-474 in binary (2's complement on 16 bits) is: 1111 1110 0010 0110

$$(-474)_{10}=(1111~1110~0010~0110)_2$$

To convert the number -474 from decimal to binary, we first convert the positive number 474 to binary, then encode the negative sign (2's complement).

1. absolute value conversion (474)

Let's divide $474$ by $2$ and note the remainder at each step:

• $474 \div 2 = 237$, remainder 0
• $237 \div 2 = 118$, remainder 1
• $118 \div 2 = 59$, remainder 0
• $59 \div 2 = 29$, remainder 1
• $29 \div 2 = 14$, remainder 1
• $14 \div 2 = 7$, remainder 0
• $7 \div 2 = 3$, remainder 1
• $3 \div 2 = 1$, remainder 1
• $1 \div 2 = 0$, remainder 1

Reading the remainders from bottom to top, the binary conversion of $474$ is :

$$(-474)_ {10} = (111011010)_2$$

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 474 (111011010)

• Inverted: 000100101

Step 2 : Add 1

• 000100101 + 1 = 000100110

Step 3 : Complete with 1s

• Add 1's to obtain a 16-bit representation: 1111 1110 0010 0110

On 16 bits, this gives :

$$(-474)_{10}=(1111~1110~0010~0110)_2$$