(both negative and non-negative) and real numbers as sequences of 0s and 1s. Int

ID: 3172283 • Letter: #

Question

(both negative and non-negative) and real numbers as sequences of 0s and 1s. Integers can be stored exactly (using binary numbers and/or 2’s

complement), but real numbers must be approximated by rational numbers. In decimal (base 10), a rational number can be expanded as, for example,

That is,

11 = 1.375. 8

11 = 1×100 +3×101 +7×102 +5×103. 8

In a similar fashion, we can expand a number in binary,

11 =···#×20 +#×21 +#×22 +#×23 +··· 8

= 1×20 +0×21 +1×22 +1×23.

Thus, we can write 11/8 as 1.011 in binary; thereby converting it to a sequence of 0s and 1s (and a

decimal—binary?—point).

(a) Is 0.01 the binary expansion of 1/2? Explain why or why not. (b) Determine the binary expansion of 25/4.

Remember that a rational number is a number that can be written as a ratio of integers, such as 1/2, 3/4, and 5/1.

(a) 0.01 = 0*0 + 0*2^-1 +1*2^-2 = 1/4

Hence 0.01 is not the binary expansion of 1.2

(b) Binary explansion of 25/4 = 6.25 = 1*2^2+1*2^1 +0*2^0 + 0*2^-1 + 1*2^-2= 110.01

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