說明
基於記憶體的有效運用，程式語言中規定了各種不同的資料型態，也因此變數所可以表達的最大整數受到限制，例如123456789123456789這樣的 整數就不可能儲存在long變數中（例如C/C++等），我們稱這為long數，這邊翻為超長整數（避免與資料型態的長整數翻譯混淆），或俗稱大數運算。解法
一個變數無法表示超長整數，則就使用多個變數，當然這使用陣列最為方便，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，讓每一個陣列元素可以儲存四個位數，也就是0到9999的數，例如：
很多人問到如何計算像50!這樣的問題，解法就是使用程式中的乘法函式，至於要算到多大，就看需求了。
如果您使用的是Java，那麼在java.lang下有BigInteger與BigDecimal可以直接進行大數運算。
由於使用陣列來儲存數值，關於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，這邊直接提供加減乘除運算的函式供作參考，以下的N為陣列長度。
實作：C Java Python Scala
void add(int *a, int *b, int *c) {
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) {
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 i, tmp, carry = 0;
for(i = N - 1; i >=0; i--) {
tmp = a[i] * b + carry;
c[i] = tmp % 10000;
carry = tmp / 10000;
}
}
void divide(int *a, int b, int *c) { // b 為除數
int i, tmp, remain = 0;
for(i = 0; i < N; i++) {
tmp = a[i] + remain;
c[i] = tmp / b;
remain = (tmp % b) * 10000;
}
}
public class BigNumber {
public static int[] add(int[] a, int[] b) {
int carry = 0;
int[] c = new int[a.length];
for(int i = a.length - 1; i >= 0; i--) {
c[i] = a[i] + b[i] + carry;
if(c[i] < 10000)
carry = 0;
else { // 進位
c[i] -= 10000;
carry = 1;
}
}
return c;
}
public static int[] subtract(int[] a, int[] b) {
int borrow = 0;
int[] c = new int[a.length];
for(int i = a.length - 1; i >= 0; i--) {
c[i] = a[i] - b[i] - borrow;
if(c[i] >= 0)
borrow = 0;
else { // 借位
c[i] += 10000;
borrow = 1;
}
}
return c;
}
public static int[] multiply(int[] a, int b) { // b 為乘數
int carry = 0;
int[] c = new int[a.length];
for(int i = a.length - 1; i >=0; i--) {
int tmp = a[i] * b + carry;
c[i] = tmp % 10000;
carry = tmp / 10000;
}
return c;
}
public static int[] divide(int[] a, int b) { // b 為除數
int remain = 0;
int[] c = new int[a.length];
for(int i = 0; i < a.length; i++) {
int tmp = a[i] + remain;
c[i] = tmp / b;
remain = (tmp % b) * 10000;
}
return c;
}
public static void main(String[] args) {
int[] a = {1234, 5678, 9910, 1923, 1124};
int[] b = {1234, 5678, 9910, 1923, 1124};
int[] c = BigNumber.add(a, b);
for(int i = 0; i < c.length; i++) {
System.out.print(c[i]);
}
System.out.println();
}
}
class BigNumber:
@staticmethod
def add(a, b):
carry = 0
c = [0] * len(a)
for i in range(len(a) - 1, -1, -1):
c[i] = a[i] + b[i] + carry
if c[i] < 10000:
carry = 0
else:
c[i] -= 10000
carry = 1
return c
@staticmethod
def subtract(a, b):
borrow = 0
c = [0] * len(a)
for i in range(len(a) - 1, -1, -1):
c[i] = a[i] - b[i] - borrow
if c[i] >= 0:
borrow = 0
else:
c[i] = c[i] + 10000
borrow = 1
return c
@staticmethod
def multiply(a, b):
carry = 0
c = [0] * len(a)
for i in range(len(a) - 1, -1, -1):
tmp = a[i] * b + carry
c[i] = tmp % 10000
carry = tmp // 10000
return c
@staticmethod
def divide(a, b):
remain = 0
c = [0] * len(a)
for i in range(len(a)):
tmp = a[i] + remain
c[i] = tmp // b
remain = (tmp % b) * 10000
return c
a = [1234, 5678, 9910, 1923, 1124]
b = [1234, 5678, 9910, 1923, 1124]
c = BigNumber.divide(a, 3)
for i in range(len(c)):
print(c[i], end="")
print()
object BigNumber {
def add(a: Array[Int], b: Array[Int]) = {
val c = new Array[Int](a.length)
var carry = 0
for(i <- a.length - 1 until (-1, -1)) {
c(i) = a(i) + b(i) + carry
carry = if(c(i) < 10000) 0
else {
c(i) -= 10000
1
}
}
c
}
def subtract(a: Array[Int], b: Array[Int]) = {
val c = new Array[Int](a.length)
var borrow = 0
for(i <- a.length - 1 until (-1, -1)) {
c(i) = a(i) - b(i) - borrow
borrow = if(c(i) >= 0) 0
else {
c(i) += 10000
1
}
}
c
}
def multiply(a: Array[Int], b: Int) = {
val c = new Array[Int](a.length)
var carry = 0
for(i <- a.length - 1 until (-1, -1)) {
val tmp = a(i) * b + carry
c(i) = tmp % 10000
carry = tmp / 10000
}
c
}
def divide(a: Array[Int], b: Int) = {
val c = new Array[Int](a.length)
var remain = 0
for(i <- 0 until a.length) {
val tmp = a(i) + remain
c(i) = tmp / b
remain = (tmp % b) * 10000
}
c
}
}
object Main {
def main(args: Array[String]) {
val a = Array(1234, 5678, 9910, 1923, 1124)
val b = Array(1234, 5678, 9910, 1923, 1124)
val c = BigNumber.add(a, b)
c.foreach(print)
}
}