Lecture 1B: Positional Numbering Systems

Question 1

Name the most popular Positional Numbering Systems (PNS).

Answer 1

Decimal, Binary, Octal, Hexadecimal.

Question 2

What PNS is Base-2?

Answer 2

Binary

Question 3

What PNS is Base-10?

Answer 3

Decimal

Question 4

What PNS is Base-16?

Answer 4

Hexadecimal

Question 5

What PNS is Base-8?

Answer 5

Octal

Question 6

How do you convert Decimal to Binary?

Answer 6

1. Divide decimal by 2.
2. Write down the quotient and the remainder.
3. Divide quotient by 2.
4. Write down the quotient and the remainder.
5. Repeat the process (1)-(4) until the quotient becomes zero.
6. Write down the binary number from the bottom (MSB) to top (LSB)

Question 7

What is Decimal 66 in Binary?

Answer 7

100010

Question 8

How do you convert Decimal to Hexadecimal?

Answer 8

1. Divide decimal by 16.
2. Write down the quotient and the remainder.
3. Divide quotient by 16.
4. Write down the quotient and the remainder.
5. Repeat the process (1)-(4) until the quotient becomes zero.
6. Write down the hexadecimal number from the bottom (MSB) to top (LSB)

Question 9

What is Decimal 524 in Hexadecimal?

Answer 9

20C

Question 10

How do you convert Binary to Octal?

Answer 10

1. Start from the right to make your groups.
2. Add zeros to the left of the last digit if you don’t have enough to make a set of three.
3. Write down the decimal representation of every group.

Question 11

What is Binary 101011 in Octal?

Answer 11

53

Question 12

How do you convert Binary to Hexadecimal?

Answer 12

1. Cut your string of binary numbers into groups of four, starting from the right.
2. Add extra zeros to the front of the first number if required.
3. Convert one 4-digit group at a time. To convert between binary and hexadecimal, you simply replace each hexadecimal digit with its 4-bit binary equivalent and vice versa.

Question 13

What is Binary 10001101011 in Hexadecimal?

Answer 13

46B

Question 14

How do you convert Hexadecimal or Octal to Binary?

Answer 14

• Look up each hex or octal digit to obtain the equivalent group of four binary digits.

Question 15

What is Hexadecimal 8B.2 in Binary?

Answer 15

10001011.0010

Question 16

What is Octal 317.2 in Binary?

Answer 16

11001111.01

Question 17

What are the advantages of Signed Magnitude Representation?

Answer 17

It is easy for people to understand.

Question 18

What are the disadvantages of Signed Magnitude Representation?

Answer 18

It is hard for computers to process.
It requires complicated computer hardware.
It allows two different representations of zero - positive zero and negative zero.

Question 19

Which bit in a Signed Magnitude Representation is the sign bit?

Answer 19

Most significant / leftmost bit.

Question 20

What are two possible sign bits in Signed Magnitude Representation?

Answer 20

1 for positive and 0 for negative.

Question 21

How do you represent a negative value in One’s Compliment?

Answer 21

Invert all the bits in the binary representation of the number. 1 becomes 0 and 0 becomes 1.

For example:
+3 is 00000011
-3 is 11111100

Question 22

What are the disadvantages of One’s Complement?

Answer 22

It has two different representations for zero: positive zero and negative zero.
Positive and negative integers need to be processed differently.

Question 23

How do you represent a negative number in Two’s Compliment

Answer 23

First, convert the number to one’s complement, then add 1 using binary arithmetic.
• You represent positive numbers just like unsigned numbers.
• To represent negative values, start with the corresponding positive number, invert all bits then add 1.

For example, using two 8 bit two's complement representation:
\+3 is 00000011 (One's Compliment)
-3 is 11111100 (One's Compliment)
then +1
-3 is 11111101 (Two's Compliment)

Question 24

How do you represent a negative number in Two’s Compliment

Answer 24

First, convert the number to one’s complement, then add 1 using binary arithmetic.
• You represent positive numbers just like unsigned numbers.
• To represent negative values, start with the corresponding positive number, invert all bits then add 1.

For example, using two 8 bit two's complement representation:
\+3 is 00000011 (One's Compliment)
-3 is 11111100 (One's Compliment)
then +1
-3 is 11111101 (Two's Compliment)

Question 25

How can you check if two’s complement is correct?

Answer 25

By adding the two numbers. The result has to be zero.

Note that the result of the addition must be of the n bits, where n is the number of bits in the inputs.

Question 26

How can you multiply a signed integer?

Answer 26

• Simply use an arithmetic shift operation.
• A left arithmetic shift inserts a zero in for the rightmost bit and shifts everything else left one bit; in effect, it multiplies it by 2.

Question 27

How can you divide a signed integer?

Answer 27

• Simply use an arithmetic shift operation.

* A right arithmetic shift shifts everything one bit to the right, but copies the sign bit; it divides by 2.

Question 28

How many bits are used in single and double floating-point representations?

Answer 28

Single: 32-bit
Double: 64-bit