How do you convert negative decimal numbers?

Convert the absolute value of the decimal number to binary first. Then, invert all the bits and add 1 to the result to get the two's complement representation.

How are fractional decimal values converted?

Multiply the fractional part by the target base $b$. Record the integer part of the product. Repeat with the new fractional part until it reaches 0. Read the recorded integer parts from top to bottom.

Can I convert decimal to hexadecimal by converting to binary first?

Yes. Convert the decimal to binary, then group the binary bits into sets of four starting from the right. Convert each four-bit group directly into its corresponding hexadecimal digit.

How to convert a decimal number to binary and hexadecimal

Converting a decimal integer to another base requires repeated division by the target base (2 for binary, 16 for hexadecimal). The remainders of these divisions, read in reverse order, form the converted number.

This method applies to any positive integer conversion from base-10 to an arbitrary target base.

The setup

Identify the decimal integer $N$ and the target base $b$ . For binary, $b = 2$ . For hexadecimal, $b = 16$ . Prepare a two-column workspace to record successive quotients and remainders.

The steps

Divide the decimal number $N$ by the base $b$ using integer division.
Record the integer quotient and the remainder.
Replace $N$ with the new quotient.
Repeat steps 1-3 until the quotient becomes 0.
Read the remainders from bottom to top (last remainder computed is the most significant digit).
For hexadecimal, convert any remainders from 10 to 15 into letters: $10=A, 11=B, 12=C, 13=D, 14=E, 15=F$ .

Checking the result

Multiply each digit of your result by $b^i$ , where $i$ is the zero-indexed position of the digit starting from the right. Sum these products. The total must equal your original decimal number $N$ .

Common errors

Reading the remainders top-to-bottom instead of bottom-to-top, resulting in a reversed string. Forgetting to map hexadecimal remainders $10-15$ to $A-F$ . Stopping the division process when the quotient is 1 instead of continuing until it is exactly 0.

Worked example

Convert the decimal number $254$ to binary and hexadecimal.

Binary Conversion ( $b=2$ ) $254 \div 2 = 127$ remainder $0$ $127 \div 2 = 63$ remainder $1$ $63 \div 2 = 31$ remainder $1$ $31 \div 2 = 15$ remainder $1$ $15 \div 2 = 7$ remainder $1$ $7 \div 2 = 3$ remainder $1$ $3 \div 2 = 1$ remainder $1$ $1 \div 2 = 0$ remainder $1$ Reading bottom to top, the binary representation is $11111110_{2}$ .

Hexadecimal Conversion ( $b=16$ ) $254 \div 16 = 15$ remainder $14$ $15 \div 16 = 0$ remainder $15$ Convert remainders to hex digits: $14 = E$ , $15 = F$ . Reading bottom to top, the hexadecimal representation is $FE_{16}$ .

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References: Digital Design by M. Morris Mano · Computer Organization and Design by David A. Patterson and John L. Hennessy