-523 in binary (2's complement on 16 bits) is: 1111 1101 1111 0101

$$(-523)_{10}=(1111~1101~1111~0101)_2$$

To convert the number -523 from decimal to binary, we first convert the positive number 523 to binary, then encode the negative sign (2's complement).

1. absolute value conversion (523)

Let's divide $523$ by $2$ and note the remainder at each step:

• $523 \div 2 = 261$, remainder 1
• $261 \div 2 = 130$, remainder 1
• $130 \div 2 = 65$, remainder 0
• $65 \div 2 = 32$, remainder 1
• $32 \div 2 = 16$, remainder 0
• $16 \div 2 = 8$, remainder 0
• $8 \div 2 = 4$, remainder 0
• $4 \div 2 = 2$, remainder 0
• $2 \div 2 = 1$, remainder 0
• $1 \div 2 = 0$, remainder 1

Reading the remainders from bottom to top, the binary conversion of $523$ is :

$$(-523)_ {10} = (1000001011)_2$$

2. Negative sign

The negative sign is represented by the 2's complement. Invert each bit and add 1 to the result:

Step 1: invert the bits of 523 (1000001011)

• Inverted: 0111110100

Step 2 : Add 1

• 0111110100 + 1 = 0111110101

Step 3 : Complete with 1s

• Add 1's to obtain a 16-bit representation: 1111 1101 1111 0101

On 16 bits, this gives :

$$(-523)_{10}=(1111~1101~1111~0101)_2$$