**The radix of the binary number system is 2.** The binary number system is a positional numeral system with a base of 2. It uses only two digits, 0 and 1, to represent all numbers. In this system, each digit's value is determined by its position or place value within the number. The radix, also known as the base, represents the number of unique digits available in the number system.

To understand the radix of the binary number system, it is essential to have a basic understanding of positional numeral systems. In a positional numeral system, the value of a digit depends on its position relative to the other digits in the number. The radix represents the number of unique digits available to represent values in the system. For example, in the decimal system (base 10), we have 10 unique digits (0-9). In the binary system (base 2), we have only 2 unique digits (0 and 1).

The binary system is widely used in computer science and digital electronics because it directly relates to the underlying hardware architecture. Computers store and process information using electronic circuits that can represent two states, typically referred to as "on" or "off," "true" or "false," or "0" and "1." These two states correspond perfectly to the two digits used in the binary system.

In the binary system, each digit is called a bit (short for binary digit). A bit can be either 0 or 1, representing the absence or presence of a particular quantity. The value of each digit in a binary number is determined by its position or place value. The rightmost digit has a place value of 2^0, the next digit to the left has a place value of 2^1, the next digit has a place value of 2^2, and so on. Each place value is a power of 2, where the exponent corresponds to the position of the digit from the rightmost digit, starting at 0.

Let's consider an example to understand the concept further. The binary number 1011 can be broken down as follows:

1 0 1 1

2^3 2^2 2^1 2^0

To convert this binary number to decimal, we calculate the decimal equivalent of each digit by multiplying it with its corresponding place value and summing them up:

(1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0)

= 8 + 0 + 2 + 1

= 11

Therefore, the binary number 1011 is equivalent to the decimal number 11.

The binary system's radix of 2 means that there are only two unique digits available to represent values: 0 and 1. When counting in the binary system, after reaching 1, the next number is represented by adding a 1 to the leftmost digit and setting all other digits to 0. For example:

0

1

10

11

100

101

110

111

1000

...

As you can see, in the binary system, there are no digits like 2, 3, 4, 5, 6, 7, 8, or 9. All numbers are represented using only 0s and 1s, and when a digit reaches 1, it carries over to the next digit and becomes 0.

The binary system is essential in various fields, particularly in computer science and digital electronics. Computers use binary representation internally, with each bit representing the state of an electronic switch. By utilizing the binary system, computers can manipulate and process data efficiently.

In conclusion, the radix of the binary number system is 2. The binary system uses two unique digits, 0 and 1, to represent all numbers. Each digit's value is determined by its position or place value within the number, with each place value corresponding to a power of 2. Understanding the binary system and its radix is crucial in fields such as computer science and digital electronics.

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