Numeral System

Have you ever looked at a computer working and wondered what goes behind it? What software is used? What language forms its base? You would be surprised to know how important the role of Mathematics is in the working of a computer. Therefore Vedantu has brought Numeral System to introduce you to the number systems needed to score excellent marks in the subject in Class 11.

A numeral system (or a system of numeration) is a writing representation of numbers in a given set, using digits or other symbols in a sorted way. The same set of symbols can be used in a different numeral system to define different numbers like “10” which represents the number ten in the decimal numeral system and the number two in the binary number system. The numeral system is also known as the number system. The number which is represented by the numeral is called its value.

Ideally, a numeral system will:

• Represent a useful set of numbers.

• Give every number represented in the numeral system a unique representation.

• display the arithmetic and the algebraic construction of the number.

For example, the usual decimal numeration gives every non-zero number a unique representation, as a finite number of sets. However when decimal representation is used for a rational number, then such numbers contain an infinite number of representations for example 3.36 is also written as 3.35999, 3.36000, 3.360, etc., all of which meant the same except for some scientific and another context where greater precision is needed.

Why Numeration Systems Exist

The need for the numeration system can be described with the help of an example by illustrating the three following primary reasons: 

First, it is necessary to tell the number of items contained in a collection or set of those items. To do that, you must have some method for counting the items. The total number of items is described as the cardinal number. For example, if a basket contains 30 oranges, then 30 would be a cardinal number since it tells how many of an item there are.

Second, Numbers can also be used to represent the sequence or order of items. For example, the individual oranges in the basket could be sequenced according to how they were picked.

Finally, numbers can be used for the identification of a particular item. Some method must be devised to keep the record of savings accounts, credit card accounts, drivers' licenses, and other kinds of records for different people separated from each other.

History

The most commonly used numeral system is the Hindu-Arabic numeral system. Two Indian mathematicians are credited with developing it. Aryabhata developed the place-value notation in the 5th century and Brahmagupta introduced the symbol for zero a century later. The numeral system which is developed by the Indians slowly spread to other surrounding countries due to their commercial and military activities. Even today, the Arabs call their numeral system “Raqam Al-Hind”. The Arabs translated the Indian numeral system and spread them to the world due to their trade links. Hence the current western numeral system is the modification of the Hindu numeral system which was developed in India, which exhibits a great similarity with Sanskrit- Devanagari notation. 

The Bases of Numeration Systems

All the numeration systems have been created by taking certain numbers as the base. The base of a numeration system is the highest number that can be counted without the repetition of any previously counted number. In the decimal system used in most parts of the modern world, the base is 10. 

The base was chosen for a numeration system that often reflects actual methods of counting used by humans. For example, the decimal system may have developed because most humans have ten fingers. After all, it is an easy way to create numbers to count off one's ten fingers at a moment.

Position in Numeral System

In a positional base b numeral system, b basic symbols (or digits) corresponding to the first b natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the given formula is defined. The symbol in the last position has its value, and as it moves to the left its value is multiplied by the base.

\[Number_{b}  = [d_{N}....d_{2} d d_{0}]_{b} = \sum _{n - 0}^{N} d_{n}b^{n} = d_{0}b^{0} + d_{1}b^{1} + d_{2}b^{2} + ... +d_{N}b^{N}\]

b - numeral system base

in - the n-th digit

n -starts from the negative number if the number has a fraction part.

N+1 - the number of digits

List of Numeral Systems

Unary numeral System: One of the simplest numeral systems is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If a symbol “+” is chosen, for example, then the number four is represented as “++++”. The unary system is limited for small numbers, although it plays an important role in data compression in computer science. 

The unary numerals are also used in neural circuits.

Binary Numeral System

In Mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or the binary numeral system, which uses only two symbols: typically "0" (zero) and "1" (one). The base-2 numeral system is a positional notation with a base of 2. Each digit is referred to as a bit. The application of binary systems is in digital computing, imperial, and customary volume.

Here is an example:

\[10101_{2} = 1 \times 2^{4} + 0 \times 2^{3} + 1 \times 2^{2} + 0 \times 2^{1} + 1 \times 2^{0} = 16 + 4 + 1 = 21\]

\[10111_{2} = 1 \times 2^{4} + 0 \times 2^{3} + 1 \times 2^{2} + 1 \times 2^{1} + 1 \times 2^{0} = 16 + 4 + 2 + 1 = 23\]

The Octal Numeral System

The octal numeral system is a base 8 numeral system that consists of any of the following eight (8) symbols or digits namely 0, 1, 2, 3, 4, 5, 6, and 7. Octal numerals can be made from binary numerals by grouping binary digits following each other continuously into a group of three. Octal is used for the compact notation of binary numbers.

Here is an example:

\[30_{8} = 3 \times 8^{1} + 0 \times 8^{0} = 24 \]

\[4307_{8} = 4 \times 8^{3} + 3 \times 8^{2} + 0 \times 8^{1} + 7 \times 8^{0} = 2247 \]

Decimal Numeral System

A decimal (dec) number system is a base 10 numeral system that consists of any of the following ten (10) symbols or digits namely 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. A decimal point positioned at any place within a decimal number is used to generate fractional numbers. The decimal numeral system is generally used by modern civilization.

To represent the value 10 we have to write it by using two digits. To represent the value 100 we have to write it by using three digits, hence the use of multiple digits to represent the higher values. Each of these upcoming digits is associated with place value. Each place value is associated with a power of ten.

Here is an example:

\[2538_{10} = 2 \times 10^{3} + 5 \times 10^{2} + 3 \times 10^{1} + 8 \times 10^{0}\]

Hexadecimal Numeral System

The Hexadecimal Numeral System is a base-16 numeral system. Therefore, the 10 digits 0 to 9 and the 6 letters A to F are used in a hexadecimal numeral system. The Hexadecimal digit A is represented by the decimal number 10 and the Hexadecimal digit F is represented by the decimal number 15.

Here is an example:

\[28_{16} = 2 \times 16^{1} + 8 \times 16^{0} = 40\]

\[28_{F} = 2 \times 16^{1} + 15 \times 16^{0} = 47\]

Conclusion

The topic of Numeral Systems is of very importance in computer sciences because the whole software behind the computer systems was developed based on these numbers systems. Thus these number systems are an introduction for you to the amazing world of software. Studying it properly will enable you to grasp more concepts in the future with the utmost ease.