# Decimal Number System and Other Numbering Systems

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1.7Decimal Number System and Other Numbering Systems

In these revision notes for Decimal Number System and Other Numbering Systems, we cover the following key points:

• What is the decimal number system? Why it is called "decimal"?
• How are the numbers written in the decimal system?
• What is the decomposed form of a number in the decimal system?
• What are the place values and classes of a number?
• What is the binary system of numbers? What are the commonalities/differences with the decimal system?
• What is the hexadecimal system of numbers? What are the commonalities/differences with the hexadecimal system of numbers.
• What other symbols we use in the hexadecimal system to represent digits?
• How do we use the basic operations in various number systems?

## Decimal Number System and Other Numbering Systems Revision Notes

The decimal system is the numbering system made of 10 basic digits we use in everyday activities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9). In this system, the units change 10 by 10.

Any number N in decimal system is written in the following form:

N = abcde....n = a × 10n-1 + b × 10n-2 + ... + (n - 1) × 101 + n × 100

where n is the number of digits and a, b, c, d... . are the digits of the number. If written in this way, we say the number N is written in its decomposed form.

Digits in decimal system numbers are organized in groups of three, where each group is called a class. From right to left we have the simple class, then the thousands class, then the millions class, then the billions one and so on. Moreover, each class contains three digits, known as place values (the corresponding position is known as a placeholder).

The decimal system is not the only number system used in practice. For example, computers use the base 2 number system to operate, where only the digits 0 and 1 are used. Other systems such as the base 5 system (digits from 0 to 4 are used), and especially the hexadecimal system (digits from 0 to 9 and the letters A, B, C, D, E and F which represent the numbers 10, 11, 12, 13, 14 and 15 respectively are used).

All mathematical operations in other systems of numbers make use of the same math rules that are used in the decimal system.

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