#1**+2 **

Here's how to convert 2231_4 to base 2 without first converting to decimal:

Start with the largest power of 4 that fits into 2231:

4^3 = 64 is the largest power of 4 less than 2231.

Divide 2231 by 64 and get the quotient 34 and the remainder 3.

Write down the quotient 34 as the first digit in the base-2 representation.

Repeat for the remaining digits:

Repeat the process with the remainder 3.

The largest power of 4 that fits into 3 is 4^1 = 4.

Divide 3 by 4 and get the quotient 0 and the remainder 3.

Write down the quotient 0 as the second digit.

Repeat with the remaining 3.

The largest power of 4 that fits into 3 is 4^0 = 1.

Divide 3 by 1 and get the quotient 3 and the remainder 0.

Write down the quotient 3 as the third digit.

Combine the digits:

The digits obtained in the previous steps, in reverse order, give the base-2 representation.

Therefore, 2231_4 in base 2 is 100111_2.

This method directly uses the powers of 4 in the base-4 representation to express the number in the base-2 system, avoiding the need for intermediate decimal conversion.

bader Feb 22, 2024