Introduction to Number Systems

Summary

This lecture introduces the concept of number systems, explaining how different bases such as decimal, binary, octal, duodecimal, and hexadecimal are used to represent quantities. It details how the base (or radix) defines the number of distinct digits in a system, such as 0-9 for decimal and 0-1 for binary. The lecture also covers weighted and unweighted number systems, illustrating their differences and providing examples. Finally, it compares how the same quantity is represented across various systems, emphasizing the relationship between base size and the number of digits required.

Takeaways

• 🔢 The number system is a set of values used to represent quantities, such as binary, octal, decimal, and hexadecimal.
• 🔟 Decimal is the most commonly used number system, consisting of ten digits (0-9).
• 🔲 The base (radix) of a number system indicates how many distinct digits it contains, such as base 10 for decimal and base 2 for binary.
• ⚫ Binary has only two digits (0 and 1), known as bits, because its base is 2.
• 🛑 Octal has eight distinct digits (0-7), and hexadecimal has sixteen distinct digits (0-9 and A-F).
• 🅰️ Hexadecimal uses A to represent 10, B for 11, and so on up to F for 15.
• ⚖️ Number systems are categorized into weighted (where positions have weights, like decimal or binary) and unweighted systems (like Gray code).
• 💠 In a weighted system, a digit’s position determines its weight, as seen in decimal numbers (e.g., 7392 = 7 * 10^3 + 3 * 10^2 + 9 * 10^1 + 2 * 10^0).
• 📉 As the base of a number system increases, fewer digits are needed to represent the same quantity, as seen when comparing binary to octal and hexadecimal.
• 🟢 The relationship between bases and digits shows that a lower base (e.g., binary) requires more digits to represent a quantity compared to a higher base (e.g., hexadecimal).

Q & A

What is a number system?

A number system is a set of values used to represent quantity.

Why is the decimal number system popular?

The decimal number system is popular because it is the most common system used in daily life for measuring distance, weighing objects, and counting money.

How many different digits are there in the decimal number system?

There are ten different digits in the decimal number system, ranging from zero to nine.

What is the base of a number system and what is it also called?

The base of a number system, also called radix, indicates how many distinct digits are in the system.

What are the two different digits in the binary number system?

The two different digits in the binary number system are 0 and 1.

How are the digits in the binary number system referred to?

In the binary number system, the digits are referred to as bits.

What are the eight distinct digits in the octal number system?

The eight distinct digits in the octal number system are 0, 1, 2, 3, 4, 5, 6, and 7.

What is the representation of the numbers 10, 11, 12, 13, 14, and 15 in the duodecimal number system?

In the duodecimal number system, the numbers 10, 11, 12, 13, 14, and 15 are represented as A, B, C, D, E, and F respectively.

What is the significance of the base in determining the number of digits required to represent a quantity?

When the base increases, the number of digits required to represent a quantity decreases.

How does the number of digits required to represent a quantity change when moving from a smaller base to a larger base?

When moving from a smaller base to a larger base, the number of digits required to represent the same quantity decreases.

What are some examples of weighted and unweighted number systems?

Examples of weighted number systems include decimal, binary, and octal, while examples of unweighted systems include Gray code and Excess-3 code.

How is the number 7392 represented in terms of powers of 10?

The number 7392 can be represented as 7 multiplied by 10 raised to power 3, plus 3 multiplied by 10 raised to power 2, plus 9 multiplied by 10 raised to power 1, plus 2 multiplied by 10 raised to power 0.