Number System Questions With Solutions

What is Number System?

Number systems are mathematical expressions of numbers in various ways. A number is a mathematical value for arithmetic calculations, item measurement, and counting. There are many classifications of numbers, including whole numbers, and rational and irrational numbers, among others. The binary, octal, decimal, and hexadecimal number systems are just a few examples of the multiple types of number systems with various characteristics.

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Types of Number Systems

In general, there are several types of number systems that do exist, but the most commonly applicable ones are namely: 

• Decimal Number System
• Binary Number System
• Octal Number System
• Hexadecimal Number System

Every number system will have a unique base or radix, which will account for the number of unique digits in the given Number System, including digit zero.

Decimal Number System

This is the most used number system. Also known as the Hindu-Arabic number system or Arabic number system.

• The radix or base of this system is 10. It constitutes the numbers 0,1,2,3,4,5,6,7,8 and 9.

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Binary Number System

The radix or base of this number system is 2 as it consists of two digits, 0 and 1. For a given binary number each digit is known as a bit. Every binary number will consist of a sequence of bits, each of which is either 0 or 1.

The binary point is the point of separation between integer and fraction parts of the given binary number. Each bit carries weight in accordance with its position from the binary point. The decimal value of the above binary number can be evaluated as given below.

Example: The binary numbers can be converted to a decimal number using the following method. 

Octal Number System

The radix or base of this number system is 8, consisting of eight digits 0,1,2,3,4,5,6,7. The octal to the decimal mapping of the given number can be done as shown below. 

 

Example: 

Hexadecimal Number System

The radix or base of this number system is 16.It consists of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. The hexadecimal to the decimal conversion of the given number can be done as shown below.

Example:

Number System Questions

With this basic knowledge, we cannot solve all the questions from number systems, we need a deep emphasis to do so, but here we focus on discussing some sample questions of number systems. This can help understand how questions are asked in various exams like GATE and PSU exams.

Q1) What is the decimal equivalent of a binary number 110110? 

1. 66
2. 48
3. 54
4. 69

Solution:

Given binary number 110110

The decimal equivalent of the given number is

={[(0×2⁰)+(1×2¹)+(1×2²)+(0×2³)+(1×2⁴)+(1×2⁵)]}

=0+2+4+0+16+32=0+2+4+0+16+32

=54

Q.2) 2’s complement representation of a 16-bit number (one sign bit and 15 magnitude bits) is FFFF Its magnitude in decimal representation is

1. 0
2. 1
3. 32,767
4. 65.535

Solution:

FFFF= 1111 1111 1111 1111

2’s complement of the above binary number = 0000 0000 0000 0001

=1

Hence the answer is an option (b)

Q.3) X, Y, and Z are the decimal integers corresponding to the 4-bit binary number 1010 considered in signed magnitude, 1’s complement, and 2’s complement representations, respectively. The 6-bit 2’s complement representation of (X+ Y + Z) is

1. 111101
2. 110101
3. 110010
4. 110011

Solution:

The given binary number is 1010.

X= signed magnitude= -2

Y= 1’s complement of 1010= 0101=-5

Z=2’s complement of 1010= 0110=-6

X+Y+Z = -2-5-6 = -13

X+Y+Z in 6 digit = 001101

Its 1’s complement= 110010

2’s complement=110011.

Q.4) A new Binary Coded Pentary (BCP) number system is proposed in which every digit of a base-5 number is represented by its corresponding 3-bit binary code. For example, the base-5 number 24 will be represented by its BCP code 010100. In this numbering system, the BCP code 100010011001 corresponds to the following number in the base-5 system. 

1. 423
2. 1324
3. 2201
4. 4231

Solution:

Given BCP code = 100010011001 = 100 010 011 001 = 4231.

Q.5) What is the hexadecimal equivalent of a binary number 110010100010101?

1. AD23
2. CA15
3. 12AD
4. FAB1

Solution: 

Given number 1100101000010101 = 1100 1010 0001 0101 = CA15.

Q.6) The range of signed decimal numbers that can be represented by 6-bit 1’s complement number is

1. -31 to +31
2. -63 to +63
3. -64 to +63
4. -32 to +31

Solution:

The range of signed decimal numbers that can be represented by n-bit 1’s complement number is = −(2ⁿ⁻¹−1)to+(2ⁿ⁻¹−1)

Given n=6

⇒ −(2⁵−1) to(2⁵−1)

= -31 to + 31.

Hence Option (a) is correct.

Q.7) Decimal 108 in hexadecimal and BCD number system is respectively 

1. 6C, 000100001000
2. FC, 100010100011
3. 9D, 110111100001
4. E2, 100111001011

Solution:

The given decimal number is 108.

⇒ Hexadecimal equivalent

16	108
	6	C
	0	6

Hexadecimal equivalent = 6C

BCD equivalent: (108)₁₀= (000100001000)_BCD

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