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* 2^5+0* 2^4+0* 2^3+0* 2^2+0* 2^1+1* 2^0`

`100001=32+0+0+0+0+1`

`100001=33`

i.e.
`(100001)_2=(33)_(10)`

b) Binary 1001001

`1001001=1* 2^6+0* 2^5+0* 2^4+1* 2^3+0* 2^2+0* 2^1+1* 2^0`

`1001001=64+0+0+8+0+0+1`

`1001001=73`

i.e.
`(1001001)_2=(73)_(10)`

c) Binary 1000000000

`1000000000 =1* 2^9+0* 2^8+0* 2^7+0* 2^6+0* 2^5+0* 2^4+0* 2^3+0* 2^2+0* 2^1+0* 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* 2^9+0* 2^8+0* 2^7+1* 2^6+1* 2^5+1* 2^4+0* 2^3+0* 2^2+0* 2^1+0* 2^0`

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

`1001110000=624`

i.e.
`(1001110000)_2=(624)_(10)`