VLSI.4.3. Decimal Number System

Summary

This video provides an in-depth explanation of the decimal number system, focusing on its base-10 structure and how positional values work for both whole numbers and decimals. Using an example of the number 1234, the script demonstrates how each digit’s value is determined by its position relative to powers of 10. The video also covers how decimal values are calculated using negative powers of 10, breaking down a number like 12.34. The core idea is to show how the decimal system applies consistently across various number systems, making complex concepts more accessible.

Takeaways

• 😀 The decimal number system is commonly used in daily transactions and is simple for mathematical operations.
• 😀 The decimal system has a base of 10, meaning it uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• 😀 The base or radix of a number system indicates the number of digits used in that system.
• 😀 In the decimal system, the value of each digit is determined by its position in the number, starting from the right with the zeroth position.
• 😀 The value of a digit is calculated by multiplying it by the base raised to the power of its position. For example, in 1,234, 1 is in the 10^3 position, 2 is in 10^2, and so on.
• 😀 The decimal number 1,234 can be broken down as 1,000 + 200 + 30 + 4.
• 😀 The positional values in a number system are always counted from the rightmost digit, starting at zero.
• 😀 For numbers with digits after the decimal point, negative powers of 10 are used to determine their value.
• 😀 In a number like 12.34, the digits before the decimal are treated as normal, while the digits after the decimal are calculated using negative powers of 10 (10^-1 for tenths, 10^-2 for hundredths, etc.).
• 😀 For example, 12.34 is calculated as 10 + 2 + 0.3 + 0.04, resulting in 12.34.
• 😀 This method of positional values applies to all number systems, with different radices, such as binary, octal, and hexadecimal, following the same principle but with different base values.

Q & A

What is the decimal number system based on?

The decimal number system is based on the number 10, which is known as the base or radic.

What are the digits used in the decimal system?

The decimal number system uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

How is the value of a number in the decimal system determined?

The value of a number in the decimal system is determined by the position of each digit, with each position corresponding to a power of 10 (e.g., 10^0, 10^1, etc.).

What is the concept of base (radic) in a number system?

The base, or radic, of a number system represents the number of digits available in the system. In the decimal system, the base is 10.

How is the value of a number like 1234 calculated in the decimal system?

The value of 1234 is calculated by summing the products of each digit with the base raised to the power of its position: 1 × 10^3 + 2 × 10^2 + 3 × 10^1 + 4 × 10^0, which equals 1234.

How do we calculate the value of digits after the decimal point in a decimal number?

The digits after the decimal point are calculated using negative powers of 10. For example, 0.34 is calculated as 3 × 10^-1 + 4 × 10^-2.

What is the value of 10 raised to the power of a negative number?

10 raised to the power of a negative number represents the reciprocal of 10 raised to the corresponding positive number. For example, 10^-1 equals 1/10 or 0.1.

How do the positions of digits before and after the decimal point differ?

The positions of digits before the decimal point are counted from right to left starting from position 0 with positive powers of 10. The positions after the decimal point are counted starting from -1 and use negative powers of 10.

Why is 10 raised to the power of 0 equal to 1?

In mathematics, any non-zero number raised to the power of 0 is equal to 1. This is a fundamental rule in exponents.

How is a number like 12.34 represented in decimal form?

The number 12.34 is represented as 1 × 10^1 + 2 × 10^0 + 3 × 10^-1 + 4 × 10^-2, which equals 12.34 when summed.