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
- | Hundredth | Tenth | Unit |
---|---|---|---|
- | - | 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)
=13+22 (Addition)
=35 (Answer)
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.
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 System | Binary 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 |
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
Decimal System | HexaDecimal System |
---|---|
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 SYSTEM | UNARY SYSTEM | BINARY SYSTEM | Octal SYSTEM | HEXADECIMAL 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.