Convert (412)₁₀ to i. Binary ii. Octal iii. Hexadecimal

To convert the decimal number 412 to binary, octal, and hexadecimal, we will use different methods for each base.

i. Conversion to Binary:

To convert a decimal number to binary, we divide the number by 2 and record the remainder. We continue dividing the quotient by 2 until we get a quotient of 0. The binary number is then read from the bottom remainder to the top.

Here's the step-by-step process for converting 412 to binary:

1. Divide 412 by 2, quotient is 206, remainder is 0.
2. Divide 206 by 2, quotient is 103, remainder is 0.
3. Divide 103 by 2, quotient is 51, remainder is 1.
4. Divide 51 by 2, quotient is 25, remainder is 1.
5. Divide 25 by 2, quotient is 12, remainder is 1.
6. Divide 12 by 2, quotient is 6, remainder is 0.
7. Divide 6 by 2, quotient is 3, remainder is 0.
8. Divide 3 by 2, quotient is 1, remainder is 1.
9. Divide 1 by 2, quotient is 0, remainder is 1.

Now, we read the remainders from bottom to top to get the binary number:

1 1 0 0 1 1 0 1 0 0

So, (412)₁₀ in binary is (1100110100)₂.

ii. Conversion to Octal:

To convert a decimal number to octal, we divide the number by 8 and record the remainder. We continue dividing the quotient by 8 until we get a quotient of 0. The octal number is then read from the bottom remainder to the top.

Here's the step-by-step process for converting 412 to octal:

1. Divide 412 by 8, quotient is 51, remainder is 4.
2. Divide 51 by 8, quotient is 6, remainder is 3.
3. Divide 6 by 8, quotient is 0, remainder is 6.

Now, we read the remainders from bottom to top to get the octal number:

6 3 4

So, (412)₁₀ in octal is (634)₈.

iii. Conversion to Hexadecimal:

To convert a decimal number to hexadecimal, we divide the number by 16 and record the remainder. We continue dividing the quotient by 16 until we get a quotient of 0. The hexadecimal number is then read from the bottom remainder to the top. In hexadecimal, we use letters A to F to represent the values 10 to 15.

Here's the step-by-step process for converting 412 to hexadecimal:

1. Divide 412 by 16, quotient is 25, remainder is 12 (which is 'C' in hexadecimal).
2. Divide 25 by 16, quotient is 1, remainder is 9.
3. Divide 1 by 16, quotient is 0, remainder is 1.

Now, we read the remainders from bottom to top to get the hexadecimal number:

1 9 C

So, (412)₁₀ in hexadecimal is (19C)₁₆.

Summary Table:

Base	Conversion Process	Result
Binary	Divide by 2	11001101002
Octal	Divide by 8	6348
Hexadecimal	Divide by 16	19C16

These are the standard methods for converting decimal numbers to other bases, and they can be applied to any decimal number to find its equivalent in binary, octal, or hexadecimal form.