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# 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.

- Division of integer part and successive quotients with base 2.
- Multiplication of fractional part and successive fractions with base 2.

**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.

- Division of integer part and successive quotients with base 16.
- Multiplication of fractional part and successive fractions with base 16.

**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)_{16}

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 × 2^{3}) + (1 × 2^{2}) + (0 × 2^{1}) + (1 × 2^{0}) + (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 2^{3} = 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 2^{4} = 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 × 8^{2}) + (4 × 8^{1}) + (5 × 8^{0}) + (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.

- Convert octal number into its equivalent binary number.
- Convert the above binary 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 65.4*C*.

## 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 × 16^{2}) + (10 × 16^{1}) + (5 × 16^{0}) + (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.

- Convert Hexa-decimal number into its equivalent binary number.
- Convert the above binary 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)_{𝟖}

Therefore, the **octal equivalent** of Hexa-decimal number 65.4*C* is 145.23.