Number Representation MCQ Quiz - Objective Question with Answer for Number Representation - Download Free PDF

The correct answer is Gray code.

Key Points

• The Gray code, also referred to as the reflected binary code, is a binary numeral system in which two consecutive numbers differ in only one bit. The unique property of Gray code is that each transition from one value to the next value involves changing only one bit.
• This system was invented by Frank Gray at Bell Labs to prevent spurious output from electromechanical switches. For instance, while switching from one position to another in standard binary code, there is a risk that switches will change at different times leading to invalid numbers, but with Gray code, since only one bit changes at a time, such misinterpretations are avoided.
• So, Gray code is also known as reflected binary code because the sequence of binary values reflects it about its midpoint. For example:
• Binary: 000, 001, 010, 011, 100, 101, 110, 111
• If we reflect this sequence, i.e., reverse it, we get:111, 110, 101, 100, 011, 010, 001, 000

• Now let's take the first half of the original sequence and the first half of the reflected sequence.

• By inverting the bits in the second half, we get the sequence of Gray codes:  000, 001, 011, 010, 110, 111, 101, 100

So, every binary number has a unique Gray code, and vice versa, attributing to why the term "reflected binary code" is used to describe the Gray code.

2's Complement - It is a type of mathematical and logical (binary) representation that helps in representing signed numbers and performing arithmetic operations such as subtraction, addition, etc.

To perform 2's complement of (1000)₂ we will perform the following steps -

1. We will perform 1's complement on (1000)₂ by flipping 1s to 0s and 0s to 1s.
   (1000)₂ ===> (0111)₂
2. Now we will add 1 to the resultant value, that is, (0111)2.
   (0111)₂ + (1)₂ ===> (1000)₂
3. Hence, we get (1000)2 back after 2's complement.

The correct answer is 171661

 Key Points

• To find the octal equivalent of a hexadecimal number, you can convert each hexadecimal digit to its binary equivalent and then group the binary digits into groups of three (since each octal digit represents three binary digits).
• Let's convert each hexadecimal digit of (F3B1)16 to binary:
  • F = 1111
  • 3 = 0011
  • B = 1011
  • 1 = 0001
• Now group the binary digits into sets of three:
  • 1111 0011 1011 0001
• Now convert each set of three binary digits to octal:
  • 001 111 001 110 110 001
• Combine these octal digits: 171661.

Therefore, the octal equivalent of (F3B1)16 is option 3) 171661.

The correct answer is 171661

 Key Points

• To find the octal equivalent of a hexadecimal number, you can convert each hexadecimal digit to its binary equivalent and then group the binary digits into groups of three (since each octal digit represents three binary digits).
• Let's convert each hexadecimal digit of (F3B1)16 to binary:
  • F = 1111
  • 3 = 0011
  • B = 1011
  • 1 = 0001
• Now group the binary digits into sets of three:
  • 1111 0011 1011 0001
• Now convert each set of three binary digits to octal:
  • 001 111 001 110 110 001
• Combine these octal digits: 171661.

Therefore, the octal equivalent of (F3B1)16 is option 3) 171661.

Key Points

•  In a 2's complement number system, the range of integers that can be represented depends on the number of bits, n.
• The most significant bit (MSB) in a 2's complement representation indicates the sign of the number.
• If the MSB is 0, the number is positive or zero. If the MSB is 1, the number is negative.
• The range of integers that can be represented with n bits in 2's complement is from -2^n-1 to 2^n-1 - 1.
• This range allows for an equal number of positive and negative values, with one additional value for negative numbers.

Important Points

•  For example, with 8 bits (n=8), the range is from -128 to 127.
• Understanding the range of 2's complement numbers is crucial for tasks involving binary arithmetic and computer architecture.

Additional Information

•  2's complement representation simplifies the implementation of arithmetic operations in computer systems.
• It allows for the use of the same addition and subtraction circuits for both signed and unsigned integers.
• It is the standard method for representing signed integers in modern computing systems.

The correct answer is 11000110

Key Points

• To convert the hexadecimal number C6 to a binary number, you can convert each hexadecimal digit to its 4-bit binary representation.
• C in hexadecimal is 12 in decimal, which is 1100 in binary.
• 6 in hexadecimal is 6 in decimal, which is 0110 in binary.
• So, the binary representation of C6 is 11000110.

Additional InformationHere are the decimal numbers 1 to 15 represented in both hexadecimal and binary forms:

•  Decimal 1: Hexadecimal 1, Binary 0001
• Decimal 2: Hexadecimal 2, Binary 0010
• Decimal 3: Hexadecimal 3, Binary 0011
• Decimal 4: Hexadecimal 4, Binary 0100
• Decimal 5: Hexadecimal 5, Binary 0101
• Decimal 6: Hexadecimal 6, Binary 0110
• Decimal 7: Hexadecimal 7, Binary 0111
• Decimal 8: Hexadecimal 8, Binary 1000
• Decimal 9: Hexadecimal 9, Binary 1001
• Decimal 10: Hexadecimal A, Binary 1010
• Decimal 11: Hexadecimal B, Binary 1011
• Decimal 12: Hexadecimal C, Binary 1100
• Decimal 13: Hexadecimal D, Binary 1101
• Decimal 14: Hexadecimal E, Binary 1110
• Decimal 15: Hexadecimal F, Binary 1111 

Calculation:

14 in binary form is represented as:

1410 = (00001110)2

Taking the 1's complement of the above, we get 11110001

Adding 1 to the 1's complement, we get the 2's complement representation of the number, i.e. 11110010

Since there is a 1 in the MSB, the number is a negative number with value -14.

∴ The 2's complement of -6410 contains 7 bits.

The correct answer is (11010)2 = (62)8

Key Points

Binary numbers and octal numbers are both used in computing. They are different ways of representing the same value - just like how "10" and "ten" are different ways of expressing the same quantity in decimal.

• Each digit of an octal number represents three binary digits because 2³ = 8. Here's the mapping:
  • "000" => "0"
  • "001" => "1"
  • "010" => "2"
  • "011" => "3"
  • "100" => "4"
  • "101" => "5"
  • "110" => "6"
  • "111" => "7"
• Now let's convert the binary numbers to their equivalent octal numbers.
  • (111 110 111)₂ = (7 6 7)₈
  • (110 110 101)₂ = (6 6 5)₈
  • (10 101 . 110)₂ = (2 5 . 6)₈
  • (11 010)₂ = (3 2)₈ - Corrupted as the corresponding octal number should be (32)₈ instead of (62)₈.

Therefore, the 4th pair, (11010)₂ = (62)₈, is not equal.

28₁₀ = (11100)₂ = (0000 0000 0001 1100)₂

-28₁₀ = 2’s complement of 0000 0000 0001 1100

2’s complement of 0000 0000 0001 1100 = 1111 1111 1110 0100

Note:

Tricks to find: 2’s complement

Concept:

1's complement of Binary: 1's complement of a Binary number is defined by the value obtained by inverting all the bit, i.e, 0 as 1 and 1 as 0.

∴ 1's complement of 1100 0110 = 0011 1001

2's complement of Binary: It is the sum of 1's complement of Binary number and 1 to the least significant bit (LSB).

∴ 2's complement = 1's complement + 1 (LSB)

Calculation:

Given Binary Number,

0011 0101 1001 1100

1's complement = 1100 1010 0110 0011

2's complement = 1's complement + 1 (LSB)

 Alternate Method

Note: A shortcut method of forming the 2's complement of a binary number is to copy bits from the right until a one-bit has been copied, then invert the remaining bits i.e, 0 as 1 and 1 as 0.

The correct answer is 171661

 Key Points

• To find the octal equivalent of a hexadecimal number, you can convert each hexadecimal digit to its binary equivalent and then group the binary digits into groups of three (since each octal digit represents three binary digits).
• Let's convert each hexadecimal digit of (F3B1)16 to binary:
  • F = 1111
  • 3 = 0011
  • B = 1011
  • 1 = 0001
• Now group the binary digits into sets of three:
  • 1111 0011 1011 0001
• Now convert each set of three binary digits to octal:
  • 001 111 001 110 110 001
• Combine these octal digits: 171661.

Therefore, the octal equivalent of (F3B1)16 is option 3) 171661.

The correct answer is 0, 1

Key Points

• In binary, we’re only allowed two symbols: 0 and 1. But using those two symbols we can create any number that a decimal system can.
• Example: 0, 1, 10, 11, 100 ....
• The base of each number system is also called the radix.
• The radix of a decimal number is ten, and the radix of binary is two.
• The radix determines how many different symbols are required in order to flesh out a number system.
• In our decimal number system, we’ve got 10 numeral representations for values between nothing and ten somethings: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
• Each of those symbols represents a very specific, standardized value.

Find the 2s complement of (-19)₁₀

Step 1 :Convert given decimal number into binary

(19)₁₀ = (010011)₂

Step 2: Take 1s complement of the binary number by converting 0 with 1 and vice-versa

1s complement : (101100)₂

Step 3: Add 1 to 1s complement

101100 + 1 = 101101

Hence correct option is "4"

Important PointsTo find 2s complement quickly

Move from the right of the binary number to left , keep bits till first "1" as it is then complement each bit.

ex : (19)₂ = 010011

2s complement : 101101

Concept:

The range of n bit 1’s complement form is from:

–( 2n-1 -1) to + (2n-1 -1)

The range of n bit 2's complement is:

-2(n - 1) to 2(n - 1) - 1.

Calculation:

For 5 bits, The range of 1’s complement form is

– (2n-1 -1) to + (2n-1 -1)

-15 to 15

Concept:

32-bit floating-point representation of a binary number in IEEE- 754 is

Calculation:

Given binary number is

00111110011011010000000000000000

Here, sign bit is 0. So, number is positive.

Exponent bits = E = 01111100 = 124 (in decimal)

Mantissa bits M = 11011010000000000000000

In IEEE-754 format, 32-bit (single precision)

(-1)^s × 1.M × 2^{E – 127}

= (-1)⁰ × 1.1101101 × 2^{124 – 127}

= 1.1101101 × 2^-3

= (1 + 2^-1 + 2^-2 + 2^-4 + 2^-5 + 2^-7) × 2-3