Question

What positive integers have the following binary representations? (a) 100001 (c) 1000000000 (b) 1001001 (d) 1001110000

Answer 1

Answer and explanation:

To find : What positive integers have the following binary representations?

Solution :

In the binary number system, the weight of each digit increases by a factor of  2.

Converting binary number into a positive integer,

We have to multiply number by its place value power of 2 and added by another term.

a) Binary 100001

$100001=1\times 2^5+0\times 2^4+0\times 2^3+0\times 2^2+0\times 2^1+1\times 2^0$

$100001=32+0+0+0+0+1$

$100001=33$

i.e. $(100001)_2=(33)_{10}$

b) Binary 1001001

$1001001=1\times 2^6+0\times 2^5+0\times 2^4+1\times 2^3+0\times 2^2+0\times 2^1+1\times 2^0$

$1001001=64+0+0+8+0+0+1$

$1001001=73$

i.e. $(1001001)_2=(73)_{10}$

c) Binary 1000000000

$1000000000 =1\times 2^9+0\times 2^8+0\times 2^7+0\times 2^6+0\times 2^5+0\times 2^4+0\times 2^3+0\times 2^2+0\times 2^1+0\times 2^0$

$1000000000 =512+0+0+0+0+0+0+0+0+0$

$1000000000 =512$

i.e. $(1000000000)_2=(512)_{10}$

d) Binary 1001110000

$1001110000=1\times 2^9+0\times 2^8+0\times 2^7+1\times 2^6+1\times 2^5+1\times 2^4+0\times 2^3+0\times 2^2+0\times 2^1+0\times 2^0$

$1001110000=512+0+0+64+32+16+0+0+0+0$

$1001110000=624$

i.e. $(1001110000)_2=(624)_{10}$