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1Conceptual Review

  1. What is a bit? How many bits are in a byte? Nibble?

Solution

A bit is the smallest unit of digital information and it can be either 0 of 1. There are 4 bits in a nibble and 8 bits in a byte.

  1. What is overflow?

Solution

When the result of an arithmetic operation is outside the range of what is representable by given number of bits.

  1. What is the range of numbers representable by nn-bit unsigned, sign-magnitude, one’s complement, two’s complement, and biased notation?

Solution

  • Unsigned: [0,2n−1][0, 2^n-1]

  • Sign-Magnitude: [−(2n−1−1),2n−1−1][-(2^{n-1} - 1), 2^{n-1} - 1]

  • One’s complement: [−(2n−1−1),2n−1−1][-(2^{n-1} - 1), 2^{n-1} - 1]

  • Two’s complement: [−2n−1,2n−1−1][-2^{n-1}, 2^{n-1} - 1]

  • Bias: [0+[0+bias,2n−1+, 2^n-1+bias]]

  1. How many ways to represent zero do these representations have, nn-bit unsigned, sign-magnitude, one’s complement, two’s complement, and biased notation?

2Short Exercises

  1. True/False: Depending on the context, the same sequence of bits may represent different things.

  1. True/False: If you interpret a NN-bit Two’s complement number as an unsigned number, negative numbers would be smaller than positive numbers.

  1. True/False: We can represent fractions and decimals in our given number representation formats (unsigned, biased, and Two’s Complement).

  1. How many numbers can be represented by an unsigned, base-4, nn-digit number.
    A. 1
    B. 2n−12^n - 1
    C. 4n4^n
    D. 4n−14^{n-1}
    E. 4n−14^n - 1

  1. How many bits are needed to represent decimal number 116 in binary?

Number Representation
Summary