Understanding Number System Bases

A number system is a way of representing numbers. The 'base' of a number system tells us how many unique digits are used to represent numbers in that system. For example, the system we use every day has a base of 10, meaning it uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

The value of a digit in a number depends on its position. Each position represents a power of the base. For a number represented as $d_n d_{n-1} ... d_1 d_0$ in base $b$, its value is:

Analyzing the Options for Base 16

The question asks for the name of the number system with a base of 16.

• Option 1: Binary number system
  The binary system has a base of 2. It uses only two digits: 0 and 1. This is fundamental in digital computers.
• Option 2: Octal number system
  The octal system has a base of 8. It uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7.
• Option 3: Hexadecimal number system
  The hexadecimal system has a base of 16. It uses sixteen symbols. These symbols are the digits 0 through 9, plus the letters A, B, C, D, E, and F, representing the decimal values 10 through 15 respectively. This system is widely used in computing, particularly for memory addresses and color codes.
• Option 4: Decimal number system
  The decimal system is the one we commonly use. It has a base of 10 and uses the digits 0 through 9.

Identifying the Correct Number System

Based on the definitions:

• Binary is base 2.
• Octal is base 8.
• Decimal is base 10.
• Hexadecimal is base 16.

Therefore, the number system with base 16 is called the Hexadecimal number system.