Digital Circuits - Base Conversions
In previous chapter, we have seen the four prominent number systems. In this chapter, let us convert the numbers from one number system to the other in order to find the equivalent value.
Decimal Number to other Bases Conversion
If the decimal number contains both integer part and fractional part, then convert both the parts of decimal number into other base individually. Follow these steps for converting the decimal number into its equivalent number of any base ‘r’.
· Do division of integer part of decimal number and successive quotients with base ‘r’ and note down the remainders till the quotient is zero. Consider the remainders in reverse order to get the integer part of equivalent number of base ‘r’. That means, first and last remainders denote the least significant digit and most significant digit respectively.
· Do multiplication of fractional part of decimal number and successive fractions with base ‘r’ and note down the carry till the result is zero or the desired number of equivalent digits is obtained. Consider the normal sequence of carry in order to get the fractional part of equivalent number of base ‘r’.
Decimal to Binary Conversion
The following two types of operations take place, while converting decimal number into its equivalent binary number.
Example
Consider the decimal number 58.25. Here, the integer part is 58 and fractional part is 0.25.
Step 1 − Division of 58 and successive quotients with base 2.
Operation | Quotient | Remainder |
58/2 | 29 | 0 (LSB) |
29/2 | 14 | 1 |
14/2 | 7 | 0 |
7/2 | 3 | 1 |
3/2 | 1 | 1 |
1/2 | 0 | 1(MSB) |
⇒(58)10 = (111010)2.
Therefore, the integer part of equivalent binary number is 111010.
Step 2 − Multiplication of 0.25 and successive fractions with base 2.
Operation | Result | Carry |
0.25 x 2 | 0.5 | 0 |
0.5 x 2 | 1.0 | 1 |
- | 0.0 | - |
⇒(.25)10 = (.01)2
Therefore, the fractional part of equivalent binary number is .01
⇒(58.25)10 = (111010.01)2
Therefore, the binary equivalent of decimal number 58.25 is 111010.01.
Decimal to Octal Conversion
The following two types of operations take place, while converting decimal number into its equivalent octal number.
· Division of integer part and successive quotients with base 8.
· Multiplication of fractional part and successive fractions with base 8.
Example
Consider the decimal number 58.25. Here, the integer part is 58 and fractional part is 0.25.
Step 1 − Division of 58 and successive quotients with base 8.
Operation | Quotient | Remainder |
58/8 | 7 | 2 |
7/8 | 0 | 7 |
⇒(58)10 = (72)8
Therefore, the integer part of equivalent octal number is 72.
Step 2 − Multiplication of 0.25 and successive fractions with base 8.
Operation | Result | Carry |
0.25 x 8 | 2.00 | 2 |
- | 0.00 | - |
⇒ (.25)10 = (.2)8
Therefore, the fractional part of equivalent octal number is .2
⇒ (58.25)10 = (72.2)8
Therefore, the octal equivalent of decimal number 58.25 is 72.2.
Decimal to Hexa-Decimal Conversion
The following two types of operations take place, while converting decimal number into its equivalent hexa-decimal number.
Example
Consider the decimal number 58.25. Here, the integer part is 58 and decimal part is 0.25.
Step 1 − Division of 58 and successive quotients with base 16.
Operation | Quotient | Remainder |
58/16 | 3 | 10=A |
3/16 | 0 | 3 |
⇒ (58)10 = (3A)16
Therefore, the integer part of equivalent Hexa-decimal number is 3A.
Step 2 − Multiplication of 0.25 and successive fractions with base 16.
Operation | Result | Carry |
0.25 x 16 | 4.00 | 4 |
- | 0.00 | - |
⇒(.25)10 = (.4)6
Therefore, the fractional part of equivalent Hexa-decimal number is .4.
⇒(58.25)10 = (3A.4)16
Therefore, the Hexa-decimal equivalent of decimal number 58.25 is 3A.4.
Binary Number to other Bases Conversion
The process of converting a number from binary to decimal is different to the process of converting a binary number to other bases. Now, let us discuss about the conversion of a binary number to decimal, octal and Hexa-decimal number systems one by one.
Binary to Decimal Conversion
For converting a binary number into its equivalent decimal number, first multiply the bits of binary number with the respective positional weights and then add all those products.
Example
Consider the binary number 1101.11.
Mathematically, we can write it as
(1101.11)2 = (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20) + (1 × 2-1) +
(1 × 2-2)
⇒ (1101.11)2 = 8 + 4 + 0 + 1 + 0.5 + 0.25 = 13.75
⇒ (1101.11)2 = (13.75)10
Therefore, the decimal equivalent of binary number 1101.11 is 13.75.
Binary to Octal Conversion
We know that the bases of binary and octal number systems are 2 and 8 respectively. Three bits of binary number is equivalent to one octal digit, since 23 = 8.
Follow these two steps for converting a binary number into its equivalent octal number.
· Start from the binary point and make the groups of 3 bits on both sides of binary point. If one or two bits are less while making the group of 3 bits, then include required number of zeros on extreme sides.
· Write the octal digits corresponding to each group of 3 bits.
Example
Consider the binary number 101110.01101.
Step 1 − Make the groups of 3 bits on both sides of binary point.
101 110.011 01
Here, on right side of binary point, the last group is having only 2 bits. So, include one zero on extreme side in order to make it as group of 3 bits.
⇒ 101 110.011 010
Step 2 − Write the octal digits corresponding to each group of 3 bits.
⇒ (101 110.011 010)2 = (56.32)8
Therefore, the octal equivalent of binary number 101110.01101 is 56.32.
Binary to Hexa-Decimal Conversion
We know that the bases of binary and Hexa-decimal number systems are 2 and 16 respectively. Four bits of binary number is equivalent to one Hexa-decimal digit, since 24 = 16.
Follow these two steps for converting a binary number into its equivalent Hexa-decimal number.
· Start from the binary point and make the groups of 4 bits on both sides of binary point. If some bits are less while making the group of 4 bits, then include required number of zeros on extreme sides.
· Write the Hexa-decimal digits corresponding to each group of 4 bits.
Example
Consider the binary number 101110.01101
Step 1 − Make the groups of 4 bits on both sides of binary point.
10 1110.0110 1
Here, the first group is having only 2 bits. So, include two zeros on extreme side in order to make it as group of 4 bits. Similarly, include three zeros on extreme side in order to make the last group also as group of 4 bits.
⇒ 0010 1110.0110 1000
Step 2 − Write the Hexa-decimal digits corresponding to each group of 4 bits.
⇒ (0010 1110.0110 1000)2 = (2E.68)16
Therefore, the Hexa-decimal equivalent of binary number 101110.01101 is (2E.68).
Octal Number to other Bases Conversion
The process of converting a number from octal to decimal is different to the process of converting an octal number to other bases. Now, let us discuss about the conversion of an octal number to decimal, binary and Hexa-decimal number systems one by one.
Octal to Decimal Conversion
For converting an octal number into its equivalent decimal number, first multiply the digits of octal number with the respective positional weights and then add all those products.
Example
Consider the octal number 145.23.
Mathematically, we can write it as
(145.23)8 = (1 × 82) + (4 × 81) + (5 × 80) + (2 × 8-1) + (3 × 8-2)
⇒ (145.23)8 = 64 + 32 + 5 + 0.25 + 0.05 = 101.3
⇒ (145.23)8 = (101.3)10
Therefore, the decimal equivalent of octal number 145.23 is 101.3.
Octal to Binary Conversion
The process of converting an octal number to an equivalent binary number is just opposite to that of binary to octal conversion. By representing each octal digit with 3 bits, we will get the equivalent binary number.
Example
Consider the octal number 145.23.
Represent each octal digit with 3 bits.
(145.23)8 = (001 100 101.010 011)2
The value doesn’t change by removing the zeros, which are on the extreme side.
⇒ (145.23)8 = (1100101.010011)2
Therefore, the binary equivalent of octal number 145.23 is 1100101.010011.
Octal to Hexa-Decimal Conversion
Follow these two steps for converting an octal number into its equivalent Hexa-decimal number.
Example
Consider the octal number 145.23
In previous example, we got the binary equivalent of octal number 145.23 as 1100101.010011.
By following the procedure of binary to Hexa-decimal conversion, we will get
(1100101.010011)2 = (65.4C)16
⇒(145.23)8 = (65.4C)16
Therefore, the Hexa-decimal equivalent of octal number 145.23 is 65.4C.
Hexa-Decimal Number to other Bases Conversion
The process of converting a number from Hexa-decimal to decimal is different to the process of converting Hexa-decimal number into other bases. Now, let us discuss about the conversion of Hexa-decimal number to decimal, binary and octal number systems one by one.
Hexa-Decimal to Decimal Conversion
For converting Hexa-decimal number into its equivalent decimal number, first multiply the digits of Hexa-decimal number with the respective positional weights and then add all those products.
Example
Consider the Hexa-decimal number 1A5.2
Mathematically, we can write it as
(1A5.2)16 = (1 × 162) + (10 × 161) + (5 × 160) + (2 × 16-1)
⇒ (1A5.2)16 = 256 + 160 + 5 + 0.125 = 421.125
⇒ (1A5.2)16 = (421.125)10
Therefore, the decimal equivalent of Hexa-decimal number 1A5.2 is 421.125.
Hexa-Decimal to Binary Conversion
The process of converting Hexa-decimal number into its equivalent binary number is just opposite to that of binary to Hexa-decimal conversion. By representing each Hexa-decimal digit with 4 bits, we will get the equivalent binary number.
Example
Consider the Hexa-decimal number 65.4C
Represent each Hexa-decimal digit with 4 bits.
(65.4C)6 = (0110 0101.0100 1100)2
The value doesn’t change by removing the zeros, which are at two extreme sides.
⇒ (65.4C)16 = (1100101.010011)2
Therefore, the binary equivalent of Hexa-decimal number 65.4C is 1100101.010011.
Hexa-Decimal to Octal Conversion
Follow these two steps for converting Hexa-decimal number into its equivalent octal number.
Example
Consider the Hexa-decimal number 65.4C
In previous example, we got the binary equivalent of Hexa-decimal number 65.4C as 1100101.010011.
By following the procedure of binary to octal conversion, we will get
(1100101.010011)2 = (145.23)8
⇒(65.4C)16 = (145.23)ퟖ
Therefore, the octal equivalent of Hexa-decimal number 65.4C is 145.23.
Binary Numbers Representation
We can make the binary numbers into the following two groups − Unsigned numbers and Signed numbers.
Unsigned Numbers
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
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.
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
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
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
It is having 7 bits. These 7 bits represent the magnitude of the number 108.
Representation of Signed Binary Numbers
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.
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.
There are three types of representations for signed binary numbers
Representation of a positive number in all these 3 forms is same. But, only the representation of negative number will differ in each form.
Example
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
Therefore, the signed binary representation of positive decimal number +108 is ퟎퟏퟏퟎퟏퟏퟎퟎ. 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
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.
Example
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.
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
Therefore, the sign-magnitude representation of -108 is 11101100.
1’s complement form
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.
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.
Example
Consider the negative decimal number -108. The magnitude of this number is 108. We know the signed binary representation of 108 is 01101100.
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
Therefore, the 1’s complement of (108)10 is (10010011)2.
2’s complement form
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.
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.
Example
Consider the negative decimal number -108.
We know the 1’s complement of (108)10 is (10010011)2
2’s compliment of (108)10 = 1’s compliment of (108)10 + 1.
= 10010011 + 1
= 10010100
Therefore, the 2’s complement of (108)10 is (10010100)2.