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

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[tex]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[/tex][tex]100001=32+0+0+0+0+1[/tex][tex]100001=33[/tex]i.e. [tex](100001)_2=(33)_{10}[/tex]b) Binary 1001001[tex]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[/tex][tex]1001001=64+0+0+8+0+0+1[/tex][tex]1001001=73[/tex]i.e. [tex](1001001)_2=(73)_{10}[/tex]c) Binary 1000000000 [tex]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[/tex][tex]1000000000 =512+0+0+0+0+0+0+0+0+0[/tex][tex]1000000000 =512[/tex]i.e. [tex](1000000000)_2=(512)_{10}[/tex]d) Binary 1001110000[tex]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[/tex][tex]1001110000=512+0+0+64+32+16+0+0+0+0[/tex][tex]1001110000=624[/tex]i.e. [tex](1001110000)_2=(624)_{10}[/tex]