₁s + ... + v_nsⁿ.

sqrt

public final Trigintaduonion sqrt()

Returns the square root of this trigintaduonion s.

The branch cut is on (-∞, 0).

Returns:: sqrt(s)

sqrt

public final Trigintaduonion sqrt(Trigintaduonion u)

Returns the square root of this trigintaduonion s.

The branch cut is on (-∞, 0).

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: sqrt(s)

exp

public final Trigintaduonion exp()

Returns Euler's number (e) raised to the power of this trigintaduonion s.

Returns:: e^s
See Also:: log()

log

public final Trigintaduonion log()

Returns the natural logarithm of this trigintaduonion s.

The branch cut is on (-∞, 0].

Returns:: log(s)
See Also:: exp()

log

public final Trigintaduonion log(Trigintaduonion u)

Returns the natural logarithm of this trigintaduonion s.

The branch cut is on (-∞, 0].

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: log(s)
See Also:: exp()

log10

public final Trigintaduonion log10()

Returns the logarithm base 10 of this trigintaduonion s.

The branch cut is on (-∞, 0].

Returns:: log₁₀(s)

log10

public final Trigintaduonion log10(Trigintaduonion u)

Returns the natural logarithm of this trigintaduonion s.

The branch cut is on (-∞, 0].

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: log₁₀(s)

sin

public final Trigintaduonion sin()

Returns the sine of this trigintaduonion s.

Returns:: sin(s)
See Also:: asin()

cos

public final Trigintaduonion cos()

Returns the cosine of this trigintaduonion s.

Returns:: cos(s)
See Also:: acos()

tan

public final Trigintaduonion tan()

Returns the tangent of this trigintaduonion s.

Returns:: tan(s)
See Also:: atan()

sec

public final Trigintaduonion sec()

Returns the secant of this trigintaduonion s.

Returns:: sec(s)
See Also:: asec()

csc

public final Trigintaduonion csc()

Returns the cosecant of this trigintaduonion s.

Returns:: csc(s)
See Also:: acsc()

cot

public final Trigintaduonion cot()

Returns the cotangent of this trigintaduonion s.

Returns:: cot(s)
See Also:: acot()

sinh

public final Trigintaduonion sinh()

Returns the hyperbolic sine of this trigintaduonion s.

Returns:: sinh(s)
See Also:: asinh()

cosh

public final Trigintaduonion cosh()

Returns the hyperbolic cosine of this trigintaduonion s.

Returns:: cosh(s)
See Also:: acosh()

tanh

public final Trigintaduonion tanh()

Returns the hyperbolic tangent of this trigintaduonion s.

Returns:: tanh(s)
See Also:: atanh()

sech

public final Trigintaduonion sech()

Returns the hyperbolic secant of this trigintaduonion s.

Returns:: sech(s)
See Also:: asech()

csch

public final Trigintaduonion csch()

Returns the hyperbolic cosecant of this trigintaduonion s.

Returns:: csch(s)
See Also:: acsch()

coth

public final Trigintaduonion coth()

Returns the hyperbolic cotangent of this trigintaduonion s.

Returns:: coth(s)
See Also:: acoth()

asin

public final Trigintaduonion asin()

Returns the principal inverse sine of this trigintaduonion s.

The branch cuts are (-∞, -1) and (1, ∞).

Returns:: asin(s)
See Also:: sin()

asin

public final Trigintaduonion asin(Trigintaduonion u)

Returns the principal inverse sine of this trigintaduonion s.

The branch cuts are (-∞, -1) and (1, ∞).

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: asin(s)
See Also:: sin()

acos

public final Trigintaduonion acos()

Returns the principal inverse cosine of this trigintaduonion s.

The branch cuts are (-∞, -1) and (1, ∞).

Returns:: acos(s)
See Also:: cos()

acos

public final Trigintaduonion acos(Trigintaduonion u)

Returns the principal inverse cosine of this trigintaduonion s.

The branch cuts are (-∞, -1) and (1, ∞).

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: acos(s)
See Also:: cos()

atan

public final Trigintaduonion atan()

Returns the principal inverse tangent of this trigintaduonion s.

The branch cuts are (-u∞, -u] and [u, u∞).

Returns:: atan(s)
See Also:: tan()

atan

public final Trigintaduonion atan(Trigintaduonion u)

Returns the principal inverse tangent of this trigintaduonion s.

The branch cuts are (-u∞, -u] and [u, u∞).

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: atan(s)
See Also:: tan()

asec

public final Trigintaduonion asec()

Returns the principal inverse secant of this trigintaduonion s.

The branch cut is (-1, 1).

Returns:: asec(s)
See Also:: sec()

asec

public final Trigintaduonion asec(Trigintaduonion u)

Returns the principal inverse secant of this trigintaduonion s.

The branch cut is (-1, 1).

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: asec(s)
See Also:: sec()

acsc

public final Trigintaduonion acsc()

Returns the principal inverse cosecant of this trigintaduonion s.

The branch cut is (-1, 1).

Returns:: acsc(s)
See Also:: csc()

acsc

public final Trigintaduonion acsc(Trigintaduonion u)

Returns the principal inverse cosecant of this trigintaduonion s.

The branch cut is (-1, 1).

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: acsc(s)
See Also:: csc()

acot

public final Trigintaduonion acot()

Returns the principal inverse cotangent of this trigintaduonion s.

The branch cut is (-u, u).

Returns:: acot(s)
See Also:: cot()

acot

public final Trigintaduonion acot(Trigintaduonion u)

Returns the principal inverse cotangent of this trigintaduonion s.

The branch cut is (-u, u).

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: acot(s)
See Also:: cot()

asinh

public final Trigintaduonion asinh()

Returns the principal inverse hyperbolic sine of this trigintaduonion s.

The branch cuts are (-u∞, -u) and (u, u∞).

Returns:: asinh(s)
See Also:: sinh()

asinh

public final Trigintaduonion asinh(Trigintaduonion u)

Returns the principal inverse hyperbolic sine of this trigintaduonion s.

The branch cuts are (-u∞, -u) and (u, u∞).

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: asinh(s)
See Also:: sinh()

acosh

public final Trigintaduonion acosh()

Returns the principal inverse hyperbolic cosine of this trigintaduonion s.

The branch cut is (-∞, 1).

Returns:: acosh(s)
See Also:: cosh()

acosh

public final Trigintaduonion acosh(Trigintaduonion u)

Returns the principal inverse hyperbolic cosine of this trigintaduonion s.

The branch cut is (-∞, 1).

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: acosh(s)
See Also:: cosh()

atanh

public final Trigintaduonion atanh()

Returns the principal inverse hyperbolic tangent of this trigintaduonion s.

The branch cuts are (-∞, -1] and [1, ∞).

Returns:: atanh(s)
See Also:: tanh()

atanh

public final Trigintaduonion atanh(Trigintaduonion u)

Returns the principal inverse hyperbolic tangent of this trigintaduonion s.

The branch cuts are (-∞, -1] and [1, ∞).

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: atanh(s)
See Also:: tanh()

asech

public final Trigintaduonion asech()

Returns the principal inverse hyperbolic secant of this trigintaduonion s.

The branch cuts are (-∞, 0] and (1, ∞).

Returns:: asech(s)
See Also:: sech()

asech

public final Trigintaduonion asech(Trigintaduonion u)

Returns the principal inverse hyperbolic secant of this trigintaduonion s.

The branch cuts are (-∞, 0] and (1, ∞).

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: asech(s)
See Also:: sech()

acsch

public final Trigintaduonion acsch()

Returns the principal inverse hyperbolic cosecant of this trigintaduonion s.

The branch cut is (-i, i).

Returns:: acsch(s)
See Also:: csch()

acsch

public final Trigintaduonion acsch(Trigintaduonion u)

Returns the principal inverse hyperbolic cosecant of this trigintaduonion s.

The branch cut is (-u, u).

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: acsch(s)
See Also:: csch()

acoth

public final Trigintaduonion acoth()

Returns the principal inverse hyperbolic cotangent of this trigintaduonion s.

The branch cut is [-1, 1].

Returns:: acoth(s)
See Also:: coth()

acoth

public final Trigintaduonion acoth(Trigintaduonion u)

Returns the principal inverse hyperbolic cotangent of this trigintaduonion s.

The branch cut is [-1, 1].

On the branch cuts, the imaginary part is a multiple of the imaginary trigintaduonion u.

Parameters:: u - a non-real trigintaduonion
Returns:: acoth(s)
See Also:: coth()

ceil

public final Trigintaduonion ceil()

Returns the ceiling of each of the coefficients of this trigintaduonion s.

Returns:: ceil(a₀) + i ceil(a₁)+ j ceil(a₂)+...+ k⁽⁷⁾ ceil(a₃₁)
See Also:: Math.ceil(double)

floor

public final Trigintaduonion floor()

Returns the floor of each of the coefficients of this trigintaduonion s.

Returns:: floor(a₀) + i floor(a₁)+ j floor(a₂)+...+ k⁽⁷⁾ floor(a₃₁)
See Also:: Math.floor(double)

rint

public final Trigintaduonion rint()

Returns each coefficient of the trigintaduonion s rounded to the nearest double equal to an integer.

Returns:: rint(a₀) + i rint(a₁)+ j rint(a₂)+...+ k''' rint(a₃₁)
See Also:: Math.rint(double)

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ca.uwaterloo.alumni.dwharder.Numbers Class Trigintaduonion

ZERO

ONE

I

J

K

E1

I1

J1

K1

E2

I2

J2

K2

E3

I3

J3

K3

E4

I4

J4

K4

E5

I5

J5

K5

E6

I6

J6

K6

E7

I7

J7

K7

NaN

Trigintaduonion

Trigintaduonion

Trigintaduonion

Trigintaduonion

Trigintaduonion

random

random

random

randomReal

randomImaginary

equals

equals

hashCode

toString

imaginaryUnits

toArray

toArray

toImagArray

isZero

isReal

isImaginary

isImaginary

isNaN

isInfinite

abs

abs2

absImag

abs2Imag

real

imag

coefficient

project

project

conjugate

argument

signum

add

add

add

subtract

negate

multiply

multiply

divide

pow

ca.uwaterloo.alumni.dwharder.Numbers
Class Trigintaduonion