大數運算

December 3, 2021

程式語言中可表達的最大整數多半受到限制,例如,Java 中 123456789123456789 就無法以 long 型態儲存,若要表示更大的整數,有些程式語言是透過程式庫,例如 Java 的 java.math.BigInteger,有些程式語言(例如 Python、Ruby、Haskell 等)直接支援大數運算,無需明確式地使用 API。

解法思路

一個型態的空間無法表示超長整數,那就使用多個空間,方式之一是使用陣列,假設程式語言的最大資料型態可以儲存至 65535的數好了,為了計算方便及符合使用十進位制的習慣,讓每個陣列元素可以儲存 0 到 9999,也就是 10000 進位。例如:

大數運算

在考慮正負數的情況下,最高位數用來標示正負數,正數最高位數會是 0000,負數最高位數會是 9999,負數採 10000 補數,例如 99 為 0000 0000 0000 0099,而 -99 為9999 9999 9999 9901,也就是用 9999 減 99 表示法每個位數,最後低數位再加 1。

由於使用陣列儲存數值,數值在運算時的加減乘除等各種運算、位數的進位或借位就必須自行定義,加、減、乘都是由低位數開始運算,而除法則是由高位數開始運算。a + b 時若 b 為負數,求 b 的補數 c 並改進行a - c;a - b 時若 b 為負數,求 b 的補數 c 並改進行 a + c,乘法與除法一律先以正數表示運算,之後再判斷正負數決定是否轉為補數。

程式實作:

#include <stdio.h> 
#include <stdlib.h>

#define N 8

void add(int*, int*, int*);
void subtract(int*, int*, int*);
void multiply(int*, int, int*);
void divide(int*, int, int*);
void toComp(int*, int*);  // 轉補數
void absv(int*, int*);    // 轉絕對值
void print(int*);         // 顯示大整數

int main(void) { 
    int a[] = {0, 0, 0, 1234, 5678, 9910, 1923, 1124};
    int b[] = {9999, 9999, 9999, 9999, 9999, 9999, 9999, 9901}; // -99
    int c1[N], c2[N], c3[N], c4[N] = {0};
        
    add(a, b, c1);       print(c1);
    subtract(a, b, c2);  print(c2);
    multiply(a, -3, c3); print(c3);
    divide(a, -3, c4);   print(c4);

    return 0; 
} 

void add(int* a, int* b, int* c) { 
    if(b[0] == 9999) {
        int comp[N] = {0};
        toComp(b, comp);
        subtract(a, comp, c);
    } else {
        int i, carry = 0; 
        for(i = N - 1; i >= 0; i--) {
            c[i] = a[i] + b[i] + carry; 
            if(c[i] < 10000) {
                carry = 0; 
            } else { // 進位 
                c[i] = c[i] - 10000; 
                carry = 1; 
            } 
        } 
    }
} 

void subtract(int* a, int* b, int* c) {
    if(b[0] == 9999) {
        int comp[N] = {0};
        toComp(b, comp);
        add(a, comp, c);
    } else {
        int i, borrow = 0; 
        for(i = N - 1; i >= 0; i--) { 
            c[i] = a[i] - b[i] - borrow; 
            if(c[i] >= 0) {
                borrow = 0;
            } else { // 借位 
                c[i] = c[i] + 10000; 
                borrow = 1; 
            } 
       } 
    }
} 

void multiply(int* a, int b, int* c) { // b 為乘數
    int op1[N] = {0}; absv(a, op1);
    int op2 = abs(b);

    int i, tmp, carry = 0; 
    for(i = N - 1; i >= 0; i--) { 
        tmp = op1[i] * op2 + carry; 
        c[i] = tmp % 10000;    
        carry = tmp / 10000; 
    }
    
    if((a[0] == 9999 && b > 0) || (a[0] == 0 && b < 0)) {
        toComp(c, c);
    }
} 

void divide(int* a, int b, int *c) {  // b 為除數 
    int op1[N] = {0}; absv(a, op1);
    int op2 = abs(b);
    
    int i, tmp, remain = 0; 
    for(i = 1; i < N; i++) { 
        tmp = op1[i] + remain; 
        c[i] = tmp / op2; 
        remain = (tmp % op2) * 10000; 
    }

    if((a[0] == 9999 && b > 0) || (a[0] == 0 && b < 0)) {
        toComp(c, c);
    }    
}

void toComp(int* src, int* comp) {
    int j;
    for(j = 0; j < N; j++) {
        comp[j] = 9999 - src[j];
    }
    comp[N - 1] += 1;
}

void absv(int* src, int* op) {
    if(src[0] == 9999) {
        toComp(src, op);
    } else {
        int i;
        for(i = 0; i < N; i++) {
            op[i] = src[i];
        }
    }
}

void print(int* c) {
    int v[N] = {0}; absv(c, v);
    
    if(c[0] == 9999) {
        printf("-");
    }
    
    int isDropZero = 1;
    int i;
    for(i = 0; i < N; i++) {
        char s[5] = {'\0'};
        sprintf(s, "%04d", v[i]);
        if(isDropZero) {
            int j;
            for(j = 0; s[j] == '0' && j < 4; j++);
            if(j < 4) {
                isDropZero = 0;
                for(; j < 4; j++) {
                    printf("%c", s[j]);
                }
            } else if(i == N - 1) {
                printf("0");
            }
        } else {
            printf(s);
        }
    }
    printf("\n");
}
import java.util.*;
import static java.lang.System.out;

public class BigInteger {
    private List<Integer> value;
    
    public BigInteger(String val) {
        // 取數字部份
        String v = val.charAt(0) == '-' ? val.substring(1) : val;
        // 每四個字元剖析為一個 int
        value = new ArrayList<>();
        for(int i = v.length() - 4; i > -4; i -= 4) {
            value.add(Integer.parseInt(v.substring(i >= 0 ? i : 0, i + 4)));
        }
        // 補位,位數以 8 為單位
        int valueLength = (value.size() / 8 + 1) * 8;
        for(int i = value.size(); i < valueLength; i++) {
            value.add(0);
        }
        // 負數轉補數表示
        value = val.charAt(0) == '-' ? toComplement(value) : value;        
    }

    private BigInteger(List<Integer> value) {
        this.value = value;
    }
    
    public BigInteger add(BigInteger that) {
        if(isNegative(that.value)) {
            return subtract(new BigInteger(toComplement(that.value)));
        }
        // 對齊位數
        int length = Math.max(value.size(), that.value.size());
        List<Integer> op1 = copyOf(value, length);
        List<Integer> op2 = copyOf(that.value, length);
        List<Integer> result = new ArrayList<>();
        
        int carry = 0;
        for(int i = 0; i < length - 1; i++) {
            int c = op1.get(i) + op2.get(i) + carry;
            if(c < 10000) {
                carry = 0;
            } else {
                c -= 10000;
                carry = 1;
            }
            result.add(c);
        }
        
        if(carry == 1) { // 溢位處理
            if(isPositive(op1)) { result.add(1); } 
            else { result.clear(); } // 負數加法運算溢位就是 0
            for(int i = 0; i < 8; i++) { result.add(0); } // 自動增加 8 位數
        } else { // 補位,正數補 0,負數補 9999
            result.add(isPositive(op1) ? 0 : 9999);
        }
        return new BigInteger(result);
    }
    
    public BigInteger subtract(BigInteger that) {
        if(isNegative(that.value)) {
            return add(new BigInteger(toComplement(that.value)));
        }
        // 對齊位數
        int length = Math.max(value.size(), that.value.size());
        List<Integer> op1 = copyOf(value, length);
        List<Integer> op2 = copyOf(that.value, length);
        List<Integer> result = new ArrayList<>();
        
        int borrow = 0;
        for(int i = 0; i < length - 1; i++) { 
            int c = op1.get(i) - op2.get(i) - borrow; 
            if(c > -1) {
                borrow = 0;
            } else { // 借位 
                c += 10000; 
                borrow = 1; 
            }
            result.add(c);
        }
        
        if(borrow == 1) { // 溢位處理
            if(isNegative(op1)) { result.add(9998); } 
            else { result.clear(); } // 正數減法運算溢位就是 0
            for(int i = 0; i < 8; i++) { result.add(9999); } // 自動增加 8 位數
        } else {  // 補位,負數補 9999,正數補 0
            result.add(isNegative(op1) ? 9999 : 0);
        }
        
        return new BigInteger(result);
    }

    // 右運算元為 int 時的乘法運算,內部使用,兩個運算元都要是正數
    private BigInteger multiply(int val, int shift) {
        List<Integer> result = new ArrayList<>();
        for(int i = 0; i < shift; i++) { result.add(0); } // 位移補 0
        int carry = 0;
        for(int i = 0; i < value.size() - 1; i++) {
            int tmp = value.get(i) * val + carry;
            result.add(tmp % 10000);
            carry = tmp / 10000;
        }
        if(carry != 0) {
            result.add(carry);
            for(int i = 0; i < 8; i++) { result.add(0); }
        } else { result.add(0); }

        return new BigInteger(result);
    }
    
    public BigInteger multiply(BigInteger that) {
        // 轉正數表示
        BigInteger op1 = isNegative(value) ? 
                           new BigInteger(toComplement(value)) : this;
        List<Integer> op2 = isNegative(that.value) ? 
                                      toComplement(that.value) : that.value;
        // 逐位運算
        List<BigInteger> rs = new ArrayList<>();
        for(int i = 0; i < op2.size() - 1; i++) {
            rs.add(op1.multiply(op2.get(i), i));
        }
        // 對逐位運算結果加總
        BigInteger result = rs.get(0);
        for(int i = 1; i < rs.size(); i++) {
            result = result.add(rs.get(i));
        }
        // 判斷正負數
        return getLast(value) + getLast(that.value) == 9999 ? 
            new BigInteger(toComplement(result.value)) : result;
    }
    
    public boolean greaterOrEquals(BigInteger that) {
        return isNegative(subtract(that).value) ? false : true;
    }
    
    private boolean islessOrEqualsToQuotient(BigInteger op1, BigInteger op2) {
        return op1.greaterOrEquals(multiply(op2)) ? true : false;
    }
    
    // 右運算元為 int 時的除法運算,內部使用,兩個運算元都要是正數
    private BigInteger divide(int that) {
        List<Integer> result = new ArrayList<>();
        int remain = 0;
        for(int i = value.size() - 1; i > -1; i--) {
            int tmp = value.get(i) + remain;
            result.add(tmp / that);
            remain = (tmp % that) * 10000;
        }
        Collections.reverse(result);
        for(int i = 0; i < 8 - (result.size() % 8); i++) {
            result.add(0);
        }
        return new BigInteger(result);
    }

    public BigInteger divide(BigInteger that) {
        // 一律先以正數表示
        BigInteger op1 = isNegative(value) ? 
                             new BigInteger(toComplement(value)) : this;
        BigInteger op2 = isNegative(that.value) ? 
                             new BigInteger(toComplement(that.value)) : that;
        
        BigInteger one = new BigInteger("1");
        BigInteger left = new BigInteger("0");
        BigInteger right = op1;

        // 二分法搜尋 x.islessOrEqualsToQuotient(op1, op2) 為 true 的最大 x 值
        while(right.greaterOrEquals(left)) {
            BigInteger x = left.add(right).divide(2);
            if(x.islessOrEqualsToQuotient(op1, op2)) {
                left = x.add(one);
            } else {
                right = x.subtract(one);
            }
        }
        BigInteger result = left.subtract(one);

        // 判斷正負數
        return getLast(value) + getLast(that.value) == 9999 ? 
            new BigInteger(toComplement(result.value)) : result;
    }
    
    public String toString() {
        // 一律以正數表示
        List<Integer> v = isNegative(value) ? toComplement(value) : value;
        StringBuilder builder = new StringBuilder();
        for(int i = v.size() - 1; i > -1; i--) {
            builder.append(String.format("%04d", v.get(i)));
        }
        // 移去前端的 0,負數補負號
        while(builder.length() > 0 && builder.charAt(0) == '0') {
            builder.deleteCharAt(0);
        }
        return builder.length() == 0 ? "0" : 
                   isNegative(value) ? builder.insert(0, '-').toString() : 
                       builder.toString();
    }
        
    private static List<Integer> toComplement(List<Integer> v) {
        List<Integer> comp = new ArrayList<>();
        for(Integer i : v) { comp.add(9999 - i); }
        comp.set(0, comp.get(0) + 1);
        return comp;
    }
    
    private static List<Integer> copyOf(
                List<Integer> original, int newLength) {
        List<Integer> v = new ArrayList<>(original);
        for(int i = v.size(); i < newLength; i++) {
            v.add(isPositive(original) ? 0 : 9999);
        }
        return v;
    }
    
    private static Integer getLast(List<Integer> list) {
        return list.get(list.size() - 1);
    }
    
    private static boolean isNegative(List<Integer> list) {
        return getLast(list) == 9999;
    }
    
    private static boolean isPositive(List<Integer> list) {
        return getLast(list) == 0;
    }
    
    private static boolean isZero(List<Integer> list) {
        for(Integer i : list) if(i != 0) {
            return false;
        }
        return true;
    }
    
    public static void main(String[] args) {
        BigInteger a = new BigInteger("9999999999999999999999999999");
        BigInteger b = new BigInteger("-2");
        out.println(a.add(b));       // 9999999999999999999999999997
        out.println(a.subtract(b));  // 10000000000000000000000000001
        out.println(a.multiply(b));  // -19999999999999999999999999998
        out.println(a.divide(b));    // -4999999999999999999999999999
    }
}
from itertools import dropwhile
from functools import reduce

class BigInt:
    def __init__(self, val):
        self.value = BigInt.parse(val) if isinstance(val, str) else val
        
    def __str__(self):
        v = BigInt.toComplement(self.value) \
                if BigInt.isNegative(self.value) else self.value
        builder = ['%04d' % v[i] for i in range(len(v) - 1, -1, -1)]
        clist = list(dropwhile(lambda c: c == '0', list(''.join(builder))))
        return '0' if len(clist) == 0 else ''.join(
              ((['-'] + clist) if BigInt.isNegative(self.value) else clist))
        
    def __add__(self, that):
        return (self - BigInt(BigInt.toComplement(that.value))) \
                if BigInt.isNegative(that.value) else self.add(that)
            
    def add(self, that):            
        length = max(len(self.value), len(that.value))
        op1 = BigInt.copyOf(self.value, length)
        op2 = BigInt.copyOf(that.value, length)
        sum = BigInt.addForEach(op1, op2, 0)
        return BigInt(
            (((sum[0:-1] + [1]) if BigInt.isPositive(op1) else []) + [0] * 8)
                if sum[-1] == 1 
                else (sum[0:-1] + [0 if BigInt.isPositive(op1) else 9999])
        )
        
    def __sub__(self, that):
        return (self + BigInt(BigInt.toComplement(that.value))) \
            if BigInt.isNegative(that.value) else self.sub(that)
        
    def sub(self, that):
        length = max(len(self.value), len(that.value))
        op1 = BigInt.copyOf(self.value, length)
        op2 = BigInt.copyOf(that.value, length)        
        remain = BigInt.subForEach(op1, op2, 0)
        return BigInt(
            ((remain[0:-1] + [9998] if BigInt.isNegative(op1) else []) 
                + [9999] * 8) if remain[-1] == 1 
                    else (remain[0:-1] + 
                        [9999 if BigInt.isNegative(op1) else 0])
        )
    
    def multiply(self, val, shift):
        product = [0] * shift + \
            BigInt.multiplyForEach(self.value, val, 0)
        return BigInt((product[0:-1] + product[-1:] + [0] * 8) \
            if product[-1] != 0 else (product[0:-1] + [0]))

    def __mul__(self, that):
        op1 = BigInt(BigInt.toComplement(self.value)) \
                if BigInt.isNegative(self.value) else self
        op2 = BigInt.toComplement(that.value) \
                if BigInt.isNegative(that.value) else that.value
        result = reduce(BigInt.__add__, 
                [op1.multiply(op2[i], i) 
                    for i in range(len(op2) - 1)], BigInt('0'))
        return BigInt(BigInt.toComplement(result.value)) \
                if self.value[-1] + that.value[-1] == 9999 else result

    def __ge__(self, that):
        return False if BigInt.isNegative((self - that).value) else True

    def isLessOrEqualsQuotient(self, op1, op2):
        return True if op1 >= (self * op2) else False
    
    def __floordiv__(self, that):
        op1 = BigInt(BigInt.toComplement(self.value)) \
                if BigInt.isNegative(self.value) else self
        op2 = BigInt(BigInt.toComplement(that.value)) \
                if BigInt.isNegative(that.value) else that
        one = BigInt('1')
        
        def quotient(left, right):
            if right >= left:
                x = (left + right).divide(2)
                l, r = ((x + one, right) 
                    if x.isLessOrEqualsQuotient(op1, op2) 
                    else (left, x - one))
                return quotient(l, r)
            else:
                return left - one
        result = quotient(BigInt('0'), op1)
        return BigInt(BigInt.toComplement(result.value)) \
                if self.value[-1] + that.value[-1] == 9999 else result
    
    @staticmethod
    def divideForEach(op, val, remain):
        if op == []:
            return []
        else:
            tmp = op[-1] + remain
            nextRemain = (tmp % val) * 10000
            return [tmp // val] + \
                BigInt.divideForEach(op[0:-1], val, nextRemain)
    
    def divide(self, that):
        result = BigInt.divideForEach(self.value, that, 0)
        return BigInt(result[::-1] + [0] * (8 - (len(result) % 8)))
        
    @staticmethod
    def parse(val):
        v = val[1:] if val[0] == '-' else val
        digits = [int(v[i if i >= 0 else 0 : i + 4]) 
                for i in range(len(v) - 4, -4, -4)]
        zeros = [0] * ((len(digits) // 8 + 1) * 8 - len(digits))
        return BigInt.toComplement(digits + zeros) \
                if val[0] == '-' else (digits + zeros)

    @staticmethod
    def addForEach(op1, op2, carry):
        if op1 == []:
            return [carry]
        else:
            s = op1[0] + op2[0] + carry
            nextCarry, c = (0, s) if s < 10000 else (1, s - 10000)
            return [c] + BigInt.addForEach(op1[1:], op2[1:], nextCarry)

    @staticmethod
    def subForEach(op1, op2, borrow):
        if op1 == []:
            return [borrow]
        else:
            r = op1[0] - op2[0] - borrow
            nextBorrow, c = (0, r) if r > -1 else (1, r + 10000)
            return [c] + BigInt.subForEach(op1[1:], op2[1:], nextBorrow)

    @staticmethod
    def multiplyForEach(op, val, carry):
        if op == []:
            return [carry]
        else:
            tmp = op[0] * val + carry
            nextCarry = tmp // 10000
            return [tmp % 10000] + \
                    BigInt.multiplyForEach(op[1:], val, nextCarry)
            
    @staticmethod
    def toComplement(v):
        c = [9999 - i for i in v]
        return [c[0] + 1] + c[1:]
    
    @staticmethod
    def copyOf(original, newLength):
        return original + [0 if BigInt.isPositive(original) else 9999 
                for i in range(len(original), newLength)]
    
    @staticmethod
    def isNegative(list):
        return list[-1] == 9999
        
    @staticmethod
    def isPositive(list):
        return list[-1] == 0

a = BigInt('99999999999999990999')
b = BigInt('-200')
print(a + b)
print(a - b)
print(a * b)
print(a // b)
class BigInt private (v: List[Int]) {
    private val value = v
    import BigInt._
    override def toString = {
        val v = if(isNegative(value)) toComplement(value) else value
        val builder = for(i <- v.size - 1 until (-1, -1)) 
                         yield "%04d".format(v(i))
        val clist = builder.flatten.dropWhile(_ == '0').toList
        if(clist.size == 0) "0" 
        else (if(isNegative(value)) '-' :: clist else clist).mkString
    }
    def + (that: BigInt): BigInt = {
       if(isNegative(that.value)) this - new BigInt(toComplement(that.value))
       else add(that)
    }
    
    private def add(that: BigInt) = {
        val length = scala.math.max(value.size, that.value.size)
        val op1 = copyOf(value, length)
        val op2 = copyOf(that.value, length)
        val sum = addForEach(op1, op2, 0)
        new BigInt(
            if(sum.last == 1) ((if(isPositive(op1)) sum.init ++ List(1)
                                else Nil) ++ (for(i <- 0 until 8) yield 0))
            else sum.init ++ List(if(isPositive(op1)) 0 else 9999)
        )
    }
    
    def - (that: BigInt) = {
        if(isNegative(that.value)) this + new BigInt(toComplement(that.value))
        else sub(that)
    }
    
    private def sub(that: BigInt) = {
        val length = scala.math.max(value.size, that.value.size)
        val op1 = copyOf(value, length)
        val op2 = copyOf(that.value, length)
        val remain = subForEach(op1, op2, 0)
        new BigInt(
            if(remain.last == 1) {
                (if(isNegative(op1)) remain.init ++ List(9998) 
                 else Nil) ++ (for(i <- 0 until 8) yield 9999)
            }
            else remain.init ++ List(if(isNegative(op1)) 9999 else 0)
        )
    }
    
    def * (that: BigInt) = {
        val op1 = if(isNegative(value)) new BigInt(toComplement(value)) 
                  else this
        val op2 = if(isNegative(that.value)) toComplement(that.value) 
                  else that.value
        val result = (for(i <- 0 until op2.size - 1) 
                        yield op1.multiply(op2(i), i)).reduce(_ + _)
        if(value.last + that.value.last == 9999) {
            new BigInt(toComplement(result.value))
        }            
        else result
    }
    
    private def multiply(v: Int, shift: Int) = {
        val product = (for(i <- 0 until shift) yield 0).toList ++
                            multiplyForEach(value, v, 0)
        new BigInt(
            if(product.last != 0) {
                (product.init ++ List(product.last) 
                    ++ (for(i <- 0 until 8) yield 0)).toList
            }
            else (product.init ++ List(0)).toList
        )
    }
    
    def >= (that: BigInt) = 
        if(isNegative((this - that).value)) false else true
    
    private def isLessOrEqualsQuotient(op1: BigInt, op2: BigInt) = 
        if(op1 >= (this * op2)) true else false
    
    def / (that: BigInt) = {
        val op1 = if(isNegative(value)) 
                      new BigInt(toComplement(value))
                  else this
        val op2 = if(isNegative(that.value)) 
                      new BigInt(toComplement(that.value))
                  else that
        val one = BigInt("1")
        
        def quotient(left: BigInt, right: BigInt): BigInt = {
            if(right >= left) {
                val x = (left + right).divide(2)
                val (l, r) = if(x.isLessOrEqualsQuotient(op1, op2)) {
                                 (x + one, right) 
                             }
                             else (left, x - one)
                quotient(l, r)
            }
            else left - one
        }
        
        val result = quotient(BigInt("0"), op1)
        if(value.last + that.value.last == 9999) {
            new BigInt(toComplement(result.value)) 
        } 
        else result
    }
    
    def divide(that: Int) = {
        val result = divideForEach(value, that, 0)
        new BigInt(result.reverse ++ 
            (for(i <- 0 until (8 - (result.size % 8))) yield 0))
    }
}
              
object BigInt {
    def apply(value: String) = {
        val v = if(value(0) == '-') value.substring(1) else value
        val digits = (for(i <- v.length - 4 until (-4, -4)) 
                yield v.substring(if(i >= 0) i else 0, i + 4).toInt).toList
        val zeros = (
            for(i <- 0 until (digits.size / 8 + 1) * 8 - digits.size)yield 0
        ).toList
        new BigInt(if(value(0) == '-') toComplement(digits ++ zeros) 
                   else (digits ++ zeros));
    }
    
    private def toComplement(value: List[Int]) = {
        val c = for(i <- value) yield 9999 - i
        (c.head + 1) :: c.tail
    }
    
    private def copyOf(original: List[Int], newLength: Int) = {
        original ++ (for(i <- original.size until newLength) 
            yield if(isPositive(original)) 0 else 9999)
    }
    
    private def addForEach(op1: List[Int], 
                           op2: List[Int], carry: Int): List[Int] = {
        if(op1 == Nil) List(carry)
        else {
            val s = op1.head + op2.head + carry
            val (nextCarry, c) = if(s < 10000) (0, s) else (1, s - 10000)
            c :: addForEach(op1.tail, op2.tail, nextCarry)
        }
    }
    
    private def subForEach(op1: List[Int], 
                           op2: List[Int], borrow: Int): List[Int] = {
        if(op1 == Nil) List(borrow)
        else {
            val r = op1.head - op2.head - borrow
            val (nextBorrow, c) = if(r > -1) (0, r) else (1, r + 10000)
            c :: subForEach(op1.tail, op2.tail, nextBorrow)
        }
    }
    
    private def multiplyForEach(op: List[Int], 
                                v: Int, carry: Int): List[Int] = {
        op match {
            case Nil        => List(carry)
            case head::tail => {
                val tmp = head * v + carry
                val nextCarry = tmp / 10000
                (tmp % 10000) :: multiplyForEach(tail, v, nextCarry)
            }
        }
    }
    
    private def divideForEach(op: List[Int], 
                              v: Int, remain: Int): List[Int] = {
        if(op == Nil) Nil
        else {
            val tmp = op.last + remain
            val nextRemain = (tmp % v) * 10000
            (tmp / v) :: divideForEach(op.init, v, nextRemain)
        }
    }

    private def isNegative(value: List[Int]) = value.last == 9999
    private def isPositive(value: List[Int]) = value.last == 0
}

val a = BigInt("99999999999999999999")
val b = BigInt("-200")
println(a)
println(b)
println(a + b)
println(a - b)
println(a * b)
println(a / b)
class BigInt
    attr_reader :value
    def initialize(val)
        @value = if val.instance_of? String
                     BigInt.parse(val) 
                 else val end
    end
    
    def to_s
        v = if BigInt.isNegative(@value)
                BigInt.toComplement(@value) 
            else @value end
        builder = (0...v.size - 1).to_a
                  .reverse.map { |i| sprintf("%04d", v[i]) }
        clist = builder.join.split(//).drop_while { |c| c == "0"}.join
        if clist.empty? 
            "0" 
        else 
            if BigInt.isNegative(@value); "-" + clist else clist end
        end
    end
    
    def +(that)
        if BigInt.isNegative(that.value)
            self - BigInt.new(BigInt.toComplement(that.value))
        else
            add(that)
        end
    end
                
    def add(that)
        length = [@value.size, that.value.size].max
        op1 = BigInt.copyOf(@value, length)
        op2 = BigInt.copyOf(that.value, length)
        sum = BigInt.addForEach(op1, op2, 0)
        BigInt.new(
            if sum[-1] == 1
                (if BigInt.isPositive(op1); sum[0...-1] + [1] 
                 else [] end) + [0] * 8 
            else 
                sum[0...-1] + [if BigInt.isPositive(op1); 0 else 9999 end]  
            end
        )
    end
    
    def -(that)
       if BigInt.isNegative(that.value)
           self + BigInt.new(BigInt.toComplement(that.value))
       else
           sub(that)
       end
    end
    
    def sub(that)
       length = [@value.size, that.value.size].max
       op1 = BigInt.copyOf(@value, length)
       op2 = BigInt.copyOf(that.value, length)
       remain = BigInt.subForEach(op1, op2, 0)
       BigInt.new(
           if remain[-1] == 1
               (if BigInt.isNegative(op1); remain[0...-1] + [9998] 
                else [] end) + [9999] * 8
           else
               remain[0...-1] + [if BigInt.isNegative(op1); 9999 else 0 end]
           end
       )
    end
    
    def multiply(val, shift)
        product = [0] * shift + BigInt.multiplyForEach(@value, val, 0)
        BigInt.new(
            if product[-1] != 0
                product[0...-1] + [product[-1]] + [0] * 8
            else
                product[0...-1] + [0]
            end
        )
    end

    def *(that)
        op1 = if BigInt.isNegative(@value)
                  BigInt.new(BigInt.toComplement(@value))
              else self end
        op2 = if BigInt.isNegative(that.value)
                  BigInt.toComplement(that.value) 
              else that.value end
        result = (0...op2.size - 1).map { 
            |i| op1.multiply(op2[i], i) }.reduce(:+)
        if @value[-1] + that.value[-1] == 9999
            BigInt.new(BigInt.toComplement(result.value))
        else result end
    end
    
    def >=(that)
        if BigInt.isNegative((self - that).value); false
        else true end
    end
    
    def isLessOrEqualsQuotient(op1, op2)
        if op1 >= (self * op2); true else false end
    end
    
    def /(that)
        op1 = if BigInt.isNegative(@value)
                  BigInt.new(BigInt.toComplement(@value))
              else self end
        op2 = if BigInt.isNegative(that.value)
                  BigInt.new(BigInt.toComplement(that.value))
              else that end
        one = BigInt.new("1")
        
        quotient = ->(left, right) {
            if right >= left
                x = (left + right).divide(2)
                l, r = *(if x.isLessOrEqualsQuotient(op1, op2)
                             [x + one, right]
                         else
                             [left, x - one]
                         end)
                quotient.call(l, r)
            else
                left - one
            end
        }
        result = quotient.call(BigInt.new("0"), op1)
        if @value[-1] + that.value[-1] == 9999
            BigInt.new(BigInt.toComplement(result.value))
        else result end
    end
    
    def divide(that)
        result = BigInt.divideForEach(@value, that, 0)
        BigInt.new(result.reverse + [0] * (8 - (result.size % 8)))
    end
                
    def self.parse(val)
        v = if val[0] == "-"; val[1, val.size] else val end
        digits = (if v.size % 4 == 0; 0 else (v.size % 4) - 4 end..v.size - 4)
                 .step(4).to_a.reverse.map { |i|
                     v[if i >= 0; i else 0 end...i + 4].to_i
                 }
        zeros = [0] * ((digits.size / 8 + 1) * 8 - digits.size)
        if val[0] == "-"; BigInt.toComplement(digits + zeros) 
        else digits + zeros end
    end
    
    def self.toComplement(v)
        c = v.map { |i| 9999 - i}
        [c[0] + 1] + c[1, c.size]
    end
    
    def self.isPositive(list)
        list[-1] == 0
    end
    
    def self.isNegative(list)
        list[-1] == 9999
    end
    
    def self.copyOf(original, newLength)
        original + (original.size...newLength).map { |i| 
            if BigInt.isPositive(original); 0 else 9999 end 
        }
    end
    
    def self.addForEach(op1, op2, carry)
        if op1 == []; [carry]
        else
            s = op1[0] + op2[0] + carry
            nextCarry, c = *(if s < 10000; [0, s] 
                             else [1, s - 10000] end)
            [c] + BigInt.addForEach(
                op1[1, op1.size], op2[1, op2.size], nextCarry)
        end
    end
    
    def self.subForEach(op1, op2, borrow)
        if op1 == []; [borrow]
        else
            r = op1[0] - op2[0] - borrow
            nextBorrow, c = *(if r > -1; [0, r] 
                              else [1, r + 10000] end)
            [c] + BigInt.subForEach(
                op1[1, op1.size], op2[1, op2.size], nextBorrow)
        end
    end
    
    def self.multiplyForEach(op, val, carry)
        if op == []; [carry]
        else
            tmp = op[0] * val + carry
            nextCarry = tmp / 10000
            [tmp % 10000] + 
                BigInt.multiplyForEach(op[1, op.size], val, nextCarry)
        end
    end
    
    def self.divideForEach(op, val, remain)
        if op == []; []
        else
            tmp = op[-1] + remain
            nextRemain = (tmp % val) * 10000
            [tmp / val] + 
                BigInt.divideForEach(op[0...-1], val, nextRemain)
        end
    end
end

a = BigInt.new("99999999999999999999")
b = BigInt.new("-200")
puts(a)
puts(b)
puts(a + b)
puts(a - b)
puts(a * b)
puts(a / b)
var BigNumber = function() {
    function apply(val) {
        var v = val.charAt(0) === '-' ? val.substring(1) : val;
        var value = [];
        for(var i = v.length - 4; i > -4; i -= 4) {
            value.push(parseInt(v.substring(i >= 0 ? i : 0, i + 4), 10));
        }
        var valueLength = (parseInt(value.length / 8) + 1) * 8;
        for(var i = value.length; i < valueLength; i++) {
            value.push(0);
        }
        return new BigNumber(val.charAt(0) === '-' ? 
                       toComplement(value) : value);
    }
    
    function BigNumber(value) {
        this.value = value;
    }
    
    BigNumber.prototype.toString = function() {
        var v = isNegative(this.value) ? 
                    toComplement(this.value) : this.value;
        var builder = [];
        for(var i = v.length - 1; i > -1; i--) {
            builder.push(new Array(5 - (v[i] + '').length).join('0') + v[i]);
        }
        var charArray = builder.join('').split('');
        while(charArray.length > 0 && charArray[0] === '0') {
            charArray.shift();
        }
        return charArray.length === 0 ? '0' : (isNegative(this.value) ? 
                   '-' + charArray.join('') : charArray.join(''));
    };
    
    BigNumber.prototype.add = function(that) {
        if(isNegative(that.value)) {
            return this.subtract(new BigNumber(toComplement(that.value)));
        }
        
        var length = Math.max(this.value.length, that.value.length);
        var op1 = copyOf(this.value, length);
        var op2 = copyOf(that.value, length);
        var result = [];
        
        var carry = 0;
        for(var i = 0; i < length - 1; i++) {
            var c = op1[i] + op2[i] + carry;
            if(c < 10000) {
                carry = 0;
            } else {
                c -= 10000;
                carry = 1;
            }
            result.push(c);
        }
        
        if(carry === 1) {
            if(isPositive(op1)) { result.push(1); }
            else { result.length = 0; }
            for(var i = 0; i < 8; i++) { result.push(0); }
        } else {
            result.push(isPositive(op1) ? 0 : 9999);
        }
        return new BigNumber(result);
    };
    
    BigNumber.prototype.subtract = function(that) {
        if(isNegative(that.value)) {
            return this.add(new BigNumber(toComplement(that.value)));
        }
    
        var length = Math.max(this.value.length, that.value.length);
        var op1 = copyOf(this.value, length);
        var op2 = copyOf(that.value, length);
        var result = [];
        
        var borrow = 0;
        for(var i = 0; i < length - 1; i++) {
            var c = op1[i] - op2[i] - borrow;
            if(c > -1) {
                borrow = 0;
            } else {
                c += 10000;
                borrow = 1;
            }
            result.push(c);
        }
        
        if(borrow === 1) {
            if(isNegative(op1)) { result.push(9998); }
            else { result.length = 0; }
            for(var i = 0; i < 8; i++) { result.push(9999); }
        } else {
            result.push(isNegative(op1) ? 9999 : 0);
        }
        return new BigNumber(result);
    };
    
    BigNumber.prototype.mul = function(val, shift) {
        var result = [];
        for(var i = 0; i < shift; i++) { result.push(0); }
        var carry = 0;
        for(var i = 0; i < this.value.length - 1; i++) {
            var tmp = this.value[i] * val + carry;
            result.push(tmp % 10000);
            carry = parseInt(tmp / 10000);
        }
        
        if(carry != 0) {
            result.push(carry);
            for(var i = 0; i < 8; i++) { result.push(0); }
        } else { result.push(0); }
        
        return new BigNumber(result);
    };
    
    BigNumber.prototype.multiply = function(that) {
        var op1 = isNegative(this.value) ? 
                      new BigNumber(toComplement(this.value)) : this;
        var op2 = isNegative(that.value) ? 
                      toComplement(that.value) : that.value;
        var rs = [];
        for(var i = 0; i < op2.length - 1; i++) {
            rs.push(op1.mul(op2[i], i));
        }
        
        var result = rs[0];
        for(var i = 1; i < rs.length; i++) {
            result = result.add(rs[i]);
        }
        
        return getLast(this.value) + getLast(that.value) === 9999 ?
            new BigNumber(toComplement(result.value)) : result;
    };
    
    BigNumber.prototype.greaterOrEquals = function(that) {
        return isNegative(
            this.subtract(that).value) ? false : true;
    };
    
    BigNumber.prototype.isLessOrEqualsQuotient = function(op1, op2) {
        return op1.greaterOrEquals(
            this.multiply(op2)) ? true : false;
    };
    
    BigNumber.prototype.div = function(that) {
        var result = [];
        var remain = 0;
        for(var i = this.value.length - 1; i > -1; i--) {
            var tmp = this.value[i] + remain;
            result.push(parseInt(tmp / that));
            remain = (tmp % that) * 10000;
        }
        result.reverse();
        for(var i = 0; i < 8 - (result.length % 8); i++) {
            result.push(0);
        }
        return new BigNumber(result);
    };
    
    BigNumber.prototype.divide = function(that) {
        var op1 = isNegative(this.value) ? 
                      new BigNumber(toComplement(this.value)) : this;
        var op2 = isNegative(that.value) ? 
                      new BigNumber(toComplement(that.value)) : that;
        
        var one = apply("1");
        var left = apply("0");
        var right = op1;
        
        while(right.greaterOrEquals(left)) {
            var x = left.add(right).div(2);
            if(x.isLessOrEqualsQuotient(op1, op2)) {
                left = x.add(one);
            } else {
                right = x.subtract(one);
            }
        }
        var result = left.subtract(one);
        
        return getLast(this.value) + getLast(that.value) === 9999 ?
            new BigNumber(toComplement(result.value)) : result;
    };
    
    function toComplement(v) {
        var comp = [];
        for(var i in v) { comp.push(9999 - v[i]); }
        comp[0] += 1
        return comp;
    };
    
    function getLast(list) {
        return list[list.length - 1];
    };
    
    function isNegative(list) {
        return getLast(list) === 9999;
    };
    
    function isPositive(list) {
        return getLast(list) === 0;
    };
    
    function copyOf(original, newLength) {
        var v = [].concat(original);
        for(var i = v.length; i < newLength; i++) {
            v.push(isPositive(original) ? 0 : 9999);
        }
        return v;
    }
    
    return apply;
}();

var a = BigNumber('9999999999999999');
var b = BigNumber('-2');
print(a);
print(b);
print(a.add(b));
print(a.subtract(b));
print(a.multiply(b));
print(a.divide(b));
data BigInt = BigInt [Int]

instance Show BigInt where
    show (BigInt value) = 
        let v = if isNegative value then toComplement value else value
            builder = [take (4 - (length \$ show (v !! i))) (repeat '0') 
                ++ show (v !! i) | i <- (reverse [0..length v - 2])]
            clist = dropWhile (== '0') \$ flatten builder
        in if length clist == 0 then "0" 
           else if isNegative value then  '-' : clist
                else clist
                              
slice from to = take (to - from) . drop from

flatten list = [item | subList <- list, item <- subList]

toComplement v = (head c + 1) : (tail c)
    where c = [9999 - i | i <- v]

isPositive list = last list == 0   
 
isNegative list = last list == 9999

copyOf original newLength = original ++ [
    if isPositive original then 0 
    else 9999 | i <- [(length original)..(newLength - 1)]]

addForEach op1 op2 carry =
    if op1 == [] then [carry]
    else
        let s = head op1 + head op2 + carry
            (nextCarry, c) = if s < 10000 then (0, s) else (1, s - 10000)
        in c : addForEach (tail op1) (tail op2) nextCarry

subForEach op1 op2 borrow =
    if op1 == [] then [borrow]
    else
        let r = head op1 - head op2 - borrow
            (nextBorrow, c) = if r > -1 then (0, r) else (1, r + 10000)
        in c : subForEach (tail op1) (tail op2) nextBorrow

multiplyForEach op val carry =
    if op == [] then [carry]
    else
        let tmp = head op * val + carry
            nextCarry = tmp `div` 10000
        in (tmp `mod` 10000) : multiplyForEach (tail op) val nextCarry

divideForEach op val remain =
    if op == [] then []
    else
        let tmp = last op + remain
            nextRemain = (tmp `mod` val) * 10000
        in (tmp `div` val) : divideForEach (init op) val nextRemain
        
bigNum val = BigInt (if head val == '-' 
                         then toComplement (digits ++ zeros) 
                     else digits ++ zeros)
    where v = if head val == '-' then tail val else val
          remain = (length v) `mod` 4
          begin = if remain == 0 then 0 else remain - 4
          digits = [read (slice (if i >= 0 then i else 0) (i + 4) v) :: Int
                      | i <- reverse [begin, begin + 4..length v - 4]]
          dLen = length digits
          zeros = replicate ((dLen `div` 8 + 1) * 8 - dLen) 0

op1@(BigInt v1) `bAdd` (BigInt v2) = 
    if isNegative v2 then op1 `bSub` (BigInt (toComplement v2))
    else v1 `ad` v2
    
v1 `ad` v2 = 
    BigInt (if last sum == 1 then
               if isPositive op1 then init sum ++ [1]
               else take 8 \$ repeat 0
            else init sum ++ [if isPositive op1 then 0 else 9999])
    where len = max (length v1) (length v2)
          op1 = copyOf v1 len
          op2 = copyOf v2 len
          sum = addForEach op1 op2 0

op1@(BigInt v1) `bSub` (BigInt v2) = 
    if isNegative v2 then op1 `bAdd` (BigInt (toComplement v2))
    else v1 `sb` v2
    
v1 `sb` v2 =
    BigInt(if last remain == 1 then
               if isNegative op1 then init remain ++ [9998]
               else take 8 \$ repeat 9999
           else init remain ++ [if isNegative op1 then 9999 else 0])
    where len = max (length v1) (length v2)
          op1 = copyOf v1 len
          op2 = copyOf v2 len
          remain = subForEach op1 op2 0

multiply (BigInt v) val shift = 
    BigInt (if last product /= 0 then 
                init product ++ [last product] ++ take 8 (repeat 0)
            else init product ++ [0])
    where product = take shift (repeat 0) ++ multiplyForEach v val 0

op1@(BigInt v1) `bMul` (BigInt v2) =
    if last v1 + last v2 == 9999 then BigInt (toComplement value)
    else result
    where p1 = if isNegative v1 then BigInt (toComplement v1)
               else op1
          p2 = if isNegative v2 then toComplement v2
               else v2
          result@(BigInt value) = 
              foldl1 (bAdd) [multiply p1 (p2 !! i) i 
                  | i <- [0..length p2 - 2]]

op1 `ge` op2 = 
    let (BigInt v) = op1 `bSub` op2
    in if isNegative v then False else True

isLessOrEquqlsQuotient x op1 op2 =
    if op1 `ge` (x `bMul` op2) then True
    else False

divI (BigInt v) that = 
    BigInt (reverse result ++ take (8 - (length result) `mod` 8) (repeat 0))
    where result = divideForEach v that 0

op1@(BigInt v1) `bDiv` op2@(BigInt v2) =
    if last v1 + last v2 == 9999 then BigInt (toComplement v)
    else result
    where p1 = if isNegative v1 then BigInt (toComplement v1) else op1
          p2 = if isNegative v2 then BigInt (toComplement v2) else op2
          one = bigNum "1"
          quotient left right =
              if right `ge` left then
                  let x = (left `bAdd` right) `divI` 2
                      (l, r) = if isLessOrEquqlsQuotient x p1 p2 then 
                                   (x `bAdd` one, right) 
                               else (left, x `bSub` one)
                  in quotient l r
              else left `bSub` one
          result@(BigInt v) = quotient (bigNum "0") p1

main = do
    print (a `bAdd` b)
    print (a `bSub` b)
    print (a `bMul` b)
    print (a `bDiv` b)
    where a = bigNum "99999999999999999999"
          b = bigNum "-200"