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Last modified on January 29th, 2024

A hexadecimal-to-binary conversion is done to convert a hexadecimal number (base 16) to its equivalent binary number (base 2). The methods used to convert a hexadecimal number to its binary counterpart are discussed below.

The hexadecimal number system has 16 digits from 0 to 9 and A to F, which are directly represented with their corresponding binary numbers using only 4 bits.

Let us convert (1F5C)_{16} into its corresponding binary number.

**Step 1:** Grouping 1F5C into individual digits, we have 1, F, 5, and C.

**Step 2:** Now, we use the conversion table below to find each hexadecimal digit’s corresponding binary number.

Hexadecimal Number | Binary Number |
---|---|

0 | 0000 |

1 | 0001 |

2 | 0010 |

3 | 0011 |

4 | 0100 |

5 | 0101 |

6 | 0110 |

7 | 0111 |

8 | 1000 |

9 | 1001 |

A | 1010 |

B | 1011 |

C | 1100 |

D | 1101 |

E | 1110 |

F | 1111 |

By converting each hexadecimal digit of (1F5C)_{16} to its equivalent binary, we get

1 | F | 5 | C |

0001 | 1111 | 0101 | 1100 |

Here, we observe that each hex digit gives us a 4-bit binary number, so we must keep track of the leading zeros to maintain the correct number of digits while writing in the binary form.

**Step 3: **On writing the equivalent binary, we get (1F5C)_{16} = (0001111101011100)_{2} or (1111101011100)_{2}.

**For Fractional Hexadecimal Numbers**

Similarly, we can convert the fractional hexadecimal values into their corresponding binary numbers.

Converting (0.A0F)_{16} into its equivalent binary, we get

**For the Integral Part:**

0 → 0000

**For the Fractional Part:**

A → 1010, 0 → 0000, and F → 1111

Thus, (0.0A0F)_{16} = (0.101000001111)_{2}

**Convert (AB) _{16} into the binary number.**

Solution:

A → 1010

B → 1011

(AB)_{16} → (10101011)_{2}.

There is another way to represent each digit in the hexadecimal number system to its corresponding binary number without using the conversion table.

Let us convert a hexadecimal number (5C)_{16} into its corresponding binary without using the conversion table.

First, we convert 5C into a decimal number, and then decimal to the corresponding binary number.

**Step 1: Hexadecimal to Decimal**

While converting (5C)_{16} to its respective decimal number, we multiply each digit (from right to left) by the corresponding powers of 16, as shown.

Hexadecimal Value | 5 | C |

Decimal Value | 5 × 16^{1} | 12 × 16^{0} |

Now, on adding the values, we get the decimal number

(5 × 16^{1}) + (12 × 16^{0}) = 80 + 12 = 92

**Step 2: Decimal to Binary**

Now, converting (92)_{10} into its binary equivalent, we get

Thus, (5C)_{16} = (1011100)_{2}.

**Convert (64) _{16} to binary number.**

Solution:

By converting (64)_{10} into its corresponding decimal, we get

(6 × 16^{1}) + (4 × 16^{0}) = 96 + 4 = 100

Now, by converting (100)_{10} into its corresponding binary, we get

Thus, (64)_{16} = (1100100)_{2}.