اعداد بزرگ

در بعضی از زبان‌های برنامه نویسی محدودیت اندازه اعداد وجود دارد مانند $C$ که در آن حداکثر تا $10^9$ می‌توانیم ذخیره کنیم. برای همین باید روشی برای ذخیره این گونه اعداد ارائه دهیم و تمام عملیات‌های مانند جمع و ضرب و ... را بر روی آن ها تعریف کنیم. برای ذخیره این اعداد از آرایه استفاده می‌کنیم و هر رقم را درون خانه ای از آرایه ذخیره می‌کنیم و عملیات ها را رقم به رقم بر روی آن ها انجام می‌دهیم. **عملیات ها ** * جمع * تفریق * ضرب * تقسیم * باقی مانده * توان * لگاریتم پیاده سازی این روش را در کد زیر می‌توان مشاهده کرد. <code c++> #include <bits/stdc++.h> using namespace std; typedef int64_t ll; typedef long long ll; typedef pair<ll,ll> lll; typedef pair<ll,int> lli; typedef pair<int,int> ii; #define EL printf("\n") #define OK printf("OK") #define pb push_back #define mp make_pair #define ep emplace_back #define X first #define Y second #define fillchar(a,x) memset(a, x, sizeof(a)) #define FOR(i,l,r) for (int i=l;i<=r;i++) #define FORD(i,r,l) for (int i=r;i>=l;i--) const int base = 1e9; typedef vector<int> BigInt; void Set(BigInt &a) { while (a.size() > 1 && a.back() == 0) a.pop_back(); } void Print(BigInt a) { Set(a); printf("%d", (a.size() == 0) ? 0 : a.back()); FORD(i,a.size()-2,0) printf("%09d", a[i]); EL; } BigInt Integer(string s) { BigInt ans; if (s[0] == '-') return ans; if (s.size() == 0) {ans.pb(0); return ans;} while (s.size()%9 != 0) s = '0'+s; for (int i=0;i<s.size();i+=9) { int v = 0; for (int j=i;j<i+9;j++) v = v*10+(s[j]-'0'); ans.insert(ans.begin(),v); } Set(ans); return ans; } BigInt Integer(char c[]) { string s = ""; FOR(i,0,strlen(c)-1) s = s + c[i]; return Integer(s); } BigInt Integer(ll x) { string s = ""; while (x > 0) s = char(x%10+'0') + s, x /= 10; return Integer(s); } BigInt Integer(int x) { return Integer( (ll) x); } void operator >> (istream &in, BigInt &a) { string s; getline(cin, s); a = Integer(s); } void operator << (ostream &out, BigInt a) { Print(a); } bool operator < (BigInt a, BigInt b) { Set(a); Set(b); if (a.size() != b.size()) return (a.size() < b.size()); FORD(i,a.size()-1,0) if (a[i] != b[i]) return (a[i] < b[i]); return false; } bool operator > (BigInt a, BigInt b) { return (b < a); } bool operator == (BigInt a, BigInt b) { return (!(a < b) && !(b < a)); } bool operator <= (BigInt a, BigInt b) { return (a < b

  return (b < a || b == a);

} bool operator < (BigInt a, int b) {

  return (a < Integer(b));

} bool operator > (BigInt a, int b) {

  return (a > Integer(b));

} bool operator == (BigInt a, int b) {

  return (a == Integer(b));

} bool operator >= (BigInt a, int b) {

  return (a >= Integer(b));

} bool operator ⇐ (BigInt a, int b) {

  return (a <= Integer(b));

} BigInt max(BigInt a, BigInt b) {

  if (a > b) return a;
  return b;

} BigInt min(BigInt a, BigInt b) {

  if (a < b) return a;
  return b;

} BigInt operator + (BigInt a, BigInt b) {

  Set(a);
  Set(b);
  BigInt ans;
  int carry = 0;
  FOR(i,0,max(a.size(), b.size())-1) {
      if (i < a.size()) carry += a[i];
      if (i < b.size()) carry += b[i];
      ans.pb(carry%base);
      carry /= base;
  }
  if (carry) ans.pb(carry);
  Set(ans);
  return ans;

} BigInt operator + (BigInt a, int b) {

  return a + Integer(b);

} BigInt operator (BigInt &a) { a a = a + 1; return a; } void operator += (BigInt &a, BigInt b) { a = a + b; } void operator += (BigInt &a, int b) { a = a + b; } BigInt operator - (BigInt a, BigInt b) { Set(a); Set(b); BigInt ans; int carry = 0; FOR(i,0,a.size()-1) { carry += a[i] - (i < b.size() ? b[i] : 0); if (carry < 0) ans.pb(carry+base), carry = -1; else ans.pb(carry), carry = 0; } Set(ans); return ans; } BigInt operator - (BigInt a, int b) { return a - Integer(b); } void operator -- (BigInt &a) { // --a a = a - 1; } void operator -= (BigInt &a, BigInt b) { a = a + b; } void operator -= (BigInt &a, int b) { a = a - b; } BigInt operator * (BigInt a, BigInt b) { Set(a); Set(b); BigInt ans; ans.assign(a.size()+b.size(), 0); FOR(i,0,a.size()-1) { ll carry = 0ll; for (int j=0;j<b.size() ) { ll s = ans[i+j] + carry + (ll)a[i]*(j<b.size()?(ll)b[j]:0ll); ans[i+j] = s%base; carry = s/base; } } Set(ans); return ans; } BigInt operator * (BigInt a, int b) { return a * Integer(b); } void operator *= (BigInt &a, BigInt b) { a = a * b; } void operator *= (BigInt &a, int b) { a = a * b; } BigInt operator / (BigInt a, BigInt b) { Set(a); Set(b); if (b == Integer(0)) return Integer("-1"); BigInt ans, cur; FORD(i,a.size()-1,0) { cur.insert(cur.begin(), a[i]); int x = 0, L = 0, R = base; while (L ⇐ R) { int mid = (L+R)»1; if (b*Integer(mid) > cur) { x = mid; R = mid-1; } else L = mid+1; } cur = cur - Integer(x-1)*b; ans.insert(ans.begin(),x-1); } Set(ans); return ans; } BigInt operator / (BigInt a, int b) { Set(a); BigInt ans; ll cur = 0ll; FORD(i,a.size()-1,0) { cur = (cur*(ll)base + (ll)a[i]); ans.insert(ans.begin(),cur/b); cur %= b; } Set(ans); return ans; } void operator /= (BigInt &a, BigInt b) { a = a / b; } void operator /= (BigInt &a, int b) { a = a / b; } BigInt operator % (BigInt a, BigInt b) { Set(a); Set(b); if (b == Integer(0)) return Integer("-1"); BigInt ans; FORD(i,a.size()-1,0) { ans.insert(ans.begin(), a[i]); int x = 0, L = 0, R = base; while (L ⇐ R) { int mid = (L+R)»1; if (b*Integer(mid) > ans) { x = mid; R = mid-1; } else L = mid+1; } ans = ans - Integer(x-1)*b; } Set(ans); return ans; } int operator % (BigInt a, int b) { Set(a); if (b == 0) return -1; int ans = 0; FORD(i,a.size()-1,0) ans = (ans*(base%b) + a[i]%b)%b; return ans; } void operator %= (BigInt &a, BigInt b) { a = a % b; } void operator %= (BigInt &a, int b) { a = a % Integer(b); } BigInt gcd(BigInt a, BigInt b) { Set(a); Set(b); while (b > Integer(0)) { BigInt r = a%b; a = b; b = r; } Set(a); return a; } BigInt lcm(BigInt a, BigInt b) { return (a*b/gcd(a,b)); } BigInt sqrt(BigInt a) { BigInt x0 = a, x1 = (a+1)/2; while (x1 < x0) { x0 = x1; x1 = (x1+a/x1)/2; } return x0; } BigInt pow(BigInt a, BigInt b) { if (b == Integer(0)) return Integer(1); BigInt tmp = pow(a, b/2); if (b%2 == 0) return tmp * tmp; return tmp * tmp * a; } BigInt pow(BigInt a, int b) { return pow(a,(Integer(b))); } int log(int n, BigInt a) { log_n(a)

  Set(a);
  int ans = 0;
  while (a > Integer(1)) {
      ans++;
      a /= n;
  }
  return ans;

}

int main() {

  BigInt B;  cin >> B;
  BigInt A = Integer("123456789");
  BigInt C = Integer(123456789ll);
  int x; x = 123456789;
  if (B <= A) cout << A - B;
  else {
      cout << "-";
      cout << B - A;
  }
  cout << A + B; Print(A + x);
  cout << A * B; Print(A * x);
  cout << A / B; Print(A / x);
  cout << A % B; printf("%d\n", A % x);
  C = ++A; ++B; C += B + x;
  Print(A); Print(B); Print(C);
  cout << max(A,B);
  cout << min(A,B);
  cout << gcd(A,B);
  cout << lcm(A,B);
  cout << sqrt(A);
  printf("%d %d %d\n", log(2,A), log(10,B), log(5,C));
  A = Integer(16); x = 12;
  cout << pow(A,B);
  cout << pow(A,x);
  return 0;

} </code>