Question

Convert the hexadecimal number directly to base 4; then convert both the original number and your answer to binary to check your result.

Please show steps and explain

Answer 1

Let us consider the decimal value 256

In order to find the hexadecimal value, we will divide the above number using base 16

The number is divided until we get the quotient as 1

Number	Remainder	Quotient
256	0	16
16	0	1

We will write the last quotient as Most Significant bit and other remainders from bottom to up

256 in decimal = (100)₁₆

Hexadecimal to Base 4 conversion

To convert Hexadecimal to base 4, we need to know both the tables.

Here is the table till decimal value 0 to 7

In Base 4, we will have the range from 0 to 3 only

Hex	Base 4
0	00
1	01
2	10
3	11

(100)₁₆ = (01 00 00)₄

Base-4 to Binary

4 can be written as 2²

Every digit in Base 4 will be written in binary using two bits

0 in base 4 = 00 in binary

1 in base 4 = 01 in binary

Therefore, (01 00 00)₄ = 00 01 00 00 00 00 = (100000000)₂

Similarly, for octal, every digit will be written in binary using three bits as 2³ =8

For Hexadecimal, every digit will be written using four bits as 2⁴ = 16

256 in binary is

Number	Remainder	Quotient
256	0	128
128	0	64
64	0	32
32	0	16
16	0	8
8	0	4
4	0	2
2	0	1

We will write the last quotient as Most Significant bit and other remainders from bottom to up

The number in binary is (100000000)₂

We have successfully got the same binary representation.

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