1313 lines
40 KiB
Dart
1313 lines
40 KiB
Dart
import 'dart:math';
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import 'package:rational/rational.dart';
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import 'models/calculation_step.dart';
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import 'calculator.dart';
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import 'parser.dart';
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/// 帮助解析一元一次方程 ax+b=cx+d 的辅助类
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class LinearEquationParts {
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final double a, b, c, d;
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LinearEquationParts(this.a, this.b, this.c, this.d);
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}
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class SolverService {
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/// 主入口方法，识别并分发任务
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CalculationResult solve(String input) {
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// 预处理输入字符串
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final cleanInput = input.replaceAll(' ', '').toLowerCase();
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// 对包含x的方程进行预处理，展开表达式
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String processedInput = cleanInput;
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if (processedInput.contains('x') && processedInput.contains('(')) {
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processedInput = _expandExpressions(processedInput);
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}
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// 1. 检查是否为二元一次方程组 (格式: ...;...)
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if (processedInput.contains(';') &&
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processedInput.contains('x') &&
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processedInput.contains('y')) {
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return _solveSystemOfLinearEquations(processedInput);
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}
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// 2. 检查是否为一元二次方程 (包含 x^2 或 x²)
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if (processedInput.contains('x^2') || processedInput.contains('x²')) {
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return _solveQuadraticEquation(processedInput.replaceAll('x²', 'x^2'));
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}
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// 3. 检查是否为一元一次方程 (包含 x 但不包含 y 或 x^2)
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if (processedInput.contains('x') && !processedInput.contains('y')) {
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return _solveLinearEquation(processedInput);
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}
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// 4. 如果都不是，则作为简单表达式计算
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try {
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return _solveSimpleExpression(input); // 使用原始输入以保留运算符
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} catch (e) {
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throw Exception('无法识别的格式。请检查您的方程或表达式。');
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}
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}
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/// ---- 求解器实现 ----
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/// 1. 求解简单表达式
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CalculationResult _solveSimpleExpression(String input) {
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final steps = <CalculationStep>[];
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// Parse the input to get LaTeX-formatted version
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final parser = Parser(input);
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final parsedExpr = parser.parse();
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final latexInput = parsedExpr.toString().replaceAll('*', '\\cdot');
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steps.add(
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CalculationStep(
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stepNumber: 1,
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title: '表达式求值',
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explanation: '这是一个标准的数学表达式，我们将直接计算其结果。',
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formula: '\$\$$latexInput\$\$',
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),
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);
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// 检查是否为特殊三角函数值，可以返回精确结果
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final exactTrigResult = getExactTrigResult(input);
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if (exactTrigResult != null) {
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return CalculationResult(
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steps: steps,
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finalAnswer: '\$\$$exactTrigResult\$\$',
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);
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}
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// 预处理输入，将三角函数的参数从度转换为弧度
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String processedInput = convertTrigToRadians(input);
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try {
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// 使用自定义解析器解析表达式
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final parser = Parser(processedInput);
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final expr = parser.parse();
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// 对表达式进行求值
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final evaluatedExpr = expr.evaluate();
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// 获取数值结果 - 需要正确进行类型转换
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double result;
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if (evaluatedExpr is IntExpr) {
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result = evaluatedExpr.value.toDouble();
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} else if (evaluatedExpr is DoubleExpr) {
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result = evaluatedExpr.value;
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} else if (evaluatedExpr is FractionExpr) {
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result = evaluatedExpr.numerator / evaluatedExpr.denominator;
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} else {
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// 如果无法完全求值为数值，尝试简化并转换为字符串
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final simplified = evaluatedExpr.simplify();
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return CalculationResult(
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steps: steps,
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finalAnswer: '\$\$${simplified.toString()}\$\$',
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);
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}
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// 尝试将结果格式化为几倍根号的形式
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final formattedResult = formatSqrtResult(result);
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return CalculationResult(
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steps: steps,
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finalAnswer: '\$\$$formattedResult\$\$',
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);
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} catch (e) {
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throw Exception('无法解析表达式: $input');
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}
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}
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/// 2. 求解一元一次方程
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CalculationResult _solveLinearEquation(String input) {
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final steps = <CalculationStep>[];
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// Parse the input to get LaTeX-formatted version
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final parser = Parser(input);
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final parsedExpr = parser.parse();
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final latexInput = parsedExpr.toString().replaceAll('*', '\\cdot');
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steps.add(
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CalculationStep(
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stepNumber: 0,
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title: '原方程',
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explanation: '这是一元一次方程。',
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formula: '\$\$$latexInput\$\$',
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),
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);
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final parts = _parseLinearEquation(input);
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final a = parts.a, b = parts.b, c = parts.c, d = parts.d;
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final newA = _rationalFromDouble(a) - _rationalFromDouble(c);
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final newD = _rationalFromDouble(d) - _rationalFromDouble(b);
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steps.add(
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CalculationStep(
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stepNumber: 1,
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title: '移项',
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explanation: '将所有含 x 的项移到等式左边，常数项移到右边。',
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formula:
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'\$\$${a}x ${c >= 0 ? '-' : '+'} ${c.abs()}x = $d ${b >= 0 ? '-' : '+'} ${b.abs()}\$\$',
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),
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);
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steps.add(
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CalculationStep(
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stepNumber: 2,
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title: '合并同类项',
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explanation: '合并等式两边的项。',
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formula:
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'\$\$${newA.toDouble().toStringAsFixed(4)}x = ${newD.toDouble().toStringAsFixed(4)}\$\$',
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),
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);
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if (newA == Rational.zero) {
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return CalculationResult(
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steps: steps,
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finalAnswer: newD == Rational.zero ? '有无穷多解' : '无解',
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);
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}
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final x = newD / newA;
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steps.add(
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CalculationStep(
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stepNumber: 3,
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title: '求解 x',
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explanation: '两边同时除以 x 的系数 ($newA)。',
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formula: '\$\$x = \\frac{$newD}{$newA}\$\$',
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),
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);
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return CalculationResult(steps: steps, finalAnswer: '\$\$x = $x\$\$');
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}
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/// 3. 求解一元二次方程 (升级版)
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CalculationResult _solveQuadraticEquation(String input) {
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final steps = <CalculationStep>[];
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final eqParts = input.split('=');
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if (eqParts.length != 2) throw Exception("方程格式错误，应包含一个 '='。");
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// Keep original equation for display
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final originalEquation = _formatOriginalEquation(input);
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// Parse coefficients symbolically
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final leftCoeffsSymbolic = _parsePolynomialSymbolic(eqParts[0]);
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final rightCoeffsSymbolic = _parsePolynomialSymbolic(eqParts[1]);
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final aSymbolic = _subtractCoefficients(
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leftCoeffsSymbolic[2] ?? '0',
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rightCoeffsSymbolic[2] ?? '0',
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);
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final bSymbolic = _subtractCoefficients(
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leftCoeffsSymbolic[1] ?? '0',
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rightCoeffsSymbolic[1] ?? '0',
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);
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final cSymbolic = _subtractCoefficients(
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leftCoeffsSymbolic[0] ?? '0',
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rightCoeffsSymbolic[0] ?? '0',
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);
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// Also get numeric values for calculations
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final leftCoeffs = _parsePolynomial(eqParts[0]);
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final rightCoeffs = _parsePolynomial(eqParts[1]);
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final a = (leftCoeffs[2] ?? 0) - (rightCoeffs[2] ?? 0);
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final b = (leftCoeffs[1] ?? 0) - (rightCoeffs[1] ?? 0);
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final c = (leftCoeffs[0] ?? 0) - (rightCoeffs[0] ?? 0);
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if (a == 0) {
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return _solveLinearEquation('${b}x+$c=0');
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}
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steps.add(
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CalculationStep(
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stepNumber: 1,
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title: '整理方程',
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explanation: r'将方程整理成标准形式 $ax^2+bx+c=0$。',
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formula: originalEquation,
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),
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);
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if (a == a.round() && b == b.round() && c == c.round()) {
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final factors = _tryFactorization(a.toInt(), b.toInt(), c.toInt());
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if (factors != null) {
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steps.add(
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CalculationStep(
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stepNumber: 2,
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title: '因式分解法 (十字相乘)',
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explanation: '我们发现可以将方程分解为两个一次因式的乘积。',
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formula: factors.formula,
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),
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);
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steps.add(
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CalculationStep(
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stepNumber: 3,
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title: '求解',
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explanation: '分别令每个因式等于 0，解出 x。',
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formula: factors.solution,
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),
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);
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steps.add(
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CalculationStep(
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stepNumber: 4,
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title: '化简结果',
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explanation: '将分数化简到最简形式，并将负号写在分数外面。',
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formula: factors.solution,
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),
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);
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return CalculationResult(steps: steps, finalAnswer: factors.solution);
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}
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}
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steps.add(
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CalculationStep(
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stepNumber: 2,
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title: '选择解法',
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explanation: '无法进行因式分解，我们选择使用求根公式法。',
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formula: '\$\$\\Delta = b^2 - 4ac\$\$',
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),
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);
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// Calculate delta symbolically
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final deltaSymbolic = _calculateDeltaSymbolic(
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aSymbolic,
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bSymbolic,
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cSymbolic,
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);
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final delta =
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_rationalFromDouble(b).pow(2) -
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Rational.fromInt(4) * _rationalFromDouble(a) * _rationalFromDouble(c);
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steps.add(
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CalculationStep(
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stepNumber: 3,
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title: '计算判别式 (Delta)',
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explanation: '\$\$\\Delta = b^2 - 4ac = $deltaSymbolic\$\$',
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formula:
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'\$\$\\Delta = $deltaSymbolic = ${delta.toDouble().toStringAsFixed(4)}\$\$',
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),
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);
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final deltaDouble = delta.toDouble();
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if (deltaDouble > 0) {
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// Pass delta directly to maintain precision
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final x1Expr = _formatQuadraticRoot(-b, delta, 2 * a, true);
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final x2Expr = _formatQuadraticRoot(-b, delta, 2 * a, false);
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steps.add(
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CalculationStep(
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stepNumber: 4,
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title: '应用求根公式',
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explanation:
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r'因为 $\Delta > 0$，方程有两个不相等的实数根。公式: $x = \frac{-b \pm \sqrt{\Delta}}{2a}$。',
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formula: '\$\$x_1 = $x1Expr, \\quad x_2 = $x2Expr\$\$',
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),
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);
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return CalculationResult(
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steps: steps,
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finalAnswer: '\$\$x_1 = $x1Expr, \\quad x_2 = $x2Expr\$\$',
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);
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} else if (deltaDouble == 0) {
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final x = -b / (2 * a);
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steps.add(
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CalculationStep(
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stepNumber: 4,
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title: '应用求根公式',
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explanation: r'因为 $\Delta = 0$，方程有两个相等的实数根。',
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formula: '\$\$x_1 = x_2 = ${x.toStringAsFixed(4)}\$\$',
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),
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);
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return CalculationResult(
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steps: steps,
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finalAnswer: '\$\$x_1 = x_2 = ${x.toStringAsFixed(4)}\$\$',
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);
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} else {
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steps.add(
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CalculationStep(
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stepNumber: 4,
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title: '判断解',
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explanation: r'因为 $\Delta < 0$，该方程在实数范围内无解，但有虚数解。',
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formula: '无实数解，有虚数解',
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),
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);
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// For complex roots, we need to handle -delta
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final negDelta = -delta;
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final sqrtNegDeltaStr = _formatSqrtFromRational(negDelta);
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final realPart = -b / (2 * a);
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final imagPartExpr = _formatImaginaryPart(sqrtNegDeltaStr, 2 * a);
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steps.add(
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CalculationStep(
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stepNumber: 5,
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title: '计算虚数根',
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explanation: '使用求根公式计算虚数根。',
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formula: r'$$x = \frac{-b \pm \sqrt{-\Delta} i}{2a}$$',
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),
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);
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return CalculationResult(
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steps: steps,
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finalAnswer:
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'\$\$x_1 = ${realPart.toStringAsFixed(4)} + $imagPartExpr, \\quad x_2 = ${realPart.toStringAsFixed(4)} - $imagPartExpr\$\$',
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);
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}
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}
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/// 4. 求解二元一次方程组
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CalculationResult _solveSystemOfLinearEquations(String input) {
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final steps = <CalculationStep>[];
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final equations = input.split(';');
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if (equations.length != 2) throw Exception("格式错误, 请用 ';' 分隔两个方程。");
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final p1 = _parseTwoVariableLinear(equations[0]);
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final p2 = _parseTwoVariableLinear(equations[1]);
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double a1 = p1[0], b1 = p1[1], c1 = p1[2];
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double a2 = p2[0], b2 = p2[1], c2 = p2[2];
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steps.add(
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CalculationStep(
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stepNumber: 0,
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title: '原始方程组',
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explanation: '这是一个二元一次方程组，我们将使用加减消元法求解。',
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formula:
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'''
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\$\$
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\\begin{cases}
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${a1}x ${b1 >= 0 ? '+' : ''} ${b1}y = $c1 & (1) \\\\
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${a2}x ${b2 >= 0 ? '+' : ''} ${b2}y = $c2 & (2)
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\\end{cases}
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\$\$
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''',
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),
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);
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final det =
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_rationalFromDouble(a1) * _rationalFromDouble(b2) -
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_rationalFromDouble(a2) * _rationalFromDouble(b1);
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if (det == Rational.zero) {
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final infiniteCheck =
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_rationalFromDouble(a1) * _rationalFromDouble(c2) -
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_rationalFromDouble(a2) * _rationalFromDouble(c1);
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return CalculationResult(
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steps: steps,
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finalAnswer: infiniteCheck == Rational.zero ? '有无穷多解' : '无解',
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);
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}
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final newA1 = _rationalFromDouble(a1) * _rationalFromDouble(b2);
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final newC1 = _rationalFromDouble(c1) * _rationalFromDouble(b2);
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final newA2 = _rationalFromDouble(a2) * _rationalFromDouble(b1);
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final newC2 = _rationalFromDouble(c2) * _rationalFromDouble(b1);
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steps.add(
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CalculationStep(
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stepNumber: 1,
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title: '消元',
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explanation: '为了消去变量 y，将方程(1)两边乘以 $b2，方程(2)两边乘以 $b1。',
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formula:
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'''
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\$\$
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\\begin{cases}
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${newA1.toDouble().toStringAsFixed(2)}x ${b1 * b2 >= 0 ? '+' : ''} ${(b1 * b2).toStringAsFixed(2)}y = ${newC1.toDouble().toStringAsFixed(2)} & (3) \\\\
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${newA2.toDouble().toStringAsFixed(2)}x ${b1 * b2 >= 0 ? '+' : ''} ${(b1 * b2).toStringAsFixed(2)}y = ${newC2.toDouble().toStringAsFixed(2)} & (4)
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\\end{cases}
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\$\$
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''',
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),
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);
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final xCoeff = newA1 - newA2;
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final constCoeff = newC1 - newC2;
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steps.add(
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CalculationStep(
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stepNumber: 2,
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title: '相减',
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explanation: '将方程(3)减去方程(4)，得到一个只含 x 的方程。',
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formula:
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'\$\$(${newA1.toDouble().toStringAsFixed(2)} - ${newA2.toDouble().toStringAsFixed(2)})x = ${newC1.toDouble().toStringAsFixed(2)} - ${newC2.toDouble().toStringAsFixed(2)} \\Rightarrow ${xCoeff.toDouble().toStringAsFixed(2)}x = ${constCoeff.toDouble().toStringAsFixed(2)}\$\$',
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),
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);
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final x = constCoeff / xCoeff;
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steps.add(
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CalculationStep(
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stepNumber: 3,
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title: '解出 x',
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explanation: '求解上述方程得到 x 的值。',
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formula: '\$\$x = $x\$\$',
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),
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);
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if (b1.abs() < 1e-9) {
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final yCoeff = b2;
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final yConst = c2 - a2 * x.toDouble();
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final y = yConst / yCoeff;
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steps.add(
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CalculationStep(
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stepNumber: 4,
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title: '回代求解 y',
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||
explanation: '将 x = ${x.toDouble().toStringAsFixed(4)} 代入原方程(2)中。',
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formula:
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'''
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\$\$
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\\begin{aligned}
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$a2(${x.toDouble().toStringAsFixed(4)}) + ${b2}y &= $c2 \\\\
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${a2 * x.toDouble()} + ${b2}y &= $c2 \\\\
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${b2}y &= $c2 - ${a2 * x.toDouble()} \\\\
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${b2}y &= ${c2 - a2 * x.toDouble()}
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\\end{aligned}
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\$\$
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''',
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),
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);
|
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steps.add(
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||
CalculationStep(
|
||
stepNumber: 5,
|
||
title: '解出 y',
|
||
explanation: '求解得到 y 的值。',
|
||
formula: '\$\$y = ${y.toStringAsFixed(4)}\$\$',
|
||
),
|
||
);
|
||
return CalculationResult(
|
||
steps: steps,
|
||
finalAnswer:
|
||
'\$\$x = ${x.toDouble().toStringAsFixed(4)}, \\quad y = ${y.toStringAsFixed(4)}\$\$',
|
||
);
|
||
} else {
|
||
final yCoeff = b1;
|
||
final yConst = c1 - a1 * x.toDouble();
|
||
final y = yConst / yCoeff;
|
||
steps.add(
|
||
CalculationStep(
|
||
stepNumber: 4,
|
||
title: '回代求解 y',
|
||
explanation: '将 x = ${x.toDouble().toStringAsFixed(4)} 代入原方程(1)中。',
|
||
formula:
|
||
'''
|
||
\$\$
|
||
\\begin{aligned}
|
||
$a1(${x.toDouble().toStringAsFixed(4)}) + ${b1}y &= $c1 \\\\
|
||
${a1 * x.toDouble()} + ${b1}y &= $c1 \\\\
|
||
${b1}y &= $c1 - ${a1 * x.toDouble()} \\\\
|
||
${b1}y &= ${c1 - a1 * x.toDouble()}
|
||
\\end{aligned}
|
||
\$\$
|
||
''',
|
||
),
|
||
);
|
||
steps.add(
|
||
CalculationStep(
|
||
stepNumber: 5,
|
||
title: '解出 y',
|
||
explanation: '求解得到 y 的值。',
|
||
formula: '\$\$y = ${y.toStringAsFixed(4)}\$\$',
|
||
),
|
||
);
|
||
return CalculationResult(
|
||
steps: steps,
|
||
finalAnswer:
|
||
'\$\$x = ${x.toDouble().toStringAsFixed(4)}, \\quad y = ${y.toStringAsFixed(4)}\$\$',
|
||
);
|
||
}
|
||
}
|
||
|
||
/// ---- 辅助函数 ----
|
||
|
||
String _expandExpressions(String input) {
|
||
String result = input;
|
||
int maxIterations = 10; // Prevent infinite loops
|
||
int iterationCount = 0;
|
||
|
||
while (iterationCount < maxIterations) {
|
||
String oldResult = result;
|
||
|
||
final powerMatch = RegExp(
|
||
r'(-?\d*\.?\d*)?\(([^)]+)\)\^2',
|
||
).firstMatch(result);
|
||
if (powerMatch != null) {
|
||
final kStr = powerMatch.group(1);
|
||
double k = 1.0;
|
||
if (kStr != null && kStr.isNotEmpty) {
|
||
k = kStr == '-' ? -1.0 : double.parse(kStr);
|
||
}
|
||
|
||
final factor = powerMatch.group(2)!;
|
||
final coeffs = _parsePolynomial(factor);
|
||
final a = coeffs[1] ?? 0;
|
||
final b = coeffs[0] ?? 0;
|
||
|
||
final newA = k * a * a;
|
||
final newB = k * 2 * a * b;
|
||
final newC = k * b * b;
|
||
|
||
final expanded =
|
||
'${newA}x^2${newB >= 0 ? '+' : ''}${newB}x${newC >= 0 ? '+' : ''}$newC';
|
||
result = result.replaceFirst(powerMatch.group(0)!, '($expanded)');
|
||
iterationCount++;
|
||
continue;
|
||
}
|
||
|
||
final factorMulMatch = RegExp(
|
||
r'\(([^)]+)\)\(([^)]+)\)',
|
||
).firstMatch(result);
|
||
if (factorMulMatch != null) {
|
||
final factor1 = factorMulMatch.group(1)!;
|
||
final factor2 = factorMulMatch.group(2)!;
|
||
print('Expanding: ($factor1) * ($factor2)');
|
||
|
||
final coeffs1 = _parsePolynomial(factor1);
|
||
final coeffs2 = _parsePolynomial(factor2);
|
||
print('Coeffs1: $coeffs1, Coeffs2: $coeffs2');
|
||
|
||
final a = coeffs1[1] ?? 0;
|
||
final b = coeffs1[0] ?? 0;
|
||
final c = coeffs2[1] ?? 0;
|
||
final d = coeffs2[0] ?? 0;
|
||
print('a=$a, b=$b, c=$c, d=$d');
|
||
|
||
final newA = a * c;
|
||
final newB = a * d + b * c;
|
||
final newC = b * d;
|
||
print('newA=$newA, newB=$newB, newC=$newC');
|
||
|
||
final expanded =
|
||
'${newA}x^2${newB >= 0 ? '+' : ''}${newB}x${newC >= 0 ? '+' : ''}$newC';
|
||
print('Expanded result: $expanded');
|
||
|
||
result = result.replaceFirst(factorMulMatch.group(0)!, expanded);
|
||
iterationCount++;
|
||
continue;
|
||
}
|
||
|
||
// Handle expressions like x(expr) or (expr)x or coeff(expr)
|
||
final termFactorMatch = RegExp(
|
||
r'([+-]?(?:\d*\.?\d*)?x?)\(([^)]+)\)',
|
||
).firstMatch(result);
|
||
if (termFactorMatch != null) {
|
||
final termStr = termFactorMatch.group(1)!;
|
||
final factorStr = termFactorMatch.group(2)!;
|
||
|
||
// Skip if the term is just a sign or empty
|
||
if (termStr == '+' || termStr == '-' || termStr.isEmpty) {
|
||
break;
|
||
}
|
||
|
||
// Parse the term (coefficient and x power)
|
||
final termCoeffs = _parsePolynomial(termStr);
|
||
final factorCoeffs = _parsePolynomial(factorStr);
|
||
|
||
final termA = termCoeffs[1] ?? 0; // x coefficient
|
||
final termB = termCoeffs[0] ?? 0; // constant term
|
||
|
||
final factorA = factorCoeffs[1] ?? 0; // x coefficient
|
||
final factorB = factorCoeffs[0] ?? 0; // constant term
|
||
|
||
// Multiply: (termA*x + termB) * (factorA*x + factorB)
|
||
final newA = termA * factorA;
|
||
final newB = termA * factorB + termB * factorA;
|
||
final newC = termB * factorB;
|
||
|
||
final expanded =
|
||
'${newA == 1
|
||
? ''
|
||
: newA == -1
|
||
? '-'
|
||
: newA}x^2${newB >= 0 ? '+' : ''}${newB}x${newC >= 0 ? '+' : ''}$newC';
|
||
result = result.replaceFirst(termFactorMatch.group(0)!, '($expanded)');
|
||
iterationCount++;
|
||
continue;
|
||
}
|
||
|
||
if (result == oldResult) break;
|
||
iterationCount++;
|
||
}
|
||
|
||
if (iterationCount >= maxIterations) {
|
||
throw Exception('表达式展开过于复杂，请简化输入。');
|
||
}
|
||
|
||
// 检查是否为方程（包含等号），如果是的话，将右边的常数项移到左边
|
||
if (result.contains('=')) {
|
||
final parts = result.split('=');
|
||
if (parts.length == 2) {
|
||
final leftSide = parts[0];
|
||
final rightSide = parts[1];
|
||
|
||
// 解析左边的多项式
|
||
final leftCoeffs = _parsePolynomial(leftSide);
|
||
final rightCoeffs = _parsePolynomial(rightSide);
|
||
|
||
// 计算标准形式 ax^2 + bx + c = 0 的系数
|
||
// A = B 转换为 A - B = 0，所以右边的系数要取相反数
|
||
final a = (leftCoeffs[2] ?? 0) - (rightCoeffs[2] ?? 0);
|
||
final b = (leftCoeffs[1] ?? 0) - (rightCoeffs[1] ?? 0);
|
||
final c = (leftCoeffs[0] ?? 0) - (rightCoeffs[0] ?? 0);
|
||
|
||
// 构建标准形式的方程
|
||
String standardForm = '';
|
||
if (a != 0) {
|
||
standardForm +=
|
||
'${a == 1
|
||
? ''
|
||
: a == -1
|
||
? '-'
|
||
: a}x^2';
|
||
}
|
||
if (b != 0) {
|
||
standardForm += b > 0 ? '+${b}x' : '${b}x';
|
||
}
|
||
if (c != 0) {
|
||
standardForm += c > 0 ? '+$c' : '$c';
|
||
}
|
||
|
||
// 移除开头的加号
|
||
if (standardForm.startsWith('+')) {
|
||
standardForm = standardForm.substring(1);
|
||
}
|
||
|
||
// 如果所有系数都为0，则方程恒成立
|
||
if (standardForm.isEmpty) {
|
||
standardForm = '0';
|
||
}
|
||
|
||
result = '$standardForm=0';
|
||
}
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
LinearEquationParts _parseLinearEquation(String input) {
|
||
final parts = input.split('=');
|
||
if (parts.length != 2) throw Exception("方程格式错误，应包含一个'='。");
|
||
|
||
final leftCoeffs = _parsePolynomial(parts[0]);
|
||
final rightCoeffs = _parsePolynomial(parts[1]);
|
||
|
||
return LinearEquationParts(
|
||
(leftCoeffs[1] ?? 0.0),
|
||
(leftCoeffs[0] ?? 0.0),
|
||
(rightCoeffs[1] ?? 0.0),
|
||
(rightCoeffs[0] ?? 0.0),
|
||
);
|
||
}
|
||
|
||
Map<int, double> _parsePolynomial(String side) {
|
||
final coeffs = <int, double>{};
|
||
|
||
// 如果输入包含括号，去掉括号
|
||
var cleanSide = side;
|
||
if (cleanSide.startsWith('(') && cleanSide.endsWith(')')) {
|
||
cleanSide = cleanSide.substring(1, cleanSide.length - 1);
|
||
}
|
||
|
||
// 扩展模式以支持 sqrt 函数
|
||
final pattern = RegExp(
|
||
r'([+-]?(?:\d*\.?\d*|sqrt\(\d+\)))x(?:\^(\d+))?|([+-]?(?:\d*\.?\d*|sqrt\(\d+\)))',
|
||
);
|
||
var s = cleanSide.startsWith('+') || cleanSide.startsWith('-')
|
||
? cleanSide
|
||
: '+$cleanSide';
|
||
|
||
for (final match in pattern.allMatches(s)) {
|
||
if (match.group(0)!.isEmpty) continue; // Skip empty matches
|
||
|
||
if (match.group(3) != null) {
|
||
// 常数项
|
||
final constStr = match.group(3)!;
|
||
final constValue = _parseCoefficientWithSqrt(constStr);
|
||
coeffs[0] = (coeffs[0] ?? 0) + constValue;
|
||
} else {
|
||
// x 的幂次项
|
||
int power = match.group(2) != null ? int.parse(match.group(2)!) : 1;
|
||
String coeffStr = match.group(1) ?? '+';
|
||
final coeff = _parseCoefficientWithSqrt(coeffStr);
|
||
coeffs[power] = (coeffs[power] ?? 0) + coeff;
|
||
}
|
||
}
|
||
return coeffs;
|
||
}
|
||
|
||
/// 解析包含 sqrt 函数的系数
|
||
double _parseCoefficientWithSqrt(String coeffStr) {
|
||
if (coeffStr.isEmpty || coeffStr == '+') return 1.0;
|
||
if (coeffStr == '-') return -1.0;
|
||
|
||
// 检查是否包含 sqrt 函数
|
||
final sqrtMatch = RegExp(r'sqrt\((\d+)\)').firstMatch(coeffStr);
|
||
if (sqrtMatch != null) {
|
||
final innerValue = int.parse(sqrtMatch.group(1)!);
|
||
|
||
// 对于完全平方数，直接返回整数结果
|
||
final sqrtValue = sqrt(innerValue.toDouble());
|
||
final rounded = sqrtValue.round();
|
||
if ((sqrtValue - rounded).abs() < 1e-10) {
|
||
// 检查是否有系数
|
||
final coeffPart = coeffStr.replaceFirst(sqrtMatch.group(0)!, '');
|
||
if (coeffPart.isEmpty) return rounded.toDouble();
|
||
if (coeffPart == '-') return -rounded.toDouble();
|
||
|
||
final coeff = double.parse(coeffPart);
|
||
return coeff * rounded;
|
||
}
|
||
|
||
// 对于非完全平方数，计算数值但保持高精度
|
||
final nonPerfectSqrtValue = sqrt(innerValue.toDouble());
|
||
|
||
// 检查是否有系数
|
||
final coeffPart = coeffStr.replaceFirst(sqrtMatch.group(0)!, '');
|
||
if (coeffPart.isEmpty) return nonPerfectSqrtValue;
|
||
if (coeffPart == '-') return -nonPerfectSqrtValue;
|
||
|
||
final coeff = double.parse(coeffPart);
|
||
return coeff * nonPerfectSqrtValue;
|
||
}
|
||
|
||
// 普通数值
|
||
return double.parse(coeffStr);
|
||
}
|
||
|
||
List<double> _parseTwoVariableLinear(String equation) {
|
||
final parts = equation.split('=');
|
||
if (parts.length != 2) throw Exception("方程 $equation 格式错误");
|
||
final c = double.tryParse(parts[1]) ?? 0.0;
|
||
|
||
double a = 0, b = 0;
|
||
final xMatch = RegExp(r'([+-]?\d*\.?\d*)x').firstMatch(parts[0]);
|
||
if (xMatch != null) {
|
||
final coeff = xMatch.group(1);
|
||
if (coeff == null || coeff.isEmpty || coeff == '+') {
|
||
a = 1.0;
|
||
} else if (coeff == '-') {
|
||
a = -1.0;
|
||
} else {
|
||
a = double.tryParse(coeff) ?? 0.0;
|
||
}
|
||
}
|
||
final yMatch = RegExp(r'([+-]?\d*\.?\d*)y').firstMatch(parts[0]);
|
||
if (yMatch != null) {
|
||
final coeff = yMatch.group(1);
|
||
if (coeff == null || coeff.isEmpty || coeff == '+') {
|
||
b = 1.0;
|
||
} else if (coeff == '-') {
|
||
b = -1.0;
|
||
} else {
|
||
b = double.tryParse(coeff) ?? 0.0;
|
||
}
|
||
}
|
||
return [a, b, c];
|
||
}
|
||
|
||
({String formula, String solution})? _tryFactorization(int a, int b, int c) {
|
||
if (a == 0) return null;
|
||
int ac = a * c;
|
||
int absAc = ac.abs();
|
||
|
||
// Try all divisors of abs(ac) and consider both positive and negative factors
|
||
for (int d = 1; d <= sqrt(absAc).toInt(); d++) {
|
||
if (absAc % d == 0) {
|
||
int d1 = d;
|
||
int d2 = absAc ~/ d;
|
||
|
||
// Try all sign combinations for the factors
|
||
// We need m * n = ac and m + n = b
|
||
List<int> signCombinations = [1, -1];
|
||
|
||
for (int sign1 in signCombinations) {
|
||
for (int sign2 in signCombinations) {
|
||
int m = sign1 * d1;
|
||
int n = sign2 * d2;
|
||
if (m + n == b && m * n == ac) {
|
||
return formatFactor(m, n, a);
|
||
}
|
||
|
||
// Also try the swapped version
|
||
m = sign1 * d2;
|
||
n = sign2 * d1;
|
||
if (m + n == b && m * n == ac) {
|
||
return formatFactor(m, n, a);
|
||
}
|
||
}
|
||
}
|
||
}
|
||
}
|
||
return null;
|
||
}
|
||
|
||
bool check(int m, int n, int b) => m + n == b;
|
||
|
||
({String formula, String solution}) formatFactor(int m, int n, int a) {
|
||
// Roots are -m/a and -n/a
|
||
int g1 = gcd(m.abs(), a.abs());
|
||
int root1Num = -m ~/ g1;
|
||
int root1Den = a ~/ g1;
|
||
|
||
int g2 = gcd(n.abs(), a.abs());
|
||
int root2Num = -n ~/ g2;
|
||
int root2Den = a ~/ g2;
|
||
|
||
String sol1 = _formatFraction(root1Num, root1Den);
|
||
String sol2 = _formatFraction(root2Num, root2Den);
|
||
|
||
// For formula, show (a x + m)(x + n/a) or simplified
|
||
String f1 = a == 1 ? 'x' : '${a}x';
|
||
f1 = m == 0 ? f1 : '$f1 ${m >= 0 ? '+' : ''} $m';
|
||
|
||
String f2;
|
||
if (n % a == 0) {
|
||
int coeff = n ~/ a;
|
||
f2 = 'x ${coeff >= 0 ? '+' : ''} $coeff';
|
||
if (coeff == 0) f2 = 'x';
|
||
} else {
|
||
f2 = 'x ${n >= 0 ? '+' : ''} \\frac{$n}{$a}';
|
||
}
|
||
|
||
String formula = '\$\$($f1)($f2) = 0\$\$';
|
||
|
||
String solution;
|
||
if (root1Num * root2Den == root2Num * root1Den) {
|
||
solution = '\$\$x_1 = x_2 = $sol1\$\$';
|
||
} else {
|
||
solution = '\$\$x_1 = $sol1, \\quad x_2 = $sol2\$\$';
|
||
}
|
||
|
||
return (formula: formula, solution: solution);
|
||
}
|
||
|
||
String _formatFraction(int num, int den) {
|
||
if (den == 0) return 'undefined';
|
||
|
||
// Handle sign: make numerator positive, put sign outside
|
||
bool isNegative = (num < 0) != (den < 0);
|
||
int absNum = num.abs();
|
||
int absDen = den.abs();
|
||
|
||
// Simplify fraction
|
||
int g = gcd(absNum, absDen);
|
||
absNum ~/= g;
|
||
absDen ~/= g;
|
||
|
||
if (absDen == 1) {
|
||
return isNegative ? '-$absNum' : '$absNum';
|
||
} else {
|
||
String fraction = '\\frac{$absNum}{$absDen}';
|
||
return isNegative ? '-$fraction' : fraction;
|
||
}
|
||
}
|
||
|
||
int gcd(int a, int b) => b == 0 ? a : gcd(b, a % b);
|
||
|
||
/// 格式化 Rational 值的平方根表达式，保持符号形式
|
||
String _formatSqrtFromRational(Rational value) {
|
||
if (value == Rational.zero) return '0';
|
||
|
||
// 处理负数（用于复数根）
|
||
if (value < Rational.zero) {
|
||
return '\\sqrt{${(-value).toBigInt()}}';
|
||
}
|
||
|
||
// 尝试将 Rational 转换为完全平方数的形式
|
||
// 例如: 4/9 -> 2/3, 9/4 -> 3/2, 25/16 -> 5/4 等
|
||
|
||
// 首先简化分数
|
||
final simplified = value;
|
||
|
||
// 检查分子和分母是否都是完全平方数
|
||
final numerator = simplified.numerator;
|
||
final denominator = simplified.denominator;
|
||
|
||
// 寻找分子和分母的平方根因子
|
||
BigInt sqrtNumerator = _findSquareRootFactor(numerator);
|
||
BigInt sqrtDenominator = _findSquareRootFactor(denominator);
|
||
|
||
// 计算剩余的分子和分母
|
||
final remainingNumerator = numerator ~/ (sqrtNumerator * sqrtNumerator);
|
||
final remainingDenominator =
|
||
denominator ~/ (sqrtDenominator * sqrtDenominator);
|
||
|
||
// 构建结果
|
||
String result = '';
|
||
|
||
// 处理系数部分
|
||
if (sqrtNumerator > BigInt.one || sqrtDenominator > BigInt.one) {
|
||
if (sqrtNumerator > sqrtDenominator) {
|
||
final coeff = sqrtNumerator ~/ sqrtDenominator;
|
||
if (coeff > BigInt.one) {
|
||
result += '$coeff';
|
||
}
|
||
} else if (sqrtDenominator > sqrtNumerator) {
|
||
// 这会导致分母，需要用分数表示
|
||
final coeffNum = sqrtNumerator;
|
||
final coeffDen = sqrtDenominator;
|
||
if (coeffNum == BigInt.one) {
|
||
result += '\\frac{1}{$coeffDen}';
|
||
} else {
|
||
result += '\\frac{$coeffNum}{$coeffDen}';
|
||
}
|
||
}
|
||
}
|
||
|
||
// 处理根号部分
|
||
if (remainingNumerator == BigInt.one &&
|
||
remainingDenominator == BigInt.one) {
|
||
// 没有根号部分
|
||
if (result.isEmpty) {
|
||
return '1';
|
||
}
|
||
} else if (remainingNumerator == remainingDenominator) {
|
||
// 根号部分约分后为1
|
||
if (result.isEmpty) {
|
||
return '1';
|
||
}
|
||
} else {
|
||
// 需要根号
|
||
String sqrtContent = '';
|
||
if (remainingDenominator == BigInt.one) {
|
||
sqrtContent = '$remainingNumerator';
|
||
} else {
|
||
sqrtContent = '\\frac{$remainingNumerator}{$remainingDenominator}';
|
||
}
|
||
|
||
if (result.isEmpty) {
|
||
result = '\\sqrt{$sqrtContent}';
|
||
} else {
|
||
result += '\\sqrt{$sqrtContent}';
|
||
}
|
||
}
|
||
|
||
return result.isEmpty ? '1' : result;
|
||
}
|
||
|
||
/// 寻找一个大整数的平方根因子
|
||
BigInt _findSquareRootFactor(BigInt n) {
|
||
if (n <= BigInt.one) return BigInt.one;
|
||
|
||
BigInt factor = BigInt.one;
|
||
BigInt i = BigInt.two;
|
||
|
||
while (i * i <= n) {
|
||
BigInt count = BigInt.zero;
|
||
while (n % (i * i) == BigInt.zero) {
|
||
n = n ~/ (i * i);
|
||
count += BigInt.one;
|
||
}
|
||
if (count > BigInt.zero) {
|
||
factor = factor * i;
|
||
}
|
||
i += BigInt.one;
|
||
}
|
||
|
||
return factor;
|
||
}
|
||
|
||
/// 格式化二次方程的根：(-b ± sqrt(delta)) / (2a)
|
||
String _formatQuadraticRoot(
|
||
double b,
|
||
Rational delta,
|
||
double denominator,
|
||
bool isPlus,
|
||
) {
|
||
final denomInt = denominator.toInt();
|
||
final denomStr = denominator == 2 ? '2' : denominator.toString();
|
||
|
||
// Format sqrt(delta) symbolically using the Rational value
|
||
final sqrtExpr = _formatSqrtFromRational(delta);
|
||
|
||
if (b == 0) {
|
||
// 简化为 ±sqrt(delta)/denominator
|
||
if (denominator == 2) {
|
||
return isPlus ? '\\frac{$sqrtExpr}{2}' : '-\\frac{$sqrtExpr}{2}';
|
||
} else {
|
||
return isPlus
|
||
? '\\frac{$sqrtExpr}{$denomStr}'
|
||
: '-\\frac{$sqrtExpr}{$denomStr}';
|
||
}
|
||
} else {
|
||
// 完整的表达式：(-b ± sqrt(delta))/denominator
|
||
final bInt = b.toInt();
|
||
|
||
// Check if b is divisible by denominator for simplification
|
||
if (bInt % denomInt == 0) {
|
||
// Can simplify: b/denominator becomes integer
|
||
final simplifiedB = bInt ~/ denomInt;
|
||
|
||
if (simplifiedB == 0) {
|
||
// Just the sqrt part with correct sign
|
||
return isPlus ? '$sqrtExpr' : '-$sqrtExpr';
|
||
} else if (simplifiedB == 1) {
|
||
// +1 * sqrt part
|
||
return isPlus ? '1 + $sqrtExpr' : '1 - $sqrtExpr';
|
||
} else if (simplifiedB == -1) {
|
||
// -1 * sqrt part
|
||
return isPlus ? '-1 + $sqrtExpr' : '-1 - $sqrtExpr';
|
||
} else if (simplifiedB > 0) {
|
||
// Positive coefficient
|
||
return isPlus
|
||
? '$simplifiedB + $sqrtExpr'
|
||
: '$simplifiedB - $sqrtExpr';
|
||
} else {
|
||
// Negative coefficient
|
||
final absB = (-simplifiedB).toString();
|
||
return isPlus ? '-$absB + $sqrtExpr' : '-$absB - $sqrtExpr';
|
||
}
|
||
} else {
|
||
// Cannot simplify, use fraction form
|
||
final bStr = b > 0 ? '${bInt}' : '(${bInt})';
|
||
final signStr = isPlus ? '+' : '-';
|
||
final numerator = b > 0
|
||
? '-$bStr $signStr $sqrtExpr'
|
||
: '(${bInt}) $signStr $sqrtExpr';
|
||
|
||
if (denominator == 2) {
|
||
return '\\frac{$numerator}{2}';
|
||
} else {
|
||
return '\\frac{$numerator}{$denomStr}';
|
||
}
|
||
}
|
||
}
|
||
}
|
||
|
||
/// 格式化复数根的虚部：sqrt(-delta)/(2a)
|
||
String _formatImaginaryPart(String sqrtExpr, double denominator) {
|
||
final denomStr = denominator == 2 ? '2' : denominator.toString();
|
||
|
||
if (denominator == 2) {
|
||
return '\\frac{\\sqrt{${sqrtExpr.replaceAll('\\sqrt{', '').replaceAll('}', '')}}}{2}i';
|
||
} else {
|
||
return '\\frac{\\sqrt{${sqrtExpr.replaceAll('\\sqrt{', '').replaceAll('}', '')}}}{$denomStr}i';
|
||
}
|
||
}
|
||
|
||
/// 格式化原始方程，保持符号形式
|
||
String _formatOriginalEquation(String input) {
|
||
// Parse the equation and convert to LaTeX
|
||
String result = input.replaceAll(' ', '');
|
||
|
||
// 确保方程格式正确
|
||
if (!result.contains('=')) {
|
||
result = '$result=0';
|
||
}
|
||
|
||
final parts = result.split('=');
|
||
if (parts.length == 2) {
|
||
try {
|
||
final leftParser = Parser(parts[0]);
|
||
final leftExpr = leftParser.parse();
|
||
final rightParser = Parser(parts[1]);
|
||
final rightExpr = rightParser.parse();
|
||
result =
|
||
'${leftExpr.toString().replaceAll('*', '\\cdot')}=${rightExpr.toString().replaceAll('*', '\\cdot')}';
|
||
} catch (e) {
|
||
// Fallback to original if parsing fails
|
||
result = result.replaceAll('sqrt(', '\\sqrt{');
|
||
result = result.replaceAll(')', '}');
|
||
}
|
||
} else {
|
||
try {
|
||
final parser = Parser(result.split('=')[0]);
|
||
final expr = parser.parse();
|
||
result = '${expr.toString().replaceAll('*', '\\cdot')}=0';
|
||
} catch (e) {
|
||
// Fallback
|
||
result = result.replaceAll('sqrt(', '\\sqrt{');
|
||
result = result.replaceAll(')', '}');
|
||
}
|
||
}
|
||
|
||
return '\$\$$result\$\$';
|
||
}
|
||
|
||
/// 解析多项式，保持符号形式
|
||
Map<int, String> _parsePolynomialSymbolic(String side) {
|
||
final coeffs = <int, String>{};
|
||
|
||
// Use a simpler approach: split by terms and parse each term individually
|
||
var s = side.replaceAll(' ', ''); // Remove spaces
|
||
if (!s.startsWith('+') && !s.startsWith('-')) {
|
||
s = '+$s';
|
||
}
|
||
|
||
// Split by + and - but be more careful about parentheses and functions
|
||
final terms = <String>[];
|
||
int start = 0;
|
||
int parenDepth = 0;
|
||
|
||
for (int i = 0; i < s.length; i++) {
|
||
final char = s[i];
|
||
|
||
if (char == '(') {
|
||
parenDepth++;
|
||
} else if (char == ')') {
|
||
parenDepth--;
|
||
}
|
||
|
||
// Only split on + or - when not inside parentheses
|
||
if (parenDepth == 0 && (char == '+' || char == '-') && i > start) {
|
||
terms.add(s.substring(start, i));
|
||
start = i;
|
||
}
|
||
}
|
||
terms.add(s.substring(start));
|
||
|
||
for (final term in terms) {
|
||
if (term.isEmpty) continue;
|
||
|
||
// Parse each term
|
||
final termPattern = RegExp(r'^([+-]?)(.*?)x(?:\^(\d+))?$|^([+-]?)(.*?)$');
|
||
final match = termPattern.firstMatch(term);
|
||
|
||
if (match != null) {
|
||
if (match.group(5) != null) {
|
||
// Constant term
|
||
final sign = match.group(4) ?? '+';
|
||
final value = match.group(5)!;
|
||
final coeffStr = sign == '+' && value.isNotEmpty
|
||
? value
|
||
: '$sign$value';
|
||
coeffs[0] = _combineCoefficients(coeffs[0], coeffStr);
|
||
} else {
|
||
// x term
|
||
final sign = match.group(1) ?? '+';
|
||
final coeffPart = match.group(2) ?? '';
|
||
final power = match.group(3) != null ? int.parse(match.group(3)!) : 1;
|
||
|
||
String coeffStr;
|
||
if (coeffPart.isEmpty) {
|
||
coeffStr = sign == '+' ? '1' : '-1';
|
||
} else {
|
||
coeffStr = sign == '+' ? coeffPart : '$sign$coeffPart';
|
||
}
|
||
|
||
coeffs[power] = _combineCoefficients(coeffs[power], coeffStr);
|
||
}
|
||
}
|
||
}
|
||
|
||
return coeffs;
|
||
}
|
||
|
||
/// 合并系数，保持符号形式
|
||
String _combineCoefficients(String? existing, String newCoeff) {
|
||
if (existing == null || existing == '0') return newCoeff;
|
||
if (newCoeff == '0') return existing;
|
||
|
||
// 简化逻辑：如果都是数字，可以相加；否则保持原样
|
||
final existingNum = double.tryParse(existing);
|
||
final newNum = double.tryParse(newCoeff);
|
||
|
||
if (existingNum != null && newNum != null) {
|
||
final sum = existingNum + newNum;
|
||
return sum.toString();
|
||
}
|
||
|
||
// 如果包含符号表达式，直接连接
|
||
return '$existing+$newCoeff'.replaceAll('+-', '-');
|
||
}
|
||
|
||
/// 减去系数
|
||
String _subtractCoefficients(String a, String b) {
|
||
if (a == '0') return b.startsWith('-') ? b.substring(1) : '-$b';
|
||
if (b == '0') return a;
|
||
|
||
final aNum = double.tryParse(a);
|
||
final bNum = double.tryParse(b);
|
||
|
||
if (aNum != null && bNum != null) {
|
||
final result = aNum - bNum;
|
||
return result.toString();
|
||
}
|
||
|
||
// 符号表达式相减
|
||
return '$a-${b.startsWith('-') ? b.substring(1) : b}';
|
||
}
|
||
|
||
/// 计算判别式，保持符号形式
|
||
String _calculateDeltaSymbolic(String a, String b, String c) {
|
||
// Delta = b^2 - 4ac
|
||
|
||
// 计算 b^2
|
||
String bSquared;
|
||
if (b == '0') {
|
||
bSquared = '0';
|
||
} else if (b == '1') {
|
||
bSquared = '1';
|
||
} else if (b == '-1') {
|
||
bSquared = '1';
|
||
} else if (b.startsWith('-')) {
|
||
final absB = b.substring(1);
|
||
bSquared = '$absB^2';
|
||
} else {
|
||
bSquared = '$b^2';
|
||
}
|
||
|
||
// 计算 4ac
|
||
String fourAC;
|
||
if (a == '0' || c == '0') {
|
||
fourAC = '0';
|
||
} else {
|
||
// 处理符号
|
||
String aCoeff = a;
|
||
String cCoeff = c;
|
||
|
||
// 如果 a 或 c 是负数，需要处理符号
|
||
bool aNegative = a.startsWith('-');
|
||
bool cNegative = c.startsWith('-');
|
||
|
||
if (aNegative) aCoeff = a.substring(1);
|
||
if (cNegative) cCoeff = c.substring(1);
|
||
|
||
String acProduct;
|
||
if (aCoeff == '1' && cCoeff == '1') {
|
||
acProduct = '1';
|
||
} else if (aCoeff == '1') {
|
||
acProduct = cCoeff;
|
||
} else if (cCoeff == '1') {
|
||
acProduct = aCoeff;
|
||
} else {
|
||
acProduct = '$aCoeff \\cdot $cCoeff';
|
||
}
|
||
|
||
// 确定 4ac 的符号
|
||
bool productNegative = aNegative != cNegative;
|
||
String fourACValue = '4 \\cdot $acProduct';
|
||
|
||
if (productNegative) {
|
||
fourAC = '-$fourACValue';
|
||
} else {
|
||
fourAC = fourACValue;
|
||
}
|
||
}
|
||
|
||
// 计算 Delta = b^2 - 4ac
|
||
if (bSquared == '0' && fourAC == '0') {
|
||
return '0';
|
||
} else if (bSquared == '0') {
|
||
return fourAC.startsWith('-') ? fourAC.substring(1) : '-$fourAC';
|
||
} else if (fourAC == '0') {
|
||
return bSquared;
|
||
} else {
|
||
String sign = fourAC.startsWith('-') ? '+' : '-';
|
||
String absFourAC = fourAC.startsWith('-') ? fourAC.substring(1) : fourAC;
|
||
return '$bSquared $sign $absFourAC';
|
||
}
|
||
}
|
||
|
||
Rational _rationalFromDouble(double value, {int maxPrecision = 12}) {
|
||
// 限制小数精度，避免无限循环小数
|
||
final str = value.toStringAsFixed(maxPrecision);
|
||
|
||
if (!str.contains('.')) {
|
||
return Rational.parse(str);
|
||
}
|
||
|
||
final parts = str.split('.');
|
||
final integerPart = parts[0];
|
||
final fractionalPart = parts[1];
|
||
|
||
final numerator = BigInt.parse(integerPart + fractionalPart);
|
||
final denominator = BigInt.from(10).pow(fractionalPart.length);
|
||
|
||
return Rational(numerator, denominator);
|
||
}
|
||
}
|