Digital-electronics 简明教程

Digital Electronics - Base Conversions

在上一章中，我们解释了数字电子产品中使用的不同类型的数字系统。在本章中，我们将解释如何将一个基数的数字转换为另一个基数的数字。

In the previous chapter, we explained the different types of number systems used in digital electronics. In this chapter, we will explain how you can convert a number from one base to another.

What is Number System Conversion?

数字系统转换是将一个基数的数字转换为另一个基数的数字的过程。数字系统转换是一个重要的概念，用于以不同的形式表示信息。在数字电子产品中，主要执行以下数字系统转换。

Number system conversion is a process of converting a number from one base to another. Number system conversion is an important concept to represent information in different forms. In digital electronics, the following number system conversions are mostly performed.

Binary to Decimal Conversion
Decimal to Binary Conversion
Binary to Octal Conversion
Octal to Binary Conversion
Octal to Decimal Conversion
Decimal to Octal Conversion
Hexadecimal to Binary Conversion
Binary to Hexadecimal Conversion
Hexadecimal to Decimal Conversion
Decimal to Hexadecimal Conversion
Octal to Hexadecimal Conversion
Hexadecimal to Octal Conversion

让我们借助示例了解每一种数字系统转换。

Let us understand each of these number system conversions with the help of examples.

Binary to Decimal Conversion

我们可以使用位置权重法将二进制数转换为其等效的十进制数。

We can convert a binary number into its equivalent decimal number by using the positional weights method.

在这种将二进制转换为十进制的方法中，给定二进制数的每个数字都乘以其位置权重。然后，将所有乘积相加以获得等效的十进制数。

In this method of binary to decimal conversion, each digit of the given binary number is multiplied by its positional weight. Then, all the products are added to obtain the equivalent decimal number.

使用位置权重法将二进制数转换为其等效十进制数的分步过程如下所述−

The step-by-step process of converting a binary number to its equivalent decimal number by using positional weights method is explained below −

Step 1 − 为每个二进制数位编写位置权重。

Step 1 − Write the positional weights for each binary digit.

Step 2 − 将每个二进制数位与其位置权重相乘。

Step 2 − Multiply each binary digit with its positional weight.

Step 3 − 将乘积项相加以获得等效十进制数。

Step 3 − Add the product terms to obtain the equivalent decimal number.

让我们考虑一些示例来理解二进制到十进制的转换。

Let us consider some examples to understand the binary to decimal conversion.

Example 1

将 (101101)2 转换为十进制等价。

Convert (101101)2 into decimal equivalent.

给定的二进制数为 (101101)2

The given binary number is (101101)2

Step 1 − 为给定的二进制数定义位置权重−

Step 1 − Defining positional weights for the given binary number −

Bits	1	0	1	1	0	1
Weights	25	24	23	22	21	20

Step 2 − 计算位和位置权重的乘积−

Step 2 − Calculating product of bits and positional weights −

Bits	Weights	Multiply	Product
1	25	1 × 32	32
0	24	0 × 16	0
1	23	1 × 8	8
1	22	1 × 4	4
0	21	0 × 2	0
1	20	1 × 1	1

Step 3 − 将所有乘积项相加以获得等效的十进制数−

Step 3 − Add all the product terms to obtain the equivalent decimal number −

Decimal Number = 32 + 0 + 8 + 4 + 0 + 1 = (45)10

因此，(101101)2 的十进制等价为 (45)10。

Hence, the decimal equivalent of (101101)2 is (45)10.

Example 2

将 (1111011)2 转换为十进制等价。

Convert (1111011)2 into decimal equivalent.

将位与位置权重相乘，我们得到，

Multiplying Bits with positional weights, we get,

Decimal Number = 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20

Decimal Number = 64 + 32 + 16 + 8 + 0 + 2 + 1 = (123)10

因此，(1111011)2 的十进制等价为 (123)10。

Hence, the decimal equivalent of (1111011)2 is (123)10.

Example 3

将 (1001.11)2 转换为十进制。

Convert (1001.11)2 into decimal.

给定的二进制数有整数和小数部分。整数部分乘以正权重，而小数部分乘以负权重，如下所示−

The given binary number has integer and fractional parts. The integer part is multiplied with positive weights, while the fractional part is multiplied with negative weights as follows −

Decimal Number = 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20 + 1 × 2-1 + 1 × 2-2

Decimal Number = 8 + 0 + 0 + 1 + 0.5 + 0.25 = (9.75)10

因此，(1001.11)2 的十进制等价数为 (9.75)10。

Thus, the decimal equivalent of (1001.11)2 is (9.75)10.

Decimal to Binary Conversion

十进制数可以通过双重翻转法转换为等价的二进制数。在该方法中，将给定十进制数的整数部分连续除以 2，并将小数部分连续乘以 2。

A decimal number can be converted to their equivalent binary number by using the double-dabble method. In this method, the integer part of the given decimal number is successively divided by 2 and the fractional part is successively multiplied by 2.

在整数部分中，从下到上读取余数将给出二进制等价数的整数部分。在小数部分中，从上到下读取进位将给出二进制等价数的小数部分。

In the integer part, the remainders read from bottom to top give the integer part of the binary equivalent. In the fractional part, the carries read from top to bottom give the fractional part of the binary equivalent.

将十进制数转换为二进制等价数遵循以下步骤 −

The following steps are followed to convert a decimal number to the binary equivalent −

Step 1 − 连续将给定十进制数的整数部分除以 2，并从下到上读取余数。

Step 1 − Divide the integer part of the given decimal number successively by 2 and read the remainders from bottom to top.

Step 2 − 连续将给定十进制数的小数部分乘以 2，并从上到下读取进位。

Step 2 − Multiply the fractional part of the given decimal number successively by 2 and read the carries from top to bottom.

让我们看一些示例来理解将十进制数转换为其等价二进制数的过程。

Let us see some examples to understand the conversion of a decimal number into its equivalent binary number.

Example 1

将 (28)10 转换为二进制等价数。

Convert (28)10 to binary equivalent.

给定的十进制数是整数。因此，我们将十进制数连续除以 2 并向上读取余数以获得等效的二进制数。

The given decimal number is an integer. Thus, we divide the decimal number successively by 2 and read the remainders upwards to obtain the equivalent binary number.

	Decimal	Remainders
2	28
2	14	0
2	7	0
2	3	1
2	1	1
	0	1

从下到上读取余数，结果将为 (11100)2。这是 (28)10 的二进制等价数。

Reading the remainders from bottom to top, the result will be (11100)2. It is the binary equivalent of (28)10.

Example 2

将 (165.75)10 转换为其等价二进制数。

Convert (165.75)10 to its equivalent binary.

给定的十进制数是整数和小数部分的混合数。因此，要获得其等价的二进制数，我们将整数和小数部分分别转换。

The given decimal number is a mixed number having both integer and fractional parts. Thus, to obtain its equivalent binary number, we convert the integer and fractional parts separately.

16510 的二进制等价数如下所示：

The binary equivalent of 16510 is obtained as follows,

	Decimal	Remainders
2	165
2	82	1
2	41	0
2	20	1
2	10	0
2	5	0
2	2	1
2	1	0
	0	1

从下到上读取余数，16510 的二进制等价数为 (10100101)2。

Reading the remainders from bottom to top, the binary equivalent of 16510 is (10100101)2.

现在，让我们转换给定数字的小数部分 (0.75)。

Now, let’s convert the Fractional Part (0.75) of the given number.

要将给定的十进制分数转换为二进制数，我们将它乘以 2，如下所示：

To convert the given decimal fraction into binary, we multiply it by 2, as follows,

Decimal	Product	Carry
0.75 × 2	1.5	1
0.5 × 2	1.0	1
0 × 2	0

从上到下读取进位，结果是 0.11。因此，(0.75)10 的二进制等价数为 (0.11)2。

Reading the carries from top to bottom, the result is 0.11. Thus, the binary equivalent of (0.75)10 is (0.11)2.

因此，(165.75)10 = (10100101.11)2

Therefore, (165.75)10 = (10100101.11)2

Binary to Octal Conversion

可以通过映射方法将二进制数转换为等效八进制数。将二进制数转换为八进制等效数的步骤如下：

A binary number can be converted into its equivalent octal number by mapping method. The conversion of a binary number to the octal equivalent is done as per the following steps −

Step 1 - 从二进制小数点开始，在二进制小数点两侧分组为每组 3 位。

Step 1 − Starting from the binary point, make groups of 3-bits on both sides of the binary point.

Step 2 - 用等效的八进制数字替换每组 3 位二进制。

Step 2 − Replace each group of 3-bit binary by the equivalent octal digit.

下表显示了每组 3 位二进制的等效八进制数字：

The following table shows the equivalent octal digital for each 3-bit binary group −

Octal	Binary
	(2)2 = 4
(2)1 = 2	(2)0 = 1
0	0
0	0
1	0
0	1
2	0
1	0
3	0
1	1
4	1
0	0
5	1
0	1
6	1
1	0
7	1
1	1

让我们借助示例来理解二进制到八进制的转换。

Let us understand the binary to octal conversion with the help of examples.

Example 1

将 (110011101.110101)2 转换为其八进制等效数。

Convert (110011101.110101)2 to its octal equivalent.

二进制到八进制的转换如下执行：

The binary to octal conversion will be performed as follows −

3-bit Group	110	011	101	.	110	101
Octal Equivalent	6	3	5	.	6	5

因此，给定二进制数的八进制等效数为 (635.65)8。

Hence, the octal equivalent of given binary number is (635.65)8.

Example 2

将 (1110111011.11101)2 转换为八进制等效数。

Convert (1110111011.11101)2 to octal equivalent.

给定二进制数到八进制数的转换如下所示：

The conversion of given binary number to octal number is given below −

3-bit Group	1	110	111	011	.	111	01
	011	110	111	011	.	111	010
Octal Equivalent	3	6	7	3	.	7	2

因此，(1110111011.11101)2 的八进制等效数为 (3673.72)8。

Hence, the octal equivalent of (1110111011.11101)2 is (3673.72)8.

Octal to Binary Conversion

我们还可以使用映射方法将八进制数转换为其等效二进制数。在此方法中，我们只需用其 3 位二进制等效数替换给定八进制数的每个数字。

We can also use the mapping method to convert an octal number into its equivalent binary number. In this method, we just replace each digital of the given octal number by its 3-bit binary equivalent.

让我们借助示例来理解八进制数到二进制等效数的转换。

Let’s understand the conversion of octal number to binary equivalent with the help of examples.

Example 1

将 (3572.126)8 转换为二进制等效数。

Convert (3572.126)8 to binary equivalent.

将给定的八进制数转换为二进制等效数如下所示：

The given octal number is converted into binary equivalent as follows −

Octal Number	3	5	7	2	.	1	2	6
3-bit Binary Equivalent	011	101	111	010	.	001	010	110

因此，(3572.126)8 的二进制等效数为 (011101111010.001010110)2。

Hence, the binary equivalent of (3572.126)8 is (011101111010.001010110)2.

Example 2

将 (364.52)8 转换为其二进制等效数。

Convert (364.52)8 to its binary equivalent.

将给定八进制数转换为其等效二进制数如下所示：

The conversion of given octal number to its equivalent binary number is given below −

Octal Number	3	6	4	.	5	2
3-bit Binary Equivalent	011	110	100	.	101	010

因此，八进制数 (364.52)8 的二进制等价物为 (011110100.101010)2。

Thus, the binary equivalent of the octal number (364.52)8 is (011110100.101010)2.

Octal to Decimal Conversion

将八进制数转换为其等效十进制数的过程与将二进制数转换为十进制数的过程相同。要将八进制数转换为其十进制等价物，我们要将八进制数的每个数字乘以其权重，然后将所有乘积项相加以获得等效的十进制数。

The conversion of an octal number to its equivalent decimal number is same as the binary to decimal conversion. To convert an octal number to its decimal equivalent, we multiply each digit of the octal number by its positional weight and then add all the product terms to obtain the equivalent decimal number.

将八进制数转换为其等效十进制数的逐步过程如下：

The step-by-step procedure to convert an octal number to its equivalent decimal number is given below −

Step 1 - 为每个八进制数字编写权重。

Step 1 − Write the positional weights for each octal digit.

Step 2 - 将每个八进制数字与其权重相乘。

Step 2 − Multiply each octal digit with its positional weight.

Step 3 − 将乘积项相加以获得等效十进制数。

Step 3 − Add the product terms to obtain the equivalent decimal number.

让我们借助示例来理解八进制数到十进制数的转换。

Let us understand the conversion of octal number to decimal number with the help of examples.

Example 1

将 (356.25)8 转换为其十进制等价物。

Convert (356.25)8 to its decimal equivalent.

给定的八进制数可以转换为等效的十进制数，如下所示：

The given octal number can be converted to equivalent decimal number as follows −

Octal Digits	Positional Weights	Multiply	Product
3	(8)2	3 × (8)2	192
5	(8)1	5 × (8)1	40
6	(8)0	6 × (8)0	6
.	.	.	.
2	(8)-1	2 × (8)-1	0.25
5	(8)-2	5 × (8)-2	0.078

将所有乘积项相加以获得等效的十进制数，

Adding all the product terms to obtain the equivalent decimal number,

(356.25)8 = 192 + 40 + 6 + 0.25 + 0.078 = (238.328)10

Example 2

将 (527.322)8 转换为其十进制等价物。

Convert (527.322)8 to its decimal equivalent.

我们可以将给定的八进制数转换为其等效的十进制数，如下所示：

We can convert the given octal number to its equivalent decimal number as follows −

Octal Digits	Positional Weights	Multiply	Product
5	(8)2	5 × (8)2	320
2	(8)1	2 × (8)1	16
7	(8)0	7 × (8)0	7
.	.	.	.
3	(8)-1	3 × (8)-1	0.375
2	(8)-2	2 × (8)-2	0.0313
2	(8)-3	2 × (8)-3	0.004

将所有乘积项相加以获得结果，

Add all the product terms to obtain the result,

(527.322)8 = 320 + 16 + 7 + 0.375 + 0.0313 + 0.004 = (343.4103)10

Decimal to Octal Conversion

我们可以将混合十进制数（有整数和小数部分）转换为其等效的八进制数。为此，我们分别转换整数和小数部分。

We can convert a mixed decimal number (having integer and fractional parts) to its equivalent octal number. For this, we convert the integer and fractional parts separately.

若要将给定十进制数的整数部分转换为八进制数，我们要以 8 为基数连续除以给定的十进制数，直到商为 0。八进制等价物是通过从下至上读取余数获得的，其中最后一个余数将是最重要的数字。

To convert the integer part of the given decimal number to octal, we divide the given decimal number successively by 8 till the quotient is 0. The octal equivalent is obtained by reading the remainders from bottom to top, where the last remainder will be the most significant digit.

若要将给定十进制数的小数部分转换为八进制数，我们要以 8 为基数连续乘以给定的十进制分数，直到乘积为 0 或达到所需的精度。等效八进制数的小数部分是通过从上至下读取进位获得的。

To convert the fractional part of the given decimal number to octal, we multiply the given decimal fraction successively by 8 till the product is 0 or the desired accuracy is obtained. The fractional part of the equivalent octal number is obtained by reading the carries from top to bottom.

让我们借助示例来理解十进制到八进制的转换。

Let’s understand the decimal to octal conversion with the help of examples.

Example

将 (589.278)10 转换为八进制。

Convert (589.278)10 to octal.

给定的十进制是一个带589整数部分和0.278小数部分的混合数。因此，我们首先将整数部分转换为八进制，然后将小数部分转换为八进制。

The given decimal number is a mixed number with 589 as integer part and 0.278 as fractional part. Thus, we first convert the integer part to octal and then we convert the fractional part to octal.

将整数部分(589)10转换为八进制 -

Converting Integer Part (589)10 to Octal −

	Decimal	Remainders
8	589
8	73	5
8	9	1
8	1	1
	0	1

从下往上读出余数，(589)10的八进制等价是(1115)8。

Reading the remainders from bottom to top, the equivalent octal of (589)10 is (1115)8.

将小数部分(0.278)10转换为八进制 -

Converting the Fractional Part (0.278)10 to Octal −

Decimal	Product	Carry
0.278 × 8	2.224	2
0.224 × 8	1.792	1
0.792 × 8	6.336	6
0.336 × 8	2.688	2

从上往下读取进位以获得等价八进制数的小数部分，结果是(0.2162)8。

Reading the carries from top to bottom to obtain the fractional part of the equivalent octal number, the result is (0.2162)8.

因此，(589.278)10的等价八进制数是(1115.2162)8。

Thus, the equivalent octal number of (589.278)10 is (1115.2162)8.

Hexadecimal to Binary Conversion

我们可以使用映射方法将十六进制数转换为其二进制等价。在这个方法中，我们会用其等价的4位二进制组替换给定十六进制数中的每一个数字。

We can convert a hexadecimal number into its equivalent binary by using the mapping method. In this method, we replace each digit of the given hexadecimal number by its equivalent 4-bit binary group.

下表显示了每个十六进制数字的等价4位二进制组 -

The following table shows the equivalent 4-bit binary group of each hexadecimal digit −

Hexadecimal	Binary
	(2)3 = 8
(2)2 = 4	(2)1 = 2
(2)0 = 1	0
0	0
0	0
1	0
0	0
1	2
0	0
1	0
3	0
0	1
1	4
0	1
0	0
5	0
1	0
1	6
0	1
1	0
7	0
1	1
1	8
1	0
0	0
9	1
0	0
1	A (10)
1	0
1	0
B (11)	1
0	1
1	C (12)
1	1
0	0
D (13)	1
1	0
1	E (14)
1	1
1	0
F (15)	1
1	1

让我们通过例子来理解十六进制数转换为二进制数的过程。

Let us understand the conversion of a hexadecimal number to binary number with the help of examples.

Example 1

将(3A94.C5D)16转换为二进制等价。

Convert (3A94.C5D)16 to binary equivalent.

给定的十六进制数可以被转换为等价二进制数，如下所示 -

The given hexadecimal number can be converted into equivalent binary number as follows −

Hexadecimal Number	3	A	9	4	.	C	5	D
	3	10	9	4	.	12	5	13
4-bit Binary Equivalent	0011	1010	1001	0100	.	1100	0101	1101

因此，给定十六进制数的二进制等价是(0011101010010100.110001011101)2。

Thus, the binary equivalent of the given hexadecimal number is (0011101010010100.110001011101)2.

Example 2

将(ABD.2E)16转换为二进制等价。

Convert (ABD.2E)16 to binary equivalent.

给定十六进制数转换为其二进制的过程如下 -

The conversion of given hexadecimal number to its binary is done as follows −

Hexadecimal Number	A	B	D	.	2	E
	10	11	13	.	2	14
4-bit Binary Equivalent	1010	1011	1101	.	0010	1110

因此，(ABD.2E)16的等价二进制是(101010111101.00101110)2。

Hence, the equivalent binary of (ABD.2E)16 is (101010111101.00101110)2.

Binary to Hexadecimal Conversion

要将给定的二进制数转换为其等价十六进制数，我们在二进制小数点两侧创建每组4位的组。然后，我们用等价十六进制数字替换每组4位二进制。

To convert a given binary number to its equivalent hexadecimal number, we create groups of 4 bits each on both sides of the binary point. Then, we replace each group of 4-bit binary by the equivalent hexadecimal digit.

让我们通过例子来理解将二进制数转换为等价十六进制的过程。

Let us understand the conversion of a binary number to its equivalent hexadecimal with the help of examples.

Example 1

将(1110111001101.111011)2转换为十六进制。

Convert (1110111001101.111011)2 to hexadecimal.

将给定的二进制数转换为十六进制等价的过程如下 -

The conversion of the given binary number to hexadecimal equivalent is done as follows −

4-bit Group	1	1101	1100	1101	.	1110	11
	0001	1101	1100	1101	.	1110	1100
Hexadecimal Equivalent	1	D	C	D	.	E	C

因此，给定二进制数的十六进制数等价物为 (1DCD.EC)16。

Thus, the hexadecimal equivalent of the given binary number is (1DCD.EC)16.

Example 2

将 (110111110111.1100)2 转换为十六进制。

Convert (110111110111.1100)2 to hexadecimal.

我们可以按如下方式将给定的二进制数转换为十六进制等价物 −

We can convert the given binary number into hexadecimal equivalent as follows −

4-bit Group	1101	1111	0111	.	1100
Hexadecimal Equivalent	D	F	7	.	C

因此，(110111110111.1100)2 的十六进制等价物为(DF7.C)16。

Thus, the hexadecimal equivalent of (110111110111.1100)2 is (DF7.C)16.

Hexadecimal to Decimal Conversion

要将十六进制数转换为其等价十进制数，我们将十六进制数中的每个数字乘以其位置权重，然后将所有积项相加得到最终结果。

To convert a hexadecimal number to its equivalent decimal number, we multiply each digit in the hexadecimal number by its positional weight and then add all the product terms to obtain the final result.

将十六进制数转换为其等价十进制数的分步过程如下所述−

The step-by-step procedure to convert a hexadecimal number to its equivalent decimal number is explained below −

Step 1 − 为每个十六进制数字写出位置权重。

Step 1 − Write the positional weights for each hexadecimal digit.

Step 2 − 将每个十六进制数字与其位置权重相乘。

Step 2 − Multiply each hexadecimal digit with its positional weight.

Step 3 − 将乘积项相加以获得等效十进制数。

Step 3 − Add the product terms to obtain the equivalent decimal number.

让我们看一些示例来理解从十六进制到十进制的转换。

Let us see some examples to understand the conversion of hexadecimal to decimal number.

Example 1

将 (5AB2.8C)16 转换为十进制等价物。

Convert (5AB2.8C)16 to decimal equivalent.

将给定的十六进制数转换为其十进制数等价物的过程如下−

The conversion of the given hexadecimal number to its decimal equivalent is given below −

Hex Digits	Decimal Equiv.	Positional Weights	Multiply	Product
5	5	(16)3	5 × (16)3	20480
A	10	(16)2	10 × (16)2	2560
B	11	(16)1	11 × (16)1	176
2	2	(16)0	2 × (16)0	2
.	.	.	.	.
8	8	(16)-1	8 × (16)-1	0.5
C	12	(16)-2	12 × (16)-2	0.0468

将所有积项相加得到等价十进制数，

Add all the product terms to obtain the equivalent decimal,

(5AB2.8C)16 = 20480 + 2560 + 176 + 2 + 0.5 + 0.0468 = (23218.5468)10

Example 2

将 (1AF.2)16 转换为十进制数。

Convert (1AF.2)16 to decimal.

给定十六进制数的十进制等价物可按如下方式获取−

The decimal equivalent of the given hexadecimal number can be obtained as follows −

Hex Digits	Decimal Equiv.	Positional Weights	Multiply	Product
1	1	(16)2	1 × (16)2	256
A	10	(16)1	10 × (16)1	160
F	15	(16)0	15 × (16)0	15
.	.	.	.	.
2	2	(16)-1	2 × (16)-1	0.125

将积项相加得到等价十进制数，

Adding the product terms to obtain the equivalent decimal number,

(1AF.2)16 = 256 + 160 + 15 + 0.125 = (431.125)10

Decimal to Hexadecimal Conversion

如果给定了具有整数部分和小数部分的混合小数，那么要将给定的十进制数转换为其等价的十六进制数，我们将整数和小数部分分别进行转换。

If a mixed decimal number is given that has integer and fraction parts. Then, to convert the given decimal number to its equivalent hexadecimal, we convert integer and fraction parts separately.

要转换整数部分，我们将十进制整数连续除以 16，直到商为 0。等价十六进制数的整数部分是通过从下到上读取余数得到的。

To convert the integer part, we successively divide the decimal integer by 16 till the quotient is 0. The integer part of the equivalent hexadecimal is obtained by reading the remainders from bottom to top.

要转换小数部分，我们将小数部分乘以 16，直到乘积为 0 或达到所需的精度。等效十六进制的小数部分是通过从上到下读取进位得到的。

To convert the fractional part, we multiply the decimal fractional number by 16 till the product is 0 or till the desired accuracy is obtained. The fractional part of the equivalent hexadecimal is obtained by reading the carries from top to bottom.

我们来看一些将十进制转换为十六进制的示例，以加深理解。

Let us see some examples to understand the decimal to hexadecimal conversion.

Example

将十进制 (524.26) 转换为十六进制。

Convert (524.26)10 to hexadecimal.

给定的十进制数是混合数。因此，我们必须分别转换其整数和小数部分。

The given decimal number is a mixed number. Hence, we have to convert its integer and fractional parts separately.

将整数部分 (524) 转换为十六进制

Converting Integer Part (524)10 to Hexadecimal −

	Decimal	Remainders
16	524
16	32	12 ©
16	2	0
	0	2

从下至上读取余数以获得十六进制等价值，结果为 (20C)16。

Reading the remainder from bottom to top to obtained the hexadecimal equivalent, the result is (20C)16.

将小数部分 (0.26) 转换为十六进制

Converting Fractional Part (0.26)10 to Hexadecimal −

Decimal	Product	Carry
0.26 × 16	4.16	4
0.16 × 16	2.56	2
0.56 × 16	8.96	8
0.96 × 16	15.36	15 (F)

从上到下读取进位以获得等效的十六进制数，结果为 (0.428F)16。

Reading the carries from top to bottom to obtain the equivalent hexadecimal number, the result is (0.428F)16.

因此，十进制数 (524.26) 的十六进制等价值是 (20C.428F)16。

Thus, the hexadecimal equivalent of the decimal number (524.26)10 is (20C.428F)16.

Octal to Hexadecimal Conversion

将八进制转换为十六进制的过程非常简单。我们首先将给定的八进制数转换为二进制，然后将二进制数转换为十六进制。

The conversion of octal to hexadecimal is very simple. We first convert the given octal number to binary and then the binary number to the hexadecimal.

将给定的八进制数转换为其等效十六进制的逐步过程如下：

The step-by-step process to convert a given octal number to its equivalent hexadecimal is given below −

Step 1 - 将给定的八进制数的每一位转换为等效的 3 位二进制数。

Step 1 − Convert each digit of the given octal number to its equivalent binary of 3-bits.

Step 2 - 将获得的二进制数每 4 位分组。

Step 2 − Make groups of 4 bits each of the obtained binary number.

Step 3 - 将每个 4 位二进制组转换为其等效的十六进制。

Step 3 − Convert each 4-bit binary group to its equivalent hexadecimal.

我们来看一些将八进制转换为十六进制的示例，以加深理解。

Let us see some examples to understand the conversion of octal to hexadecimal.

Example 1

将八进制 (742.35) 转换为十六进制。

Convert (742.35)8 to hexadecimal.

给定的八进制数转换为十六进制的说明如下：

The conversion of given octal number to hexadecimal is explained below −

Octal Digits	3-bit Binary	4-bit Binary	Hex Digits
7	111	0001	1
4	100	1110	E
2	010	0010	2
.	.	.	.
3	011	0111	7
5	101	0100	4

因此，给定八进制数的十六进制等价值是 (1E2.74)16。

Thus, the hexadecimal equivalent of the given octal number is (1E2.74)16.

Example 2

将八进制 (1523.742) 转换为十六进制。

Convert (1523.742)8 to hexadecimal.

Solution

下表演示了给定的八进制转换为十六进制的过程 -

The following table demonstrates the conversion of given octal number to hexadecimal −

Octal Digits	3-bit Binary	4-bit Binary	Hex Digits
1	001	0000	0
5	101	0011	3
2	010	0101	5
3	011	0011	3
.	.	.	.
7	111	1111	F
4	100	0001	1
2	010	0000	0

因此，给定八进制数的十六进制等价数为 (353.F1)16。

Hence, the hexadecimal equivalent of the given octal number is (353.F1)16.

Hexadecimal to Octal Conversion

十六进制到八进制转换可以按照上面解释的八进制到十六进制转换的方式进行。要将给定的十六进制数转换为八进制数，我们首先将给定的十六进制数转换为二进制数，然后将二进制数转换为八进制数。

The hexadecimal to octal conversion can be perform in the same way as the octal to hexadecimal as explained above. To convert a given hexadecimal number to octal number, we first convert the given hexadecimal number to binary and then the binary number to the octal.

将十六进制数转换为其等效八进制数的分步程序如下所述 -

The step-by-step procedure to convert a hexadecimal number to its equivalent octal number is explained below −

Step 1 - 将每个十六进制数字转换为其等效的 4 位二进制数。

Step 1 − Convert each hexadecimal digit to its equivalent 4-bit binary.

Step 2 - 对获得的二进制数进行分组，每组 3 位。

Step 2 − Make groups of 3-bits each of the obtained binary number.

Step 3 - 将每个 3 位二进制组转换为其等效的八进制数。

Step 3 − Convert each 3-bit binary group to its equivalent octal number.

以下示例演示了将给定的十六进制数转换为其等效八进制数的方法。

The following examples demonstrate the method of converting a given hexadecimal number to its equivalent octal number.

Example 1

将 (B3A9.5F)16 转换为八进制。

Convert (B3A9.5F)16 to octal.

将给定的十六进制数转换为其等效八进制数的过程如下所述 -

The conversion of the given hexadecimal number to its equivalent octal number is explained below −

Hex Digits			B	3	A	9	.	5	F
4-bit Binary			1011	0011	1010	1001	.	0101	1111
3-bit Binary	001	011	001	110	101	001	.	010	111	110
Octal Digits	1	3	1	6	5	1	.	2	7	6

因此，给定十六进制数的八进制等价数为 (131651.276)8。

Thus, the octal equivalent of the given hexadecimal number is (131651.276)8.

Example 2

将 (AC.F)16 转换为八进制。

Convert (AC.F)16 to octal.

将给定的十六进制数转换为其等效八进制数的过程如下所示 -

The conversion of given hexadecimal number to its equivalent octal number is demonstrated below −

Hex Digits		A	C	.	F
		10	12	.	15
4-bit Binary		1010	1100	.	1111
3-bit Binary	010	101	100	.	111	100
Octal Digits	2	5	4	.	7	4

因此，给定十六进制数的八进制等价数为 (254.74)8。

Hence, the octal equivalent of the given hexadecimal number is (254.74)8.

Conclusion

我们可以将给定的数字从一个进制转换为另一个进制。数字系统转换是数字电子领域中的一项重要操作，因为它允许使用不同的格式表示相同的信息。在本章中，我们借助示例解释了所有可能的数字系统转换。

We can convert a given number from one base to another. Number system conversion is one the important operations in the field of digital electronics, as it allows to represent the same information in different formats. In this chapter, we explained all the possible number system conversions with the help of examples.