 # Practice No. System Questions and Answers

NUMBER AND NUMBER SYSTEM(S)

A number is an arithmetic value that is denoted through words, symbols, or figures. The system(s) employed for the representation of these numbers is known as the number system. You may expect questions from the number system topic in your Quantitative Aptitude or Reasoning sections of the paper in exams for Bank PO, NABARD Grade A & B, SBI-PO, IBPS SO, RBI, Railways and SSC.

Several number systems such as the Decimal number system, Roman number system, Ionian number system and the Abjad number systems have been developed and modified through the centuries depending on their need and application. These number systems are used in a variety of applications such as counting, coding, measurement, etc. Let us study further to understand these number or numeral systems in detail.

TYPES OF NUMBER SYSTEMS

• DECIMAL NUMBER SYSTEM (Base 10)
Developed by Aryabhata in the 5th century BCE, the Decimal number system is the most widely employed in numerical representations. Also known as the Indo-Arabic Numeral system or Base 10 number system, it consists of integers and non-integer digits from 0 through 9 and a combination of these digits is used to express a numerical value. The value of the number is based on the digits combined to put it together. Based on the principle of place-value notation, the value of figures increases from left to right as units, tens, thousands and so on where a figure’s place value is achieved by multiplying the previous place with 10. For example, in the number 53, 3 is in the unit’s place and 5 is in the tens place. To achieve the hundredth value, the 53 needs to be multiplied by 10 to become 530. Now, 5 is in the hundredth place, 3 in the tens’ and 0 in the unit’s place.

-HundredthTenthUnit
-  - 5 3
x  - 1 0
= 5 3 0

Let us look at a more complex calculation for more clarity.

Solve using BODMAS rule;

22 - 54 ÷ 6 + 11 × 2 (Divide)

=22-9+11x2 (Multiplication)

=22- 9+ 22 (Subtraction)

• UNARY NUMBER SYSTEM

The simplest system of categorizing numbers is the Unary number system wherein a symbol is chosen and repeated several times over to convey its value. Thus, in this positional notational system, the number ‘N’ is depicted by repeating the symbol N number of times. Let us try to understand with an example. If ‘l’ is used to indicate the number 1, the number 5 will be denoted as lllll. It is interesting to note, this system does not utilize 0 in any form. Basic operations like addition and subtraction can be done easily but multiplication and division are inefficient. It is most commonly used while tallying marks. While the Unary system is used in some computer applications such as in Golomb coding, due to its overly-simplified nature the scope of its application is limited.

• BINARY NUMBER SYSTEM (Base 2)

As the name suggests, the Binary number system makes use of two numerical figures only, i.e., 0 and 1. A combination of these 2 numbers is exercised to demonstrate any numerical value. For example, as 0 and 1 are written to denote their values, 2 can be written as 10, 3 as 11, 4 as 100 and such. Also known as the Base 2 number system, it is used to express binary quantities and is also extensively applied in digital electronics.

Decimal SystemBinary System

0

0000

1

0001

2

0010

3

0011

4

0100

5

0101

6

0110

7

0111

8

1000

9

1001

10

1010

11

1011

12

1100

13

1101

14

1110

15

1111

• OCTAL NUMBER SYSTEM (Base 8)

Using the numbers from 0 through 7, the octal number system is used in a range of computer applications. All numbers have the base power of 8 and are depicted with an 8 suffix just as binary numbers are indicted with a base of 2. For example, 348, 798 and 38 are examples of octal numbers. In this system, every place has a power of eight.
For example, 78248= 1x78+1x88+ 1x28+ 1x48+ 1x88

It must be noted that numbers beyond the 7 (from other number systems) such as 68, 93, or 5394B are not octal numbers. Since it has fewer symbols, it has the advantage of not requiring extra symbols such as in the Hexadecimal system (see below) for depicting values and thus considerably reduces errors in computation.

Consisting of 16 (hex) base numbers, this method uses a combination of denoted numerical figures and alphabets. The numbers 0 through 9 are expressed using numerical figures whereas the numbers 10, 11, 12, 13, 14 and 15 are represented as A, B, C, D, E and F respectively.
8732911 = 8540EF

(8732911)10 = (8540EF)16

0

0

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

A

11

B

12

C

13

D

14

E

15

F

This positional numeral system provides computer programmers and system designers with more representational values which not significantly reduces the chances for errors but also makes programming work easy. With a variety of numbers and alphabets available, codes can be written faster and more easily as compared to the Binary system.

DECIMAL SYSTEMUNARY SYSTEMBINARY SYSTEMOctal SYSTEMHEXADECIMAL SYSTEM

0

-

0000

0

0

1

l

0001

1

1

2

ll

0010

2

2

3

lll

0011

3

3

4

llll

0100

4

4

5

lllll

0101

5

5

6

llllll

0110

6

6

7

lllllll

0111

7

7

8

llllllll

1000

10

8

9

lllllllll

1001

11

9

10

llllllllll

1010

12

A

11

lllllllllll

1011

13

B

12

llllllllllll

1100

14

C

13

lllllllllllll

1101

15

D

14

llllllllllllll

1110

16

E

15

llllllllllllllll

1111

17

F

There are and have been several other number systems developed by societies in the past as well which have varying degrees of utilization now. For example, the Sexagesimal or Babylonian Cuneiform system was developed in ancient Sumerian culture and is in use even today but in a much-modified form. The ancient Egyptian method used hieroglyphic symbols based on multiples of 10. The ancient Roman number system was entirely based on Latin alphabets for indication. In modern times, only 7 of these symbols are still utilized with fixed numerical values.

NUMBER SYSTEM CONVERSION

As we have studied above, numbers or arithmetic values may be indicated using any numeral system depending on the need for its application. However, when the need for its application changes, the number system used for representations may very well be changed from one system to another.

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