How do you store an arbitrarily large integer value in memory?

I have to store an integer value that is larger than the maximum value for the long datatype. How would I store and manipulate this value in memory?

Please illustrate it through an example, if possible.

Solution 1:

Think about storing a numbers as sequences of decimal digits using a struct like this:

struct num {
    int ndigits;
    char d[MAXDIGITS];
};

For example, the number 123456 could be initialized as

struct num n = { 6, { 6, 5, 4, 3, 2, 1 } };

The reversed digit order turns out to be important for easy calculation. In particular, the place value of n.d[i] is n.d[i] * 10^i.

Now, a few questions:

• How would you add one to a num?
• How would you add an arbitrary single digit to a num?
• How would you add two nums together?
• How would you multiply a num by two?
• How would you multiply a num by a single digit?
• How would you multiply a num by 10?
• How would you multiply two nums together? HINT: Do some pencil and paper multiplications and see how they work.

If you work through this sequence of questions, you should be able to write a function for each step, and re-use those functions to answer the later questions, and end up with a very simple and unoptimized long (well, up to MAXDIGIT digits) integer package for addition and multiplication of positive numbers.

Other questions:

• How do you generalize num to represent negative numbers as well as positive?
• How do you divide one num by another (ignoring remainders)? This is trickier than multiplication, but again, start by doing a few pencil and paper long divisions and think carefully about what you do.

Solution 2:

Possible solutions:
1) Define custom integer type that is large enough to hold that value. 128-bit integer is large enough to hold 98474737475747374739399.
2) Use any available bignum library.

Solution 3:

I won't give you the code, but I can make a couple of suggestions for approaches to take:

1. Try storing the value as a character string and converting to perform calculations
2. Try breaking the value up into multiple integers representing a portion of the value
3. Look up existing libraries that may take care of this for you

Good luck

Solution 4:

Robert Lafore - Object-Oriented Programming in C++, 4th Edition :

// verylong.cpp
// implements very long integer type
#include "verylong.h"          //header file for verylong
//--------------------------------------------------------------
void verylong::putvl() const           //display verylong
   {
   char temp[SZ];
   strcpy(temp,vlstr);                 //make copy
   cout << strrev(temp);               //reverse the copy
   }                                   //and display it
//--------------------------------------------------------------
void verylong::getvl()                 //get verylong from user
   {
   cin >> vlstr;                       //get string from user
   vlen = strlen(vlstr);               //find its length
   strrev(vlstr);                      //reverse it
   }
//--------------------------------------------------------------
verylong verylong::operator + (const verylong v) //add verylongs
   {
   char temp[SZ];
   int j;
                       //find longest number
   int maxlen = (vlen > v.vlen) ? vlen : v.vlen;
   int carry = 0;                      //set to 1 if sum >= 10
   for(j = 0; j<maxlen; j++)           //for each position
      {
      int d1 = (j > vlen-1)   ? 0 : vlstr[j]-'0';   //get digit
      int d2 = (j > v.vlen-1) ? 0 : v.vlstr[j]-'0'; //get digit
      int digitsum = d1 + d2 + carry;               //add digits
      if( digitsum >= 10 )             //if there's a carry,
         { digitsum -= 10; carry=1; }  //decrease sum by 10,
      else                             //set carry to 1
         carry = 0;                    //otherwise carry is 0
      temp[j] = digitsum+'0';          //insert char in string
      }
   if(carry==1)                        //if carry at end,
      temp[j++] = '1';                 //last digit is 1
   temp[j] = '\0';                     //terminate string
   return verylong(temp);              //return temp verylong
   }
//--------------------------------------------------------------
verylong verylong::operator * (const verylong v)  //multiply 
   {                                              //verylongs
   verylong pprod;                     //product of one digit
   verylong tempsum;                   //running total
   for(int j=0; j<v.vlen; j++)         //for each digit in arg
      {
      int digit = v.vlstr[j]-'0';      //get the digit
      pprod = multdigit(digit);        //multiply this by digit
      for(int k=0; k<j; k++)           //multiply result by
         pprod = mult10(pprod);        //   power of 10
      tempsum = tempsum + pprod;       //add product to total
      }
   return tempsum;                     //return total of prods
   }
//--------------------------------------------------------------
verylong verylong::mult10(const verylong v) const //multiply
   {                                              //arg by 10
   char temp[SZ];
   for(int j=v.vlen-1; j>=0; j--)      //move digits one 
      temp[j+1] = v.vlstr[j];          //   position higher
   temp[0] = '0';                      //put zero on low end
   temp[v.vlen+1] = '\0';              //terminate string
   return verylong(temp);              //return result
   }
//--------------------------------------------------------------
verylong verylong::multdigit(const int d2) const 
   {                                   //multiply this verylong
   char temp[SZ];                      //by digit in argument
   int j, carry = 0;
   for(j = 0; j<vlen; j++)             //for each position
      {                                //   in this verylong
      int d1 = vlstr[j]-'0';           //get digit from this
      int digitprod = d1 * d2;         //multiply by that digit
      digitprod += carry;              //add old carry
      if( digitprod >= 10 )            //if there's a new carry,
         {
         carry = digitprod/10;         //carry is high digit
         digitprod -= carry*10;        //result is low digit
         }
      else
         carry = 0;                    //otherwise carry is 0
      temp[j] = digitprod+'0';         //insert char in string
      }
   if(carry != 0)                      //if carry at end,
      temp[j++] = carry+'0';           //it's last digit
   temp[j] = '\0';                     //terminate string
   return verylong(temp);              //return verylong
   }

Verylong class header

// verylong.h
// class specifier for very long integer type
#include <iostream>
#include <string.h>         //for strlen(), etc.
#include <stdlib.h>         //for ltoa()
using namespace std;

const int SZ = 1000;
        //maximum digits in verylongs

class verylong
   {
   private:
      char vlstr[SZ];       //verylong number, as a string
      int vlen;             //length of verylong string
      verylong multdigit(const int) const;   //prototypes for
      verylong mult10(const verylong) const; //private functions
   public:
      verylong() : vlen(0)             //no-arg constructor
         { vlstr[0]='\0'; }
      verylong(const char s[SZ])       //one-arg constructor
         { strcpy(vlstr, s); vlen=strlen(s); }   //for string
      verylong(const unsigned long n)  //one-arg constructor
         {                                       //for long int
         ltoa(n, vlstr, 10);           //convert to string
         strrev(vlstr);                //reverse it
         vlen=strlen(vlstr);           //find length
         }  
      void putvl() const;              //display verylong
      void getvl();                    //get verylong from user
      verylong operator + (const verylong); //add verylongs
      verylong operator * (const verylong); //multiply verylongs
   };