Online calculator: Tips and tricks #9: Big numbers

Tips and tricks series

This article may rely on knowledge you should get from previous articles, so you may want to check them first.

Browser JavaScript implementation has a limit of integer number exact representation: 9007199254740991 or 2⁵³-1 or roughly 9 quadrillion. To overcome this limit big number libraries can be applied.
To work with big numbers PLANETCALC has a special input set: Big number and Big number array. The big number library underlying these inputs supports operations in an integer field, complex integer fields, and real, rational, or complex fractional ring. General Number output can be used to represent this library calculation results. The library is globally available in a calculator source code by PCR. prefix.
This library is internally based on big-integer library, which has a good performance, but PCR library was designed in idea to have a usable interface, the same overall supported fields $\mathbb{Z},\mathbb{R},\mathbb{Q},\mathbb{C}$ rather than performance. So consider using the bigInt library directly, if your calculator requires the best performance. The Number output supports bigInt types as well.

The calculator below uses Big number input, Number output, and PCR library to calculate GCD using extended Euclidean algorithm:

Extended Euclidean algorithm

First integer

Second integer

Greatest Common Divisor

Coefficient for bigger integer

Coefficient for smaller integer

PCR big number library overview by the Euclidean algorithm code example

The calculator has two big integer inputs, defined in the calculator editor as shown in the picture below:

So first and second input variables are defined as PCR.integer by input editor. You may also define PCR numbers variables in code, see the calculator code fragment, which defines a placeholder for the result with three PCR integers:

var euklid = {
    gcd: PCR.integer(1),
    x: PCR.integer(0),
    y: PCR.integer(0)
};

PCR integers has a number of arithmetic operations. The code below demonstrates some of them (integer division: divmod, subtraction: sub and multiplication: mul).

function gcd (a, b, holder) {
    if (a.eq(0) ) {
        holder.x = PCR.integer(0); 
        holder.y = PCR.integer(1);
        return b;
    }
    var q = b.divmod( a );
    var d = gcd (q.r, a, holder);
    var tx = holder.x;
    holder.x = holder.y.sub( 
        q.q.mul( holder.x ) );
    holder.y = tx;
    return d;
}

Besides arithmetics, PCR library has a number of compare operations defined over all PCR number types. The code fragment below demonstrates greater than (gt) operation:

euklid.gcd =gcd(first.gt( second ) ? 
second : first, first.gt( second ) 
? first : second, euklid);

To report the result to users just SetValue to a general Number output:

res.SetValue(euklid.gcd);
coef1.SetValue(euklid.y);
coef2.SetValue(euklid.x);

The Number output can be configured to display either real or rational numbers. The rational numbers mode output having PCR big number value can be used for the precise number representation as a simple fraction. See the Result set switch on the following picture:

PCR big number library reference

Initialization

Function	Example	Description
PCR.integer	var one = PCR.integer(1);	Big integer number definition
PCR.rational	var quarter = PCR.rational(1,4);	Rational number definition
PCR.complex	var complex = PCR.complex(7,3); //7+3i	Complex number definition
PCR.real	var real = PCR.real(3.1415926);	Real number definition
PCR.parse	var complexFraction = PCR.parse("1/3+i");	Parses a string into PCR number.

Operations

Function	Example	Description
add	var two = PCR.integer(1).add(1);	Adds two PCR numbers.
sub	var twoThird = PCR.integer(1).sub(PCR.rational(1,3));	Subtract two PCR numbers
neg	var minusOne = PCR.integer(1).neg();	Negates a PCR number
mul	var complex = PCR.complex(7,3).mul(2); //14+6i	Multiply two PCR numbers
div	var quarter = PCR.rational(1,2).div(2);	Divide two PCR numbers
divmod	var twoAndOne = PCR.integer(5).div(2); // returns {q:2,r:1}	Integer division with modulo
mod	var one = PCR.integer(5).mod(2);	Integer division modulo
idiv	var two = PCR.integer(5).idiv(2);	Integer division
abs	var one = PCR.integer(-1).abs();	Absolute value
gcd	var two = PCR.integer(6).gcd(8);	Greatest common divisor

Conversions

Function	Example	Description
toString	var fraction = PCR.rational(1/3).toString(); // returns "1/3"	Converts PCR number into string.
toFixed	var fraction = PCR.rational(1/3).toFixed(2); // returns "0.33"	Floating point string representation with fixed decimal digits.
toTeX	var fraction = PCR.rational(1/3).toTeX(); // returns "\frac{1}{3}"	LaTeX representation.

Comparsions

Function	Example	Description
isZero	var zero = PCR.rational(1/2).sub(0.5).isZero();	Compare a PCR number with zero.
compare	var greater = PCR.rational(1/2).compare(0.3);	Compare two values, returns 1 if the left operand is greater, returns -1 - if the left operand is less, and 0 - if both operands are equal .
eq	var equal = PCR.rational(1/2).eq(0.5);	Equality.
gt	var greater = PCR.rational(1/2).gt(0.3);	Greater than.
gte	var greater = PCR.rational(1/2).gte(0.3);	Greater than or equal.
lt	var greater = PCR.rational(1/2).lt(0.3);	Less than.
lte	var greater = PCR.rational(1/2).lte(0.3);	Less than or equal.

PLANETCALC Online calculators

Tips and tricks #9: Big numbers

This page exists due to the efforts of the following people:

Anton

Timur

Tips and tricks series

Extended Euclidean algorithm

PCR big number library overview by the Euclidean algorithm code example

PCR big number library reference

Initialization

Operations

Conversions

Comparsions

Similar calculators

Comments

PLANETCALC Online calculators

Tips and tricks series

Extended Euclidean algorithm

PCR big number library overview by the Euclidean algorithm code example

PCR big number library reference

Initialization

Operations

Conversions

Comparsions

Similar calculators

Comments

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