المپدیا

اعداد بزرگ

در بعضی از زبان های برنامه نویسی محدودیت اندازه اعداد وجود دارد مانند $C++$ که در آن حداکثر تا $10^9$ می توانیم ذخیره کنیم. برای همین باید روشی برای ذخیره این گونه اعداد ارائه دهیم و تمام عملیات های مانند جمع و ضرب و … را بر روی آن ها تعریف کنیم.

برای ذخیره این اعداد از آرایه استفاده می کنیم و هر رقم را درون خانه ای از آرایه ذخیره می کنیم و عملیات ها را رقم به رقم بر روی آن ها انجام می دهیم.

عملیات ها

• جمع
• تفریق
• ضرب
• تقسیم
• باقی مانده
• توان
• لگاریتم

پیاده سازی این روش را در کد زیر می توان مشاهده کرد.

#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 || a == b);
}
 
bool operator >= (BigInt a, BigInt 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() || carry > 0;j++) {
            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;
}