漫画：如何实现大整数相加？

亦凉 2023-07-13 03:08 11阅读 0赞

![format_png][]

![format_png 1][]

**—————  第二天  —————**

![format_png 2][]

![format_png 3][]

![format_png 4][]

![format_png 5][]

![format_png 6][]

![format_png 7][]

![format_png 8][]

————————————

![format_png 9][]

![format_png 10][]

![format_png 11][]

![format_png 12][]

![format_png 13][]

![format_png 14][]

![format_png 15][]

![format_png 16][]

![format_png 17][]

![format_png 18][]

![format_png 19][]

![format_png 20][]

在程序中列出的 “竖式” 究竟是什么样子呢？我们以 426709752318 + 95481253129 为例，来看看大整数相加的详细步骤：

**第一步**，把整数倒序存储，整数的个位存于数组0下标位置，最高位存于数组长度-1下标位置。之所以倒序存储，更加符合我们从左到右访问数组的习惯。

![format_png 21][]

**第二步**，创建结果数组，结果数组的最大长度是较大整数的位数+1，原因很明显。

![format_png 22][]

**第三步**，遍历两个数组，从左到右按照对应下标把元素两两相加，就像小学生计算竖式一样。

例子中，最先相加的是数组A的第1个元素8和数组B的第1个元素9，结果是7，进位1。把7填充到Result数组的对应下标，进位的1填充到下一个位置：

![format_png 23][]

第二组相加的是数组A的第2个元素1和数组B的第2个元素2，结果是3，再加上刚才的进位1，把4填充到Result数组的对应下标：

![format_png 24][]

第三组相加的是数组A的第3个元素3和数组B的第3个元素1，结果是4，把4填充到Result数组的对应下标：

![format_png 25][]

第四组相加的是数组A的第4个元素2和数组B的第4个元素3，结果是5，把5填充到Result数组的对应下标：

![format_png 26][]

以此类推......一直把数组的所有元素都相加完毕：

![format_png 27][]

**第四步**，把Result数组的全部元素再次逆序，去掉首位的，就是最终结果：

![format_png 28][]

![format_png 29][]

![format_png 30][]

/**
    
    * 大整数求和
    
    * @param bigNumberA  大整数A
    
    * @param bigNumberB  大整数B
    
    */
    
    public static String bigNumberSum(String bigNumberA, String bigNumberB) {
    
       //1.把两个大整数用数组逆序存储，数组长度等于较大整数位数+1
    
       int maxLength = bigNumberA.length() > bigNumberB.length() ? bigNumberA.length() : bigNumberB.length();
    
       int[] arrayA = new int[maxLength+1];
    
       for(int i=0; i< bigNumberA.length(); i++){
    
           arrayA[i] = bigNumberA.charAt(bigNumberA.length()-1-i) - '0';
    
       }
    
       int[] arrayB = new int[maxLength+1];
    
       for(int i=0; i< bigNumberB.length(); i++){
    
           arrayB[i] = bigNumberB.charAt(bigNumberB.length()-1-i) - '0';
    
       }
    
       //2.构建result数组，数组长度等于较大整数位数+1
    
       int[] result = new int[maxLength+1];
    
       //3.遍历数组，按位相加
    
       for(int i=0; i<result.length; i++){
    
           int temp = result[i];
    
           temp += arrayA[i];
    
           temp += arrayB[i];
    
           //判断是否进位
    
           if(temp >= 10){
    
               temp = temp-10;
    
               result[i+1] = 1;
    
           }
    
           result[i] = temp;
    
       }
    
       //4.把result数组再次逆序并转成String
    
       StringBuilder sb = new StringBuilder();
    
       //是否找到大整数的最高有效位
    
       boolean findFirst = false;
    
       for (int i = result.length - 1; i >= 0; i--) {
    
           if(!findFirst){
    
               if(result[i] == 0){
    
                   continue;
    
               }
    
               findFirst = true;
    
           }
    
           sb.append(result[i]);
    
       }
    
       return sb.toString();
    
    }
    
    
    
    public static void main(String[] args) {
    
       System.out.println(bigNumberSum("426709752318", "95481253129"));
    
    }

![format_png 31][]

![format_png 32][]

![format_png 33][]

如何优化呢？

我们之前是把大整数按照每一个十进制数位来拆分，比如较大整数的长度有50位，那么我们需要创建一个51位的数组，数组的每个元素存储其中一位。

![format_png 34][]

我们真的有必要把原整数拆分得那么细吗？显然不需要，只需要拆分到**可以被直接计算**的程度就够了。

int类型的取值范围是 -2147483648——2147483647，最多有10位整数。为了防止溢出，我们可以把大整数的每**9**位作为数组的一个元素，进行加法运算。（这里也可以使用long类型来拆分，按照int范围拆分仅仅是提供一个思路）

![format_png 35][]

如此一来，占用空间和运算次数，都被压缩了9倍。

![format_png 36][]

![format_png 37][]

—————END—————

[format_png]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5MVJBQmljVlA2RW1pYVVUdVYxZDh4ak1QNnJkcEVmSVlkazludlFVVHRzVUxlazkxMGE2c0ZvbWcvNjQw?x-oss-process=image/format,png
[format_png 1]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5dWoyeEpoakNuMFRuUUZqOTJSbEpxYXlpY1Nwa2tUYjk3TVRJdnBuUHF5NEFKSDZaamljQ3JpYnhnLzY0MA?x-oss-process=image/format,png
[format_png 2]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR29JaWFrSUpkSVZWcnhsaWE4M0ZVbmd2aWE2VFNsVG9USWd4bEI0TzRBWEhaNXVrNXFEUmw2UnlHVWR0V2ljdnRvR095RnFGa2RwemN3bnVBLzY0MA?x-oss-process=image/format,png
[format_png 3]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR29JaWFrSUpkSVZWcnhsaWE4M0ZVbmd2aWFZTzQ1cklHMURGb0E2V09DY1FXV2JtcFBpYWliZ1VleVBwVDlzd3ZOUmFvZGxQMUhlaGZwMk5Cdy82NDA?x-oss-process=image/format,png
[format_png 4]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5ckZ6anVDRlRxQldIUU5HcTZEQWliRHBQR3h3aWFPMHVKdFJaZGhXNFJVWHphOTNLTmhQV2pDYVEvNjQw?x-oss-process=image/format,png
[format_png 5]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5cmppYk5sYlJZRFVUWndTUUdsWFUwOHlxSnFSdGFNTXl2VnNqRWRLT0J6dzNPc1dYVWlhTU5CbUEvNjQw?x-oss-process=image/format,png
[format_png 6]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5TFlRZzVoajdxd1h1d3Z1aHBEUnNpYnI1dG5XejVlSGZUWmljaWJ5TEc2NGV3MEdrdGlhaWM3MFlPNlEvNjQw?x-oss-process=image/format,png
[format_png 7]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5NWZQdlQwNjdsS2FGcHhGOHpXcVZvWldQdHUzU2oyQWljWU9XUG5zNUI2dFFnZ2UzZUI2aWN6ekEvNjQw?x-oss-process=image/format,png
[format_png 8]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5ZENyOHpRc3VVOVVUZWw2alZMTHhVOENXdFp0ek9icURVeTdaWVhnSHRObVYwM1F0aWFiVzZLdy82NDA?x-oss-process=image/format,png
[format_png 9]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR29JaWFrSUpkSVZWcnhsaWE4M0ZVbmd2aWFCSFE1bHFIU2hGM1dNaWNkRFV4a2NFZHNOVXBna3VWd2Uwdm1TRm5xOFI4b0JOallmdllyMHVBLzY0MA?x-oss-process=image/format,png
[format_png 10]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR29JaWFrSUpkSVZWcnhsaWE4M0ZVbmd2aWFBVWtTdW9Hd0xZYVZFdVFJUUdyR01CaWFNd2ljaG9lZG4xNmY5SHoxZlgzeWI1MUZueW04YkZsUS82NDA?x-oss-process=image/format,png
[format_png 11]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5aWNINmFGdXJNYlVMajhOQ3JoMGFhUFZUZWU5bXVFbGh5ZWljRXFadWtYQksyV2VTbU5yMngwVEEvNjQw?x-oss-process=image/format,png
[format_png 12]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5ZFVvRDdnbEpwWWVoSWliVkQ0QzVCZ1VhNnVhV2NTbXFpYTBvUjA1RHNNbkdTWEhJc2V5VGxCQlEvNjQw?x-oss-process=image/format,png
[format_png 13]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5NG9YTU8wWGpvMFpMZ3d0Y2xWVzJZWHZOTEZKU0JodndhU0h6STNyNmR0Mk1IYndFWHo4Z3JRLzY0MA?x-oss-process=image/format,png
[format_png 14]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X3BuZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5UDRNaWE2TXFuSjVrM1lHN0ZSZDJFVU0zVHhvTk96RUs4eHo5amliOFdoZWdVUGRMOWljdUE4UEFBLzY0MA?x-oss-process=image/format,png
[format_png 15]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5QXVtcmlhRmVaTFV3WGhkQ0FLOUNBZ3Z6TGljaWNCT05GNWJLSjlsMmpZY2dNSXlKRE9XSVpJM1dnLzY0MA?x-oss-process=image/format,png
[format_png 16]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5Z2RBYzJPZU1kTkRUSGdpYWliNjcyQTRuam9PQUgwZktvdTBXU1lMTG1RamliSDV2VEFpYUxRb3JNQS82NDA?x-oss-process=image/format,png
[format_png 17]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5Z3YyOGdCcE1PRmJTVkx3T2pIbkNFeUwzb29YVVNKVkh4VHAzcUlMV0xYclpUYVV5STZrVXhRLzY0MA?x-oss-process=image/format,png
[format_png 18]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5aHc2Skpqc1RPVHB3aHc2UzRJVUxwdkVPN2o3dGlhZ2ljelBZWmFERzRTaG5FSGprMEg0SndHNHcvNjQw?x-oss-process=image/format,png
[format_png 19]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BEYmYyUW91ZnNsM2RwQ2lhakEyR2d5SjdEVlQ3ZUEzRWFpYllJTUtZejlBTFhYVlRIMDhXNEdldjVZNEhlQmoxRmljSVNjV0lJY3lSOFEvNjQw?x-oss-process=image/format,png
[format_png 20]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR29pYUlpYnM5OHVRUkh4aFg1TWxFRTdIUlVUN1EwWE5vWDZORFVrTjBsR1FwVUNpYVNFb3dTQzNZSWt0TGlhd2ZCcUxKcWRiTkJxSTdQdVVRLzY0MA?x-oss-process=image/format,png
[format_png 21]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X3BuZy9OdE81c2lhbEpaR29pYUlpYnM5OHVRUkh4aFg1TWxFRTdIUjRUZU9sZHBEdnpTMlI0aGZldHVQWFRWYjVhd0w4MmE3SHRJY1ZoQmRYRTZrOVFFU3dtcjZQdy82NDA?x-oss-process=image/format,png
[format_png 22]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X3BuZy9OdE81c2lhbEpaR29pYUlpYnM5OHVRUkh4aFg1TWxFRTdIUjlhTkFJVGw5Y05YbDQ5WmxPNjJXcDFXaWNIaWJwaWJpY0c1TEI3cGlhRnU2Qm5TSm96dXh3ZU1TMjB3LzY0MA?x-oss-process=image/format,png
[format_png 23]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X3BuZy9OdE81c2lhbEpaR29pYUlpYnM5OHVRUkh4aFg1TWxFRTdIUlI3eWFpY21KUHQzZ2s1ZTFvRXFpYm9oV0FjTXg1RzE5TzZSVTcyVkFBS0V3aG9PZ0lXeGgyRkxnLzY0MA?x-oss-process=image/format,png
[format_png 24]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X3BuZy9OdE81c2lhbEpaR29pYUlpYnM5OHVRUkh4aFg1TWxFRTdIUnpmZnNsaE9BOFA1OE9nblcxcWljZ2VRMXpvMVg3VDg5QkFDNUZjSnN5djcwWUltQkZMZ1pkeUEvNjQw?x-oss-process=image/format,png
[format_png 25]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X3BuZy9OdE81c2lhbEpaR29pYUlpYnM5OHVRUkh4aFg1TWxFRTdIUjhyWWxQaG5yVXQwVHowN1ZmanhyUVlFRjVaa3lWRXlCR1dFTTlUcmw3eGdaV1lWck9ZbW5rZy82NDA?x-oss-process=image/format,png
[format_png 26]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X3BuZy9OdE81c2lhbEpaR29pYUlpYnM5OHVRUkh4aFg1TWxFRTdIUkRsU1FSeUF2Rjk2b3c0eUlidlNzaWNidUplYWFkY1VocEtpY2lhRWliR1JsdXhpYWZXaWNzYUpNZE95dy82NDA?x-oss-process=image/format,png
[format_png 27]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X3BuZy9OdE81c2lhbEpaR29pYUlpYnM5OHVRUkh4aFg1TWxFRTdIUk45anpxRjBuOW9qWWtMcDJqRzIzUnE0TUFKTjd4blVaaDBFc1BSY3ppYzA4b2VMMFZkN2ZmdkEvNjQw?x-oss-process=image/format,png
[format_png 28]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X3BuZy9OdE81c2lhbEpaR3BPUFpXOExZR1ZSWGZGUnZ6NGtkaWNEeDhLbXc5NkVtVFZPSTBmZ3g2MEhycFMwVERGQksza3NCaWM4M3pUSEc0WTg5Y3VRY3ZWZVNFZy82NDA?x-oss-process=image/format,png
[format_png 29]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR29pYUlpYnM5OHVRUkh4aFg1TWxFRTdIUmVodklKNUI4R1hXUm1ITjZzajZCN1dLaWNubE9HWFNqcU1ydVA0OEhIMTVjMXM0aDVTR2NDYXcvNjQw?x-oss-process=image/format,png
[format_png 30]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR29pYUlpYnM5OHVRUkh4aFg1TWxFRTdIUmhQeVc1MWx5b3FGUU9JeldpYUhKVFF3QVBxWDN0OGVaNXF0bE1BNlNGVTdtQUZvQjRsdGdwcHcvNjQw?x-oss-process=image/format,png
[format_png 31]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR29pYUlpYnM5OHVRUkh4aFg1TWxFRTdIUmEyYklQNnNjdElxc0tjTUR1aWFWdFBQVTZKWTFpYWE3NXpqQ2ZoSHE2WjB2UlhhVTZqNkpnU0JnLzY0MA?x-oss-process=image/format,png
[format_png 32]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR29pYUlpYnM5OHVRUkh4aFg1TWxFRTdIUm1Td3FPUTd5Y0pENmljbFJKRDZEb1lPUDdpYkI3THhKdlVWeHhmMll6cHdudXVHQkFycGU4aWFaZy82NDA?x-oss-process=image/format,png
[format_png 33]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BPUFpXOExZR1ZSWGZGUnZ6NGtkaWNETXh6TllvWnU2OGZ2eXJkTWd6YkdMT1E0YUJ0NHlGT3dMVzhzQ0ZGUjBpY2pzUWxNZDNUTENRdy82NDA?x-oss-process=image/format,png
[format_png 34]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X3BuZy9OdE81c2lhbEpaR3BPUFpXOExZR1ZSWGZGUnZ6NGtkaWNENXh5ZGVnOGljT1NibmVIQ1l6cnpscjNhNE1SbEx1NVAzc0dUNXo1MVZVZDlOZUdoUXM3M1l4US82NDA?x-oss-process=image/format,png
[format_png 35]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X3BuZy9OdE81c2lhbEpaR3BPUFpXOExZR1ZSWGZGUnZ6NGtkaWNESWJ2NFBaSkRLNXlCMTdRWERUQVRYckNGVjJVTFYxc1I0emZkRGd3ejBKMGljU3BLN2NoVmliTkEvNjQw?x-oss-process=image/format,png
[format_png 36]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BPUFpXOExZR1ZSWGZGUnZ6NGtkaWNEMkg3YldQSXdRTnZsakdJRWtFWXplaWE1WVhXQ2RGb3pQaWE3eWFhNm5wNkhpYXpKTWVXQ2cwOFNRLzY0MA?x-oss-process=image/format,png
[format_png 37]: https://imgconvert.csdnimg.cn/aHR0cHM6Ly9tbWJpei5xcGljLmNuL21tYml6X2pwZy9OdE81c2lhbEpaR3BPUFpXOExZR1ZSWGZGUnZ6NGtkaWNEaWJ6ZE11ZktxVHZncWxFUjhoQlZZamxDdHBSSzlJZndmbkxvVWVPaWJXM292bkxJeTVKSkZSR1EvNjQw?x-oss-process=image/format,png