Digital-electronics 简明教程

Binary Numbers Representation

我们可以将二进制数字分为两组 Unsigned numbers 和 Signed numbers 。阅读本章以了解如何表示带符号和不带符号的二进制数。我们还将解释如何找出带符号二进制数的 1 和 2 的补码形式。

We can categorize binary numbers into two groups Unsigned numbers and Signed numbers. Read this chapter to learn how signed and unsigned binary numbers are represented. We will also explain how you can find out the 1’s and 2’s complement form of signed binary numbers.

Unsigned Numbers

无符号数仅包含数字的大小。它们没有任何符号。这意味着所有无符号二进制数都是正数。在十进制数系统中，为了表示正数，在数字前面放置正号是可选的。因此，如果在数字前面未赋值正号，则所有正数，包括零，都可以视为无符号数。

Unsigned numbers contain only magnitude of the number. They don’t have any sign. That means all unsigned binary numbers are positive. As in decimal number system, the placing of positive sign in front of the number is optional for representing positive numbers. Therefore, all positive numbers including zero can be treated as unsigned numbers if positive sign is not assigned in front of the number.

Signed Numbers

带符号数包含数字的符号和大小。通常，符号放在数字前面。所以，我们必须考虑正数的正符号，负数的负符号。因此，如果在数字前面指派了相应符号，则所有数字都可以视为带符号数。

Signed numbers contain both sign and magnitude of the number. Generally, the sign is placed in front of number. So, we have to consider the positive sign for positive numbers and negative sign for negative numbers. Therefore, all numbers can be treated as signed numbers if the corresponding sign is assigned in front of the number.

如果符号位为零，则表示二进制数为正。同样，如果符号位为一，则表示二进制数为负。

If sign bit is zero, which indicates the binary number is positive. Similarly, if sign bit is one, which indicates the binary number is negative.

Representation of Un-Signed Binary Numbers

无符号二进制数中存在的位数容纳数字的 magnitude 。这意味着，如果无符号二进制数包含 ‘N’ 位，则所有 N 位都表示该数字的大小，因为它没有任何符号位。

The bits present in the un-signed binary number holds the magnitude of a number. That means, if the un-signed binary number contains ‘N’ bits, then all N bits represent the magnitude of the number, since it doesn’t have any sign bit.

Example

考虑 decimal number 108 。此数字的二进制等价物为 1101100 。这是无符号二进制数的表示形式。

Consider the decimal number 108. The binary equivalent of this number is 1101100. This is the representation of unsigned binary number.

(108)10 = (1101100)2

它有 7 位。这 7 位表示数字 108 的大小。

It is having 7 bits. These 7 bits represent the magnitude of the number 108.

Representation of Signed Binary Numbers

带符号二进制数字的最重要位（MSB）用于指示数字的符号。因此，它也被称为 sign bit 。正符号用符号位中的“0”表示。类似地，负符号用符号位中的“1”表示。

The Most Significant Bit (MSB) of signed binary numbers is used to indicate the sign of the numbers. Hence, it is also called as sign bit. The positive sign is represented by placing ‘0’ in the sign bit. Similarly, the negative sign is represented by placing ‘1’ in the sign bit.

如果带符号二进制数字包含“N”位，则只有（N-1）位表示数字的大小，因为一位（MSB）用于表示数字的符号。

If the signed binary number contains ‘N’ bits, then (N-1) bits only represent the magnitude of the number since one bit (MSB) is reserved for representing sign of the number.

带符号二进制数字有三种 types of representations

There are three types of representations for signed binary numbers

  1. Sign-Magnitude form

  2. 1’s complement form

  3. 2’s complement form

以这 3 种形式表示正数是相同的。但是，只有负数的表示在每种形式中才会不同。

Representation of a positive number in all these 3 forms is same. But, only the representation of negative number will differ in each form.

考虑 positive decimal number +108 。该数字大小的二进制等效值为 1101100。这 7 位表示数字 108 的大小。因为它是一个正数，所以将符号位视为零，它位于大小的最左侧。

Consider the positive decimal number +108. The binary equivalent of magnitude of this number is 1101100. These 7 bits represent the magnitude of the number 108. Since it is positive number, consider the sign bit as zero, which is placed on left most side of magnitude.

(+108)10 = (01101100)2

因此，正十进制数 +108 的 signed binary representation 为 𝟎𝟏𝟏𝟎𝟏𝟏𝟎𝟎。所以，同一表示在正十进制数 +108 的符号大小形式、1 的补码形式和 2 的补码形式中都是有效的。

Therefore, the signed binary representation of positive decimal number +108 is 𝟎𝟏𝟏𝟎𝟏𝟏𝟎𝟎. So, the same representation is valid in sign-magnitude form, 1’s complement form and 2’s complement form for positive decimal number +108.

Sign-Magnitude form

在符号大小形式中，MSB 用于表示数字的 sign ，剩余位表示数字的 magnitude 。因此，只需在无符号二进制数的最左侧放入符号位。此表示类似于带符号十进制数表示。

In sign-magnitude form, the MSB is used for representing sign of the number and the remaining bits represent the magnitude of the number. So, just include sign bit at the left most side of unsigned binary number. This representation is similar to the signed decimal numbers representation.

考虑 negative decimal number -108 。该数字的大小为 108。我们知道 108 的无符号二进制表示为 1101100。它有 7 位。所有这些位表示大小。

Consider the negative decimal number -108. The magnitude of this number is 108. We know the unsigned binary representation of 108 is 1101100. It is having 7 bits. All these bits represent the magnitude.

因为给定的数字是负数，所以将符号位视为一个，它位于大小的最左侧。

Since the given number is negative, consider the sign bit as one, which is placed on left most side of magnitude.

(−108)10 = (11101100)2

因此，-108 的符号大小表示为 11101100 。

Therefore, the sign-magnitude representation of -108 is 11101100.

1’s complement form

可以通过对带符号二进制数进行 complementing all the bits 得到一个数字的 1 的补码。因此，正数的 1 的补码得到负数。类似地，负数的 1 的补码得到正数。

The 1’s complement of a number is obtained by complementing all the bits of signed binary number. So, 1’s complement of positive number gives a negative number. Similarly, 1’s complement of negative number gives a positive number.

这意味着，如果你对包括符号位的二进制数执行两次 1 的补码，那么你将得到原来的带符号二进制数。

That means, if you perform two times 1’s complement of a binary number including sign bit, then you will get the original signed binary number.

考虑 negative decimal number -108 。该数字的大小为 108。我们知道 108 的带符号二进制表示为 01101100。

Consider the negative decimal number -108. The magnitude of this number is 108. We know the signed binary representation of 108 is 01101100.

它有 8 位。该数字的 MSB 为零，表示正数。零的补码为一，反之亦然。所以，为了得到负数，用 1 替换 0，用 0 替换 1。

It is having 8 bits. The MSB of this number is zero, which indicates positive number. Complement of zero is one and vice-versa. So, replace zeros by ones and ones by zeros in order to get the negative number.

(−108)10 = (10010011)2

因此， 1’s complement of (108)10 为 (10010011)2 。

Therefore, the 1’s complement of (108)10 is (10010011)2.

2’s complement form

可以通过对带符号二进制数进行 adding one to the 1’s complement 补码来获得一个二进制数的 2 的补码。因此，正数的 2 的补码给出了负数。类似地，负数的 2 的补码给出了正数。

The 2’s complement of a binary number is obtained by adding one to the 1’s complement of signed binary number. So, 2’s complement of positive number gives a negative number. Similarly, 2’s complement of negative number gives a positive number.

这意味着，如果你对一个包括符号位的二进制数执行两次 2 的补码，那么你将获得原来的带有符号的二进制数。

That means, if you perform two times 2’s complement of a binary number including sign bit, then you will get the original signed binary number.

考虑 negative decimal number -108 。

Consider the negative decimal number -108.

我们知道 ( 108 ) 10 的 1 的补码是 ( 10010011 ) 2

We know the 1’s complement of (108)10 is (10010011)2

(108)10 的 2 的补码 = (108)10 的 1 的补码 + 1。

2’s compliment of (108)10 = 1’s compliment of (108)10 + 1.

10010011 + 1

10010100

因此， 2’s complement of (108)10 是 (10010100)2 。

Therefore, the 2’s complement of (108)10 is (10010100)2.