1. Find the decimal representation of the binary numbers 110001 and 01010101. 2. Find the binary representation of the decimal numbers 31 and 99 . 3. Find the last 4 bits of: (a) the sum of the binary numbers 0111 and 1001 (b) the difference of the binary numbers 1001 and 0111 (c) the product of the binary numbers 0111 and 1001 4. Suppose we are given n=5 bits for storing integer numbers in the signmagnitude notation. Give the 5-bit representations of 6 and 13. 5. What numbers are represented by (i) 10000 in sign-magnitude notation, and (ii) 00000 in excess notation? 6. Suppose we are given n=6 bits for storing integer numbers in the two's complement notation. (a) Find the two's complement representation of 13. (b) Find the two's complement representation of 13. (c) Find the number represented, in the two's complement notation, by the bit pattern 111001. (d) Find the negation of the bit pattern 111001 in the two's complement notation. 7. Suppose we are given 16 bits for storing floating-point numbers, and we have decided to allocate the leading bit for the sign, the following 5 bits for the exponent, and the last 10 bits for the mantissa (half precision). (i) Find the floating-point number represented by the bit pattern 1100111100000000 . (ii) Give the half precision floating point representation of +11/4. 8. Give a proof by induction that for all integers n1,2n1=i=0n12i. 9. Let N be the set of non-negative integers and for nN define f:NN by setting f(n)=2n. Show that (i) f is a function from N to N, (ii) that f is injective, and (iii) that f is not surjective