ED2. Sistemi di Numerazione

Summary

This tutorial explores the fundamentals of numeral systems, focusing on the decimal, binary, and hexadecimal systems. It explains how numbers are represented using positional values and bases: 10 for decimal, 2 for binary, and 16 for hexadecimal. Through examples, the video highlights how the significance of each digit changes based on its position in the number, as well as the conversion of binary numbers to hexadecimal. The tutorial offers a clear understanding of how these systems function and their practical applications, especially in digital electronics and computing.

Takeaways

• 😀 A number system is a set of symbols and rules that help represent numbers.
• 😀 The decimal system uses 10 symbols (0-9) and is a positional system where the value of a symbol depends on its position.
• 😀 In the decimal system, each digit can be represented as a power of 10, with the least significant digit on the right.
• 😀 The binary system is based on 2 symbols (0 and 1) and is also a positional system with a base of 2.
• 😀 With binary, the number of bits determines how many unique values can be represented (e.g., 2 bits represent 4 values).
• 😀 The hexadecimal system uses 16 symbols (0-9 and A-F) and is commonly used in computing for easier representation of binary data.
• 😀 Each hexadecimal digit corresponds to four binary digits (bits), making binary data more compact.
• 😀 The decimal number 123 can be broken down as 1 * 10^2 + 2 * 10^1 + 3 * 10^0.
• 😀 In binary, increasing the number of bits increases the number of values that can be represented (e.g., 3 bits = 8 values).
• 😀 The hexadecimal system is a base-16 system and is particularly useful in digital electronics and computing.
• 😀 The tutorial hints at future discussions on how to convert between different number systems, such as from binary to hexadecimal.

Q & A

What is a numeral system?

A numeral system is a set of symbols and rules used to represent numbers. It consists of digits, rules for their representation, and operations that define how numbers are manipulated within the system.

What are the three main elements that define a numeral system?

The three main elements of a numeral system are: the digits (symbols used to represent numbers), the rules for their representation, and the relationships that govern operations like addition, subtraction, multiplication, etc.

Why is the decimal system called 'positional'?

The decimal system is called positional because the value of each digit in a number depends on its position in the number. The rightmost digit represents the least significant value, and each digit’s value increases as we move leftward, based on powers of 10.

How is the number 123 represented in terms of powers of 10?

The number 123 can be represented as 1 * 10² + 2 * 10¹ + 3 * 10⁰, where each digit is multiplied by 10 raised to the power corresponding to its position from right to left.

What makes the binary system different from the decimal system?

The binary system is based on two symbols, 0 and 1, as opposed to the decimal system's 10 symbols. Like the decimal system, it is also positional, meaning each bit's value depends on its position, but it uses powers of 2 instead of powers of 10.

How many values can be represented with 3 bits in the binary system?

With 3 bits, you can represent 8 different values, ranging from 000 to 111 in binary, which equals 0 to 7 in decimal.

What is the relationship between binary and hexadecimal systems?

The hexadecimal system is based on 16 symbols (0-9 and A-F) and is directly related to the binary system. Each hexadecimal digit represents exactly four binary digits (bits), making it easier to represent long binary numbers in a more compact form.

Why is the hexadecimal system often used in electronics and computing?

The hexadecimal system is commonly used in electronics and computing because it is more compact than binary for representing large values. Since one hexadecimal digit corresponds to four binary digits, it simplifies the reading and writing of binary data.

How do you convert the binary number '11100110' to hexadecimal?

To convert '11100110' from binary to hexadecimal, group the binary digits into chunks of four (from right to left). '1110' is E, and '0110' is 6, so the hexadecimal representation is E6.

What is the significance of base 2 and base 16 in numeral systems?

Base 2 (binary) is fundamental to digital electronics and computing because computers operate using binary logic. Base 16 (hexadecimal) is used to simplify the representation of binary data, with each hexadecimal digit representing four binary bits.