VLSI.4.4.Binary Number System

Summary

This video explains the binary number system, where digits can only be `0` or `1`. It compares binary numbers to electrical switches, which have two states (on/off). The video demonstrates how binary numbers are converted to decimal using the position of each bit as powers of 2. Examples are provided, showing how to convert both whole numbers and numbers with decimal points. The process involves summing the values of the bits, with negative powers used for bits after the decimal point. This method of conversion highlights the essential relationship between binary and decimal systems in digital computing.

Takeaways

• 😀 The binary number system is based on only two digits: 0 and 1.
• 😀 Binary numbers closely relate to electrical switches, which have two states: ON/OFF or OPEN/CLOSED.
• 😀 In digital systems, binary digits can also be interpreted as logical 0 and logical 1.
• 😀 The radix (base) of the binary number system is 2, unlike the decimal system which has base 10.
• 😀 Each digit in a binary number is called a bit, and its position determines its value.
• 😀 The rightmost bit in a binary number is the least significant bit (LSB) with a power of 2⁰.
• 😀 The leftmost bit is the most significant bit (MSB) and has the highest power of 2.
• 😀 Converting a binary number to decimal involves multiplying each bit by 2 raised to its position and summing the results.
• 😀 Binary numbers with fractional parts use negative powers of 2 for digits after the decimal point.
• 😀 Digits after the binary point represent values such as 2⁻¹ (0.5), 2⁻² (0.25), and so on.
• 😀 Binary-to-decimal conversion applies the same positional value concept used in the decimal system.
• 😀 Example conversions show that binary numbers can represent both whole numbers and fractional decimal values accurately.

Q & A

What is the binary system, and how does it relate to electrical switches?

The binary system is a number system that uses only two digits: 0 and 1. It is similar to the concept of an electrical switch, which has two states: on (1) and off (0). In the binary system, these two states represent logical 0 and logical 1.

What are the two main values used in the binary system?

The two main values used in the binary system are 0 and 1. These can also be referred to as logical zero and logical one.

How is the value of a binary number determined?

The value of a binary number is determined by the position of each digit (bit), with each position representing a power of 2. Starting from the rightmost bit (the least significant bit), each position is assigned an increasing power of 2, beginning with 2^0.

What does 'most significant bit' (MSB) mean in binary numbers?

The most significant bit (MSB) is the leftmost bit in a binary number. It has the highest value and represents the largest power of 2 in the number.

How is a binary number converted into decimal?

To convert a binary number into decimal, each digit is multiplied by 2 raised to the power of its position (starting from 0 on the rightmost bit). The results are then added together to get the final decimal value.

What is the conversion of the binary number 11010110 to decimal?

The binary number 11010110 converts to decimal by calculating each bit’s position: 1 × 2^7 = 128, 1 × 2^6 = 64, 0 × 2^5 = 0, 1 × 2^4 = 16, 0 × 2^3 = 0, 1 × 2^2 = 4, 1 × 2^1 = 2, and 0 × 2^0 = 0. Adding them together: 128 + 64 + 16 + 4 + 2 = 214.

How does binary-to-decimal conversion work for numbers with digits after the decimal point?

For binary numbers with digits after the decimal point, each digit is assigned a negative power of 2. The positions after the decimal point are treated as 2^-1, 2^-2, 2^-3, and so on. Each digit is multiplied by its corresponding power of 2, and the results are added to get the decimal value.

What is the decimal equivalent of the binary number 1101.1010?

To convert the binary number 1101.1010 to decimal: Before the decimal point, 1 × 2^3 = 8, 1 × 2^2 = 4, 0 × 2^1 = 0, 1 × 2^0 = 1. After the decimal point, 1 × 2^-1 = 0.5, 0 × 2^-2 = 0, 1 × 2^-3 = 0.125. Adding these values: 8 + 4 + 0 + 1 + 0.5 + 0 + 0.125 = 13.625.

What is the importance of the position of digits in binary numbers?

The position of digits in binary numbers determines their value. Each position corresponds to a power of 2, with the rightmost bit representing 2^0, the next representing 2^1, and so on. The position of each bit significantly affects the overall value of the binary number.

Can binary numbers have digits after the decimal point, and how are they represented?

Yes, binary numbers can have digits after the decimal point. These are represented using negative powers of 2, with the first digit after the decimal point representing 2^-1, the second representing 2^-2, and so on.