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

Explanation:

The Radix of a Binary Number

Definition: In numerical systems, the term "radix" refers to the base of a number system. It is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, the decimal system has a radix of 10 because it uses 10 unique digits (0 through 9).

Radix of Binary Numbers: The binary number system is a numeral system that uses only two unique digits, 0 and 1. Hence, the radix of the binary number system is 2. Binary is the foundation of all modern computing systems and digital electronics because it aligns well with the two-state nature of electronic components (on and off).

Understanding Binary Numbers:

Binary numbers represent values using two symbols: 0 and 1. Each digit in a binary number is referred to as a "bit" (binary digit). The value of each position in a binary number is a power of 2, starting from the rightmost digit (least significant bit, or LSB) and increasing to the leftmost digit (most significant bit, or MSB). For example:

• The binary number 101 represents the decimal value: (1 × 2²) + (0 × 2¹) + (1 × 2⁰) = 4 + 0 + 1 = 5.

Practical Applications:

• The binary system is the basis of computer programming and hardware design. Computers process data in binary because it is easier to implement in digital circuits, where only two states are needed: on (1) and off (0).
• Binary is also used in data transmission, coding systems, and digital communication protocols.

Correct Option Analysis:

The correct option is:

Option 3: 2

This option correctly identifies the radix of a binary number system, which is 2. Binary numbers use only two unique symbols (0 and 1), making the radix or base of the system equal to 2.

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 1: 1

This option is incorrect because a radix of 1 would mean the numeral system uses only one unique digit. Such a system is not practical for representing numbers, as it would fail to distinguish between different values.

Option 2: 0

This option is incorrect because a radix of 0 is not defined or meaningful in numeral systems. A base of 0 would imply no symbols are available for representing numbers, which is impossible.

Option 4: 8

This option is incorrect because a radix of 8 corresponds to the octal numeral system, not the binary system. The octal system uses eight unique digits (0 through 7) to represent numbers.

Option 5: No answer provided

While this option is left blank, the correct answer remains option 3, as explained above.

Conclusion:

The radix of the binary number system is 2, reflecting the two unique digits (0 and 1) used to represent numbers. Understanding the concept of radix is crucial for working with various numeral systems, including binary, decimal, octal, and hexadecimal. This knowledge is fundamental in computing, electronics, and digital communication, where binary plays a pivotal role.

Calculation: To determine the maximum binary count of a 5-bit counter, we need to understand that a binary counter with 'n' bits can count from 0 up to 2ⁿ - 1. For a 5-bit counter: n = 5 Maximum count = 2⁵ - 1 =  32 - 1 = 31 Thus, the maximum count of a 5-bit binary counter is 31 in decimal. Final Answer: The maximum binary count of a 5-bit counter is 31.

The binary number (0.1101)2 can be converted into a decimal number as follows: Starting from the binary point, we move to the right and each position's value is 1/2^n where n is the position number. So, for the given binary number (0.1101)2: 1 * (1/2^1) + 1 * (1/2^2) + 0 * (1/2^3) + 1 * (1/2^4) = 0.5 + 0.25 + 0 + 0.0625 = 0.8125 So, the decimal equivalent of (.1101)2 is 0.8125. So, the correct answer is 4) 0.8125.

The correct answer is 2KB
Important PointsIn digital storage, the smallest unit of measurement is a byte. A byte is a unit of information that can represent a single character or a small amount of data. It is comprised of 8 bits.

Given the options:

2 KB (kilobytes) - 1 kilobyte is equal to 1024 bytes. Therefore, 2 KB is equal to 2 * 1024 = 2048 bytes.

2049 Bytes - This option is stated as 2049 bytes, which is greater than 2048 bytes.

0.5 MB (megabytes) - 1 megabyte is equal to 1024 kilobytes, which is equal to 1024 * 1024 = 1,048,576 bytes.

Therefore, 0.5 MB is equal to 0.5 * 1024 * 1024 = 524,288 bytes.

3 GB (gigabytes) - 1 gigabyte is equal to 1024 megabytes,

which is equal to 1024 * 1024 * 1024 = 1,073,741,824 bytes. Therefore, 3 GB is equal to 3 * 1024 * 1024 * 1024 = 3,221,225,472 bytes.

Comparing the options, we find that 2048bytes means 2KB  is the smallest storage capacity mentioned. 

Addition of unsigned and signed number

Step 1: Convert the unsigned decimal numbers into their binary equivalent.

Step 2: Find 1's complement of the signed number.

Step 3: Find 2's complement of the signed number by adding 1 to 1's complement

Step 4: Add unsigned and signed numbers.

Step 5: If the MSB is 1, the number obtained is negative.

Calculation

16 can be represented in a binary number system as given below:

16 = 0010000

83 can be represented in a binary number system as given below:

1's complement = 0101100

2's complement = 1's complement + 1

2's complement =  0101100 + 0000001

2's complement = 0101101

16 + (−83) = 0010000 + 0101101

16 + (−83) = 111101

MSB is one, this indicates the number is negative.

Converting this 2's results into binary form

16 + (−83) = −10000112

Rules for Binary subtraction are:

1-1= 0

0-1= 1 (with borrow 1)

1-0= 1

0-0= 0

 

1 0 0 1 0 0 0 0
-  1 1 1 1 0 0 1
0 0 0 1 0 1 1 1

Step 1: 0 – 1 = Borrow to make 10 – 1 = 1.
Step 2: 1 – 0 = 1.
Step 3: 1 – 0 = 1.
Step 4: 1 – 1 = 0.
Step 5: 0 – 1 = Borrow to make 10 – 1 = 1.
Step 6: 1 – 0 = 1.
Step 7: 1 – 0 = 1.

Remember: When zero takes 1 as carry from its left side number, '0' will become '10' which is equal to '2' (2-1=1) and if that '10' further gives carry then it will become '1' not '0'.

The correct answer is 'option 2'

Concept

Divide 127 by 2. Use the integer quotient obtained in this step as the dividend for the next step. Repeat the process until the quotient becomes 0.

Solution:

Dividend	Remainder
127/2	1
63/2	1
31/2	1
15/2	1
7/2	1
3/2	1
1/2	1

Write the remainder from bottom to top i.e. in the reverse chronological order.

This will give the binary equivalent of 127. 

Therefore, the binary equivalent of decimal number 127 is 1111111.

Concept:

Decimal to binary:

• Take decimal number as dividend.

• Divide the number by 2.
• Get the integer quotient for the next iteration.

• Get the remainder (it will be either 0 or 1 because of divisor 2).
• Repeat the steps until the quotient is equal to 0
• Write the remainders in reverse order (which will be equivalent binary number of given decimal number).

Decimal to binary: (fractional part)

• Take decimal number as multiplicand.
• Multiple this number by 2 (2 is base of binary so multiplier here).
• Store the value of integer part of result in an array (it will be either 0 or 1 because of multiplier 2).
• Repeat the above two steps until the number became zero.
• Write these resultant integer part

 

Calculation:

Binary of  57:

Division	Remainder (R)
57 / 2 = 28	1
28 / 2 = 14	0
14 / 2 = 7	0
7 / 2 = 3	1
3 / 2 = 1	1
1 / 2 = 0	1

Now, write remainder from bottom to up (in reverse order), this will be 111001 which is equivalent binary number of decimal integer 57.

 

Convert decimal fractional number 0.375 into binary number.

Here, decimal fraction: 0.375

Multiplication	Resultant integer part (R)
0.375 x 2= 0.750	0
0.750 x 2 = 1.50	1
0.50 x 2= 1.00	1
0.00 x 2= 0	0

Now, write these resultant integer part, this will be 0.0110 which is equivalent binary fractional number of decimal fractional 0.375.

∴ 57.375 can be written as 111001.011 in binary

Hence, option (1) is correct.

The correct answer is 1367.

• The equivalent octal number of 759 is 1367.

Key Points

• To convert decimal number 759 to the octal following are the steps:
  • Divide 759 by 8 keeping note of the quotient and the remainder.
  • Continue dividing the quotient by 8 until you get a quotient of zero.
  • Later, write out the remainders in the reverse order to get the octal equivalent of decimal number 759.
• 759 / 8 = 94 with remainder 7
• 94 / 8 = 11 with remainder 6
• 11 / 8 = 1 with remainder 3
• 1 / 8 = 0 with remainder 1
• Hence, the number is 1367.

The decimal equivalent of the binary number 101.101 is,

= 1 × 22 + 0 × 21 + 1 × 20 + 1 × 2-1 + 0 × 2-2 + 1 × 2-3

= 4 + 0 + 1 + 0.5 + 0 + 0.125  

= 5.625

Formula:

If we covert x decimal to binary, divide x successively by 2 until the quotient is 0.

Calculation:

Divide 23 successively by 2 until the quotient is 0:

23/2 = 11, remainder is 1 ^(LSB)
11/2 = 5, remainder is 1
5/2 = 2, remainder is 1
2/2 = 1, remainder is 0
1/2 = 0, remainder is 1 _(MSB)

Read from the bottom (MSB) to top (LSB) as 10111

∴ 10111 is the binary equivalent of decimal number 23

Given:

Binary number = 10101

Calculation:

According to the question, we have

Converting the binary number to decimal number

⇒ Decimal number = 1 x 24 + 0 x 23 + 1 x 22 + 0 x 21 + 1 x 20

⇒ 16 + 0 + 4 + 0 + 1

⇒ 21

∴ 21 is the decimal equivalent of the binary number 10101

Given:

Binary number = 101010

Calculation:

According to the question, we have

Converting the binary number to decimal number

⇒ Decimal number = 1 x 25 + 0 x 24 + 1 x 2³ + 0 x 22 + 1 x 2¹ + 0 x 20

⇒ 32 + 0 + 8 + 0 + 2 + 0

⇒ 42

∴ 42 is the decimal equivalent of the binary number (101010)2

The correct answer is option 4.

Key Points

• A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression that uses only two symbols: typically "0" and "1". The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit or binary digit.
•  Decimal system, also called Hindu-Arabic number system or Arabic number system, in mathematics, a positional numeral system employing 10 as the base and requiring 10 different numerals, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It also requires a dot to represent decimal fractions.

Decimal system to binary system:

(0)₁₀=(0)₂

(1)10=(1)2

(2)10=(10)2

(3)10=(11)2

(4)10=(100)2

(5)10=(101)2

(6)10=(110)2

(7)10=(111)2

(8)10=(1000)2

(9)10=(1001)2

(10)10=(1010)2....

Hence the correct answer is two.

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,

1010101

1's complement = 0101010

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