Question

What is the value of the least significant bit in the bit patterns represented by the following hexadecimal notations? a. \(0 x 9 \mathrm{~A}\) b. \(0 \times 90\) c. \(0 \mathrm{x} 1 \mathrm{~B}\) d. \(0 x 6 E\)

Short answer

a. 0; b. 0; c. 1; d. 0.

Step-by-step solution

Convert Hexadecimal to Binary for a

To determine the least significant bit (LSB) of the hexadecimal number \(0x9A\), first convert the hexadecimal to binary. \(9\) translates to \(1001\) and \(A\) (which is \(10\) in decimal) translates to \(1010\). Therefore, \(0x9A\) is \(10011010\) in binary.

Identify the LSB for a

The LSB is the rightmost bit in the binary representation from Step 1. For \(10011010\), the LSB is \(0\).

Convert Hexadecimal to Binary for b

Convert \(0x90\) to binary. \(9\) translates to \(1001\) and \(0\) is \(0000\), so \(0x90\) is \(10010000\) in binary.

Identify the LSB for b

The LSB is the rightmost bit of \(10010000\). Therefore, the LSB is \(0\).

Convert Hexadecimal to Binary for c

Convert \(0x1B\) to binary. \(1\) translates to \(0001\) and \(B\) (which is \(11\) in decimal) translates to \(1011\), making \(0x1B\) equal \(00011011\) in binary.

Identify the LSB for c

The LSB is the rightmost bit of \(00011011\). Therefore, the LSB is \(1\).

Convert Hexadecimal to Binary for d

Convert \(0x6E\) to binary. \(6\) translates to \(0110\) and \(E\) (which is \(14\) in decimal) translates to \(1110\), so \(0x6E\) is \(01101110\) in binary.

Identify the LSB for d

The LSB is the rightmost bit of \(01101110\). Therefore, the LSB is \(0\).

Key concepts

Least Significant Bit (LSB)

The least significant bit, often abbreviated as LSB, is the bit position in a binary number that holds the smallest value. It is the rightmost bit in a binary sequence. In essence, it represents the value of either 0 or 1, depending on whether the position is occupied or vacant by a binary digit.
This bit plays a crucial role because, during conversions and calculations, it can determine the parity or the uniqueness of data. Let's say you have the binary number 1011; here the LSB is '1'. If the LSB changes, it can completely change the binary number's decimal (or hexadecimal) equivalent.

• In simple terms, think of the LSB as the smallest part of a domino effect in binary operations.
• Changing the LSB in the binary system can affect the odd/even status of the number in decimal form.

Hexadecimal Notation

Hexadecimal notation, identified by the prefix '0x', is a base-16 number system used in computing and digital electronics. It includes numbers from 0 to 9 and letters A to F, where A stands for 10 and F stands for 15 in decimal.
This notation is particularly useful because it offers a more human-friendly way to represent binary-coded values. Where a binary sequence becomes unreadable due to its length, hexadecimal condenses it into fewer characters without losing any information.

• Each digit in hexadecimal corresponds to a 4-bit binary sequence.
• It simplifies the representation of large binary numbers, making it ideal for memory addresses and color codes.

For instance, the hexadecimal number 'A3' translates to '10100011' in binary. This shows how each hex digit directly aligns with a binary group of four.

Binary Representation

Binary representation is the foundation of digital computing systems. In this system, every number is expressed using a combination of 0s and 1s, also known as 'bits'.
The binary system is a base-2 number system, which means each position in the sequence represents a power of 2, starting from 2^0 at the least significant bit.

• Each bit in a binary number increases the value it represents exponentially based on its position.
• Binary sequences are fundamental as they directly map to electronic states: on or off.

Binary numbers provide an efficient way for computers to perform calculations and store information. It's important since all data on computers - text, images, or videos - is eventually broken down into binary form. For conversions, understanding the binary sequence can help easily decipher the LSB or the values during hexadecimal to binary transitions.