Digital-electronics 简明教程
Signed Binary Arithmetic
在本章中，我们将讨论基本的算术运算，可以使用 2 的补码方法对任何两个带符号二进制数执行。 basic arithmetic operations 是加法和减法。
In this chapter, let us discuss about the basic arithmetic operations, which can be performed on any two signed binary numbers using 2’s complement method. The basic arithmetic operations are addition and subtraction.
Addition of two Signed Binary Numbers
考虑两个带符号二进制数 A 和 B，它们以 2 的补码形式表示。我们可以执行这两个数的 addition ，它类似于两个无符号二进制数的加法。但是，如果结果和包含符号位的进位，则丢弃（忽略）它以获得正确值。
Consider the two signed binary numbers A & B, which are represented in 2’s complement form. We can perform the addition of these two numbers, which is similar to the addition of two unsigned binary numbers. But, if the resultant sum contains carry out from sign bit, then discard (ignore) it in order to get the correct value.
如果结果和为正，则可以直接找到它的绝对值。但是，如果结果和为负，则对其取 2 的补码以获得绝对值。
If resultant sum is positive, you can find the magnitude of it directly. But, if the resultant sum is negative, then take 2’s complement of it in order to get the magnitude.
Example 1
让我们使用 2 的补码方法执行两个十进制数 +7 and +4 的 addition 。
Let us perform the addition of two decimal numbers +7 and +4 using 2’s complement method.
下面显示了 5 位表示的 2’s complement +7 和 +4。
The 2’s complement representations of +7 and +4 with 5 bits each are shown below.
\\( (+7) {10} \: = \: (00111) {2} \\)
\mathrm{(+7){10} \: = \: (00111){2}}
\\( (+4) {10} \: = \: (00100) {2} \\)
\mathrm{(+4){10} \: = \: (00100){2}}
这两个数的加法是
The addition of these two numbers is
\mathrm{(+7) {10} \: + \: (+4) {10} \: = \: (00111) {2} \: + \: (00100) {2}}
\mathrm{(+7){10} \: + \: (+4){10} \: = \: (00111){2} \: + \: (00100){2}}
\mathrm{\Rightarrow \: (+7) {10} \: + \: (+4) {10} \: = \: (01011)_{2}}
\mathrm{\Rightarrow \: (+7){10} \: + \: (+4){10} \: = \: (01011)_{2}}
结果和包含 5 位。因此，符号位没有溢出。符号位 '0' 表示结果和为 positive 。因此，和的大小在十进制数系统中为 11。所以，两个正数相加会得到另一个正数。
The resultant sum contains 5 bits. So, there is no carry out from sign bit. The sign bit ‘0’ indicates that the resultant sum is positive. So, the magnitude of sum is 11 in decimal number system. Therefore, addition of two positive numbers will give another positive number.
Example 2
让我们使用 2 的补码法对两个十进制数 -7 和 -4 执行 addition 。
Let us perform the addition of two decimal numbers -7 and -4 using 2’s complement method.
-7 和 -4 的 2’s complement 表示（每 5 位）如下所示。
The 2’s complement representation of -7 and -4 with 5 bits each are shown below.
\mathrm{(−7) {10} \: = \: (11001) {2}}
\mathrm{(−7){10} \: = \: (11001){2}}
\mathrm{(−4) {10} \: = \: (11100) {2}}
\mathrm{(−4){10} \: = \: (11100){2}}
这两个数的加法是
The addition of these two numbers is
\mathrm{(−7) {10} \: + \: (−4) {10} \: = \: (11001) {2} \: + \: (11100) {2}}
\mathrm{(−7){10} \: + \: (−4){10} \: = \: (11001){2} \: + \: (11100){2}}
\mathrm{\Rightarrow \: (−7) {10} \: + \: (−4) {10} \: = \: (110101)_{2}}
\mathrm{\Rightarrow \: (−7){10} \: + \: (−4){10} \: = \: (110101)_{2}}
结果和包含 6 位。在这种情况下，溢出是符号位导致的。因此，我们可以将其移除。
The resultant sum contains 6 bits. In this case, carry is obtained from sign bit. So, we can remove it
移除溢出后的结果和为 (−7)10 + (−4)10 = (10101)2 。
Resultant sum after removing carry is (−7)10 + (−4)10 = (10101)2.
符号位 '1' 表示结果和为 negative 。因此，对其采用 2 的补码，我们得到结果和为十进制数系统中的 11。所以，两个负数相加会得到另一个负数。
The sign bit ‘1’ indicates that the resultant sum is negative. So, by taking 2’s complement of it we will get the magnitude of resultant sum as 11 in decimal number system. Therefore, addition of two negative numbers will give another negative number.
Subtraction of two Signed Binary Numbers
考虑以 2 的补码形式表示的两个带符号二进制数 A 和 B。我们知道正数的 2 的补码会得到一个负数。因此，每当我们需要从数 A 中减去数 B 时，就采用 B 的 2 的补码并将其加到 A 中。因此， mathematically 我们可以将其写为
Consider the two signed binary numbers A & B, which are represented in 2’s complement form. We know that 2’s complement of positive number gives a negative number. So, whenever we have to subtract a number B from number A, then take 2’s complement of B and add it to A. So, mathematically we can write it as
A - B = A + (2’s complement of B)
A - B = A + (2’s complement of B)
类似地，如果我们需要从数 B 中减去数 A，就采用 A 的 2 的补码并将其加到 B 中。因此， mathematically 我们可以将其写为
Similarly, if we have to subtract the number A from number B, then take 2’s complement of A and add it to B. So, mathematically we can write it as
B - A = B + (2’s complement of A)
B - A = B + (2’s complement of A)
因此，两个带符号二进制数的减法类似于两个带符号二进制数的加法。但是，我们必须采用要减去的数的 2 的补码。这就是 2 的补码技术的 advantage 。遵循两个带符号二进制数加法的相同规则。
So, the subtraction of two signed binary numbers is similar to the addition of two signed binary numbers. But, we have to take 2’s complement of the number, which is supposed to be subtracted. This is the advantage of 2’s complement technique. Follow, the same rules of addition of two signed binary numbers.
Example 1
让我们使用 2 的补码法对两个十进制数 +7 and +4 执行 subtraction 。
Let us perform the subtraction of two decimal numbers +7 and +4 using 2’s complement method.
这两个数的减法是
The subtraction of these two numbers is
\mathrm{(+7) {10} \: − \: (+4){10} \: = \: (+7) {10} \: + \: (−4)_{10}}
\mathrm{(+7){10} \: − \: (+4){10} \: = \: (+7){10} \: + \: (−4)_{10}}
2’s complement 中用 5 位表示 +7 和 -4，如下所示。
The 2’s complement representation of +7 and -4 with 5 bits each are shown below.
\\( (+7) {10} \: = \: (00111) {2} \\)
\mathrm{(+7){10} \: = \: (00111){2}}
\mathrm{(+4) {10} \: = \: (11100) {2}}
\mathrm{(+4){10} \: = \: (11100){2}}
\mathrm{\Rightarrow \: (+7) {10} \: + \: (+4) {10} \: = \: (00111) {2} \: + \: (11100) {2} \: = \: (00011)_{2}}
\mathrm{\Rightarrow \: (+7){10} \: + \: (+4){10} \: = \: (00111){2} \: + \: (11100){2} \: = \: (00011)_{2}}
在此，从符号位获得了进位。因此，我们可以删除它。除去进位后的结果总和为
Here, the carry obtained from sign bit. So, we can remove it. The resultant sum after removing carry is
\mathrm{(+7) {10} \: + \: (+4) {10} \: = \: (00011)_{2}}
\mathrm{(+7){10} \: + \: (+4){10} \: = \: (00011)_{2}}
符号位“0”表示结果总和为 positive 。因此，在十进制数系统中的大小为 3。所以，两个十进制数 +7 和 +4 的差为 +3。
The sign bit ‘0’ indicates that the resultant sum is positive. So, the magnitude of it is 3 in decimal number system. Therefore, subtraction of two decimal numbers +7 and +4 is +3.
Example 2
我们使用 2 的补码法执行 subtraction of 两个十进制数 +4 和 +7 。
Let us perform the subtraction of two decimal numbers +4 and +7 using 2’s complement method.
这两个数的减法是
The subtraction of these two numbers is
\mathrm{(+4) {10} \: − \: (+7) {10} \: = \: (+4) {10} \: + \: (−7) {10}}
\mathrm{(+4){10} \: − \: (+7){10} \: = \: (+4){10} \: + \: (−7){10}}
2’s complement 中用 5 位表示 +4 和 -7，如下所示。
The 2’s complement representation of +4 and -7 with 5 bits each are shown below.
\\( (+4) {10} \: = \: (00100) {2} \\)
\mathrm{(+4){10} \: = \: (00100){2}}
\mathrm{(-7) {10} \: = \: (11001) {2}}
\mathrm{(-7){10} \: = \: (11001){2}}
\mathrm{\Rightarrow \: (+4) {10} \: + \: (-7) {10} \: = \: (00100) {2} \: + \: (11001) {2} \: = \: (11101)_{2}}
\mathrm{\Rightarrow \: (+4){10} \: + \: (-7){10} \: = \: (00100){2} \: + \: (11001){2} \: = \: (11101)_{2}}
在此，未从符号位得到进位。符号位“1”表示结果总和为 negative 。所以，通过计算 2 的补码，小数点后位数在十进制数系统中得到结果总和的大小为 3。因此，两个十进制数 +4 和 +7 的差为 -3。
Here, carry is not obtained from sign bit. The sign bit ‘1’ indicates that the resultant sum is negative. So, by taking 2’s complement of it we will get the magnitude of resultant sum as 3 in decimal number system. Therefore, subtraction of two decimal numbers +4 and +7 is -3.