Chapter 2: Bits, Data Types, and Operations

Question 1

Presence of voltage is represented as _____ and absence of voltage as _____.

Answer 1

“1” and “0” (each 0 and 1 is a “bit”)

Question 2

“bit” is short for ______.

Answer 2

binary digit

Question 3

If we are limited to 8 bits, we can differentiate at most _______ different values.

Answer 3

256 (i.e. 2^8)

Question 4

In general, with k bits, we can distinguish at most _____ distinct items.

Answer 4

2^k

Question 5

unsigned integer

Answer 5

(positive only) a data type with many uses in a computer, such as: counting, and identifying different memory locations in the computer (i.e. like an address).

Question 6

signed integers

Answer 6

negative and positive

Question 7

In binary representation, all positive integers have a leading ______.

Answer 7

0

Question 8

In binary representation, all negative integers have a leading ______.

Answer 8

1

Question 9

Three possible representations of signed integers:

Answer 9

signed magnitude, 1’s complement, 2’s complement (used on almost every computer manufactured today)

Question 10

In 2’s complement representation, the choice of representation for negative integers is based on ______.

Answer 10

the wish to keep the logic circuits as simple as possible.

i. e. with ALU, the result of adding a negative and positive integer of the magnitude will always be 0.
(i. e. since 00101 is +5, 11011 is chosen for -5).

Question 11

Basic mechanism to do addition used by almost all computers:

Answer 11

arithmetic and logic unit (ALU); has 2 inputs and 1 output.

Question 12

T/F: The carry can always be ignored when dealing with 2’s complement arithmetic

Answer 12

True

Question 13

If we have A and want to find -A, do the following:

Answer 13

1) Find the complement of A by flipping all of the bits

2) then add 1 (00001) to the complement of A

Question 14

Binary to Decimal conversion (8-bits):

Answer 14

bits are: a7 to a0
1) If a7 is 0 then the integer is positive, and we can evaluate magnitude; if a7 is 1, then negative and need to first find 2’s complement of the positive number with the same magnitude.
2) Magnitude is: a6 · 2^6 + … + a0 · 2^0
simply add the powers of 2 that have a coefficient of 1.
3) if original number was negative, affix minus sign in front

Question 15

Decimal to Binary conversion (8 bits):

Answer 15

given a decimal integer value N,
1) find the binary representation of the magnitude of N by forming the equation: N = a6 · 2^6 + … + a0 · 2^0.
Repeat the following until the left side of the equation is 0, each iteration will produce the value of one coefficient a:
a) Subtract the rightmost digit from both sides of the equation (1 if N is odd, 0 if N is even)
b) divide both sides of the equation by 2

2) if the original number N is positive, append a leading 0 sign bit
3) if the original number N is negative, append a leading 0 sign bit, then form the negative of this 2’s complement representation.

Question 16

A positive binary digit is odd if _____________ and even if ______________.

Answer 16

odd if: the rightmost digit is 1

even if: the rightmost digit is 0

Question 17

Adding a number to itself (provided there are enough bits to represent the result) is equivalent to…

Answer 17

…shifting the representation one bit position to the left.
i.e. 59 + 59 is 00111011 + 00111011 = 01110110

Question 18

T/F: In the same way that leading 0s do not affect the value of a positive number, leading 1s do not affect the value of a negative number.

Answer 18

True

Question 19

In order to add representations of different lengths, it is first necessary to _______________________. Which is referred to as ______________.

Answer 19

represent them with the same number of bits
i.e. 000001101 + 111011 should be written as 000001101 + 111111011
referred to as: Sign-EXTension (SEXT)

Question 20

Overflow

Answer 20

a carry out of the leading digit.
i.e. if adding two positive numbers causes overflow, the result will have a leading 1 as the sign bit
if adding two negative numbers causes overflow, the result will have a leading 0 as the sign bit

Question 21

T/F: Overflow - the sum of a negative number and a positive number never presents a problem

Answer 21

True
Because their sum will be a number which if positive, will have a lower magnitude (less positive) than the original positive number (because a negative number is being added to it), and vice versa.

Question 22

A logical variable can have one of two values:

Answer 22

0 (false) or 1 (true)

Question 23

Bit mask

Answer 23

a binary pattern that enables the bits to be separated into 2 parts (the part you care about and the part you want to ignore)
i.e. the bit mask 00000011 ANDed with any 8-bit pattern will be 00000000, 00000001, 00000010, or 00000011
this highlights the 2 right-most bits that are relevant.

Question 24

Binary logical functions

Answer 24

Requires 2 source operands
AND
OR
XOR (exclusive-OR :D )

Question 25

Unary logical functions

Answer 25

Operates on only 1 source operand

NOT (complement operation: output is formed by complementing (inverting) the input)

Question 26

XOR

Answer 26

Exclusive OR
output is 1 if the two source operands are different
0 if the sources are the same

Truth Table
0 XOR 0 : 0
0 XOR 1 : 1
1 XOR 0 : 1
1 XOR 1 : 0

Question 27

How to determine if 2 patterns are identical:

Answer 27

If the output of the XOR is all 0s

because the XOR function produces a 0 only if the corresponding pair of bits is identical

Question 28

Bit Vector

Answer 28

complex system made up of “busy” (0) and “available” (1) units

Question 29

Floating Point Data Type

Answer 29

Allocates some bits to the range of values, and the rest for precision (except for the sign bit).

Question 30

ASCII

Answer 30

8-bit code for transferring character code between main CPU and the input (i.e. keyboard) and output (i.e. monitor) devices.
American Standard Code for Information Interchange

Question 31

Hexadecimal Notation

Answer 31

base 16
Useful for dealing with long strings of binary digits without making errors.
Mainly just used as a convenience for humans for fewer copying errors.
Reduces the number of digits by a factor of 4.