Article From:https://www.cnblogs.com/scuwangjun/p/9124610.html

# integer calculation

Today do LeetCode see a problem, flip integer, one line of code.`if ((newResult - tail) / 10 != result)`

It means that if you go beyond the scope, you will have some losses. It is a little unclear. Just reviewed the original code, the reverse code and the complement code. Next, let’s look at how the computer is calculated from two questions.

## 1. why Integer.MAX_VALUE + 1 = Integer.MIN_VALUE

We can test it with java code:

```
System.out.println(Integer.MAX_VALUE + 1);
System.out.println(Integer.MIN_VALUE);
Print:-2147483648
-2147483648
That is -2^31
```

We know that the range of values with symbolic integers is**[-2^31 ~ 2^31-1]**，In computer, it is stored in the form of complement. First, we will review what is complement, inversion and original code.

`The original number of the positive integer is the same as the complement and the symbol bit is fixed to 0.The symbol bit of a negative integer is 1:Counter code = original code (except the first symbol bit)Complement = reverse code + 1Take -1 as an example:Original code: 1000000000000000000000000000001Counter code: 11111111111111111111111111111110Complement: 11111111111111111111111111111111The inside of the computer is the causeWith the complement code stored and calculated, the process of -1-1 is-1 complement 11111111111111111111111111111111 - 1Get the complement of -2: 111111111111111111111Eleven billion one hundred and eleven million one hundred and eleven thousand one hundred and tenThen -2's reverse code: 11111111111111111111111111111101Then the original code of -2: 10000000000000000000000000000010`

With this foundation, let’s look at why`Integer.MAX_VALUE + 1 = Integer.MIN_VALUE`

:

`Integer.MAX_VALUEThe binary is:01111111111111111111111111111111+1 obtained10000000000000000000000000000000This is a complement, according to the complement, you can get the counter code:11111111111111111111111111111111Note: it may be found that 10000000000000000000000000000000 - 1 should be01111111111111111111111111111111 ah, because the sign bits of negative integer remain unchanged, so the symbol bit is changed from 0 to 1..And then the original code is obtained.10000000000000000000000000000000Because 00000000000000000000000000000000 has been expressed as 0, and the computer specifies that 10000000000000000000000000000000 represents the minimum value of a negative integer, that is, Integer.MIN_VALUE.So Integer.MIN_VALUE'sComplement: 10000000000000000000000000000000Counter code: 111111111111111111111111111Eleven thousand one hundred and elevenOriginal code: 10000000000000000000000000000000And then look at Integer.MIN_VALUE + 1Complement: 10000000000000000000000000000001Counter code: 10000000000000000000000000000000Original code: 11111111111111111111111111111111`

## 2. why Integer.MAX_VALUE * 10 = -10

`Integer.MAX_VALUE * 10Inside the computer, there are 10 Integer.MAX_VALUE additivity (it seems to be...).The binary of Integer.MAX_VALUE is:01111111111111111111111111111111firstFirst look at Integer.MAX_VALUE * 2.01111111111111111111111111111111 +01111111111111111111111111111111 = =11111111111111111111111111111110 (complement)11111111111111111111111111111101 (counter code)One hundred trillion00000000000000010 (original code)So Integer.MAX_VALUE * 2 = -2You can use System.out.println (Integer.MAX_VALUE * 2).Verify it!Next, you can calculate Integer.MAX_VALUE * 10 step by step, that is, 10 Integer.MAX_VALUE additivity.The final result is as follows:11111111111111111111111111110110 (complement)11111111111111111111111111110101 (counter code)10000000000000000000000000001010 (original code)That is, -10.`

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