-647 in binary (2's complement on 16 bits) is: 1111 1101 0111 1001

$$(-647)_{10}=(1111~1101~0111~1001)_2$$

To convert the number -647 from decimal to binary, we first convert the positive number 647 to binary, then encode the negative sign (2's complement).

1. absolute value conversion (647)

Let's divide $647$ by $2$ and note the remainder at each step:

• $647 \div 2 = 323$, remainder 1
• $323 \div 2 = 161$, remainder 1
• $161 \div 2 = 80$, remainder 1
• $80 \div 2 = 40$, remainder 0
• $40 \div 2 = 20$, remainder 0
• $20 \div 2 = 10$, remainder 0
• $10 \div 2 = 5$, remainder 0
• $5 \div 2 = 2$, remainder 1
• $2 \div 2 = 1$, remainder 0
• $1 \div 2 = 0$, remainder 1

Reading the remainders from bottom to top, the binary conversion of $647$ is :

$$(-647)_ {10} = (1010000111)_2$$

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 647 (1010000111)

• Inverted: 0101111000

Step 2 : Add 1

• 0101111000 + 1 = 0101111001

Step 3 : Complete with 1s

• Add 1's to obtain a 16-bit representation: 1111 1101 0111 1001

On 16 bits, this gives :

$$(-647)_{10}=(1111~1101~0111~1001)_2$$