Question

Express $1000$ using base $2$ and exponents.

• A. $1000={\left(1111101000\right)}_2$.
• B. $1000={\left(0111101000\right)}_2$.
• C. $1000={\left(1001101000\right)}_2$.
• D. $1000={\left(1111101011\right)}_2$.

Answer 1

The correct option is A $1000={\left(1111101000\right)}_2$.

Consider $1000$. We observe that $512=2^9,256=2^8,128=2^7$ and $2^6=64$. Thus your get the sum
$512+256+128+64=960$
We are still short by $40$ to reach $1000$. But $40=32+8=2^5+2^3$. Thus you obtain
$1000=2^9+2^8+2^7+2^6+2^5+2^3$
$=1\cdot 2^9+1\cdot 2^8+1\cdot 2^7+1\cdot 2^6+1\cdot 2^5+0\cdot 2^4+0\cdot 2^3+0\cdot 2^2+0\cdot 2^1+0\cdot 2^0$
You observe that this is precisely the binary representation of $1000$. We write this $1000={\left(1111101000\right)}_2$.