Units

Objectives

• The bit is a fundamental unit of information (either 0 or 1)
• A byte is a group of 8 bits
• Explain the least significant bit and most significant bit in a binary number
• Know how to identify whether a binary number is odd or even
• Calculate the range of values for a given number of bits
• Know the common names for multiples of bytes

In the last section we saw the binary digit, or bit, is the fundamental unit of information for digital systems, it is either a \(0\) or a \(1\). One bit can have two possible values: 1 or 0.

Bits are described in groups of 8 bits, known as a byte (which as we'll see later gives enough space to score a single character like 'h' or 'W').

MSB and LSB

In binary notation, MSB (Most Significant Bit) and LSB (Least Significant Bit) refer to the positions of the bits within a binary number, indicating their relative importance in terms of value.

Most Significant Bit (MSB):

• The MSB is the leftmost bit in a binary representation.
• It holds the highest value in the binary place value system.
• Changing the MSB can significantly affect the overall value of the binary number.
• In an 8-bit binary number, the MSB is the bit in the 128's place (2^7).

Least Significant Bit (LSB):

• The LSB is the rightmost bit in a binary representation.
• It holds the lowest value in the binary place value system.
• Changing the LSB has the smallest impact on the overall value of the binary number.
• In an 8-bit binary number, the LSB is the bit in the 1's place (2^0).

Here's an example of an 8-bit binary number with the MSB and LSB labeled:

MSB                                LSB
 ↓                                  ↓
 1    0    1    1    0    0    1    0
2^7  2^6  2^5  2^4  2^3  2^2  2^1  2^0

In this example, the leftmost bit (1) is the MSB, and the rightmost bit (0) is the LSB. The MSB and LSB are essential concepts when working with binary numbers, especially in contexts where the relative importance of bits in a number is significant, such as in digital systems and data encoding.

Odd or Even

For binary numbers the LSB determines whether a number is odd or even. If the LSB is 1 the number is odd, if the LSB is 0 the number is even. For example:

• \(00000101_2\): The LSB is a \(1\) so it must be an odd number. (It's the decimal value \(5_{10}\).)
• \(01010100_2\): The LSB is a \(0\) so it must be an even number. (It's the decimal value \(84_{10}\).)

Note

Although not technically required get into the habit of writing out ALL binary numbers in multiples of 8, as bytes, with leading zeros.

Ranges of values

Note:

• One bit can have two distinct values (\(2^1\)), ranging between \(0\) and \(1\)
• Two bits can have four distinct values (\(2^2\)), ranging between \(0\) and \(3\)
• Three bits provide eight (\(2^3\)), randing between \(0\) and \(7\)

Thus, one byte can have \(256\) distinct values ranging between \(0\) and \(255\) (which as we'll see later provides enough space to store values representing characters, digits and punctuation) because \(2^8\) is \(256\).

REMEMBER

For a given number of bits:

• The maximum number is: \(2^{n-1}\). For instance, if we have 3 bits the maximum value is: \(2^{3-1} = 7\)
• The number of numbers is \(2^n\). For unsigned integers the minimum and maximum values are \(0\) and \(2^{n-1}\) respectively where \(n\) is the number of bits.

Warning

• Do not get confused between the maximum value (2n-1) and number of values (2n) that can be represented for a given number of bits.
• The number of bits available determines the number, the range, of values that can be represented in that number of bits.

Byte multiples

Bytes are small so they are most often used to measure the amount of data, the characters in a text document or pixels in an image file, or the amount of storage available:

Unit	Equivalent (bytes)
1 kilobyte (KB)	1,024
1 megabyte (MB)	1,048,576
1 gigabyte (GB)	1,073,741,824
1 terabyte (TB)	1,099,511,627,776
1 petabyte (PB)	1,125,899,906,842,624

Note

Can you find out the names of even bigger byte multiples? How big is a Zoogolplex?

You will also meet the term nibble to represent half a byte (which is an attempt at humour by computer scientists) and the word which is the amount of data a processor can handle. The most common word sizes are 32-bits and 64-bits. The precise amount of data in a word is a bit fuzzy as some processors might have different word sizes for different tasks but will usually hold for the number of bits the CPU can process in one chunk.

To provide some context for these larger values, 1TB is roughly equivalent to the amount of information as all the books in a large library and 1PB would create (roughly) a stack of CDs a mile high!

These values are used to define amounts of storage space available and useful to note that hard drive manufacturer's will usually use the decimal system to define storage e.g. 500GB to represent 5,000,000,000 bytes (that's \(5 \times 10^9\) bytes). However, your operating system will define the size of the drive using binary, \(5 \times 2^{30} = 465 GB\), it seems 35GB of storage space has been lost!

Whilst the above definitions remain as accepted terms they were actually replaced back in 2000. To be up-to-date (!) we should all be using the following¹

Unit	Short form	Magnitude
Kibibyte	KiB	\(2^{10}\)
Mebibyte	MiB	\(2^{20}\)
Gibibyte	GiB	\(2^{30}\)
Tebibyte	TiB	\(2^{40}\)
Pebibyte	PiB	\(2^{50}\)

These have been slow to get adopted, perhaps because:

• they sound a bit ridiculous
• hard drive manufacturers do not use them
• it's hard to break out of a tradition

Questions

1. Define a bit and explain its fundamental role in digital systems.

A bit is a binary digit that can be either 0 or 1. It is the smallest unit of data in digital systems, representing a binary choice.

A bit is a unit of measurement for data transfer speed. It is used to calculate network bandwidth.

A bit is a memory storage unit that can hold multiple types of data. It is used to represent complex data structures.

A bit is a type of storage medium used in hard drives. It is fundamental to data retrieval processes.

A bit is the smallest unit of data in digital systems, representing a binary choice of 0 or 1.

Submit

2. Describe the composition of a byte and explain why it is a fundamental unit in digital systems.

A byte is composed of 4 bits. It is fundamental because it represents a single character in most encoding systems.

A byte is composed of 8 bits. It is fundamental because it can represent 256 different values, which is suitable for encoding characters.

A byte is composed of 16 bits. It is fundamental because it is the basic unit for data transfer in memory.

A byte is composed of 2 bits. It is fundamental because it is used to store floating-point numbers in calculations.

A byte is composed of 8 bits and is fundamental because it can represent 256 different values, making it suitable for encoding characters.

Submit

3. Given a binary number (e.g., $11011010_2$), identify the Most Significant Bit (MSB) and the Least Significant Bit (LSB).

In $11011010_2$, the MSB is 1 and the LSB is 0.

In $11011010_2$, the MSB is 0 and the LSB is 1.

In $11011010_2$, the MSB is 1 and the LSB is 1.

In $11011010_2$, the MSB is 0 and the LSB is 0.

In the binary number $11011010_2$, the MSB is the leftmost bit (1) and the LSB is the rightmost bit (0).

Submit

4. Explain how to determine whether a binary number is odd or even based on the value of its Least Significant Bit (LSB).

If the LSB is 1, the binary number is even. If the LSB is 0, the binary number is odd.

If the LSB is 0, the binary number is even. If the LSB is 1, the binary number is odd.

If the LSB is 1, the binary number is neither odd nor even. If the LSB is 0, the number is even.

If the LSB is 0, the binary number is neither odd nor even. If the LSB is 1, the number is odd.

A binary number is even if the LSB is 0 and odd if the LSB is 1.

Submit

5. Convert the binary number $10101001_2$ to decimal.

$10101001_2$ is 169 in decimal.

$10101001_2$ is 85 in decimal.

$10101001_2$ is 133 in decimal.

$10101001_2$ is 255 in decimal.

The binary number $10101001_2$ converts to 169 in decimal.

Submit

6. Convert the decimal number $37_{10}$ to binary.

$37_{10}$ is $100101_2$.

$37_{10}$ is $100111_2$.

$37_{10}$ is $110010_2$.

$37_{10}$ is $101010_2$.

The decimal number $37_{10}$ converts to $100101_2$ in binary.

Submit

7. Given a certain number of bits, calculate the maximum value that can be represented. For example, if given 5 bits, what is the maximum value?

The maximum value with 5 bits is 31.

The maximum value with 5 bits is 32.

The maximum value with 5 bits is 25.

The maximum value with 5 bits is 15.

With 5 bits, the maximum value that can be represented is $2^5 - 1 = 31$.

Submit

8. Explain the difference between the maximum value ($2^{n-1}$) and the number of values ($2^n$) that can be represented for a given number of bits.

The maximum value ($2^{n-1}$) is half of the total number of values ($2^n$).

The maximum value ($2^{n-1}$) is the same as the number of values ($2^n$).

The maximum value ($2^{n-1}$) is double the number of values ($2^n$).

The maximum value ($2^{n-1}$) is the square root of the number of values ($2^n$).

The maximum value that can be represented ($2^{n-1}$) is half of the total number of values ($2^n$) that can be represented.

Submit

9. Provide the equivalent number of bytes for 1 kilobyte (KB), 1 megabyte (MB), and 1 gigabyte (GB).

1 KB = 1024 bytes, 1 MB = 1024 KB, 1 GB = 1024 MB.

1 KB = 1000 bytes, 1 MB = 1000 KB, 1 GB = 1000 MB.

1 KB = 1024 bytes, 1 MB = 1000 KB, 1 GB = 1000 MB.

1 KB = 1000 bytes, 1 MB = 1024 KB, 1 GB = 1024 MB.

1 kilobyte (KB) equals 1024 bytes, 1 megabyte (MB) equals 1024 KB, and 1 gigabyte (GB) equals 1024 MB.

Submit

10. Differentiate between a kibibyte (KiB) and a kilobyte (KB), considering their magnitudes and usage in the binary and decimal systems.

1 KiB = 1024 bytes, 1 KB = 1000 bytes. KiB is used in binary systems, KB in decimal systems.

1 KiB = 1000 bytes, 1 KB = 1024 bytes. KiB is used in decimal systems, KB in binary systems.

1 KiB = 1024 bytes, 1 KB = 1024 bytes. Both are used interchangeably in binary and decimal systems.

1 KiB = 1000 bytes, 1 KB = 1000 bytes. Both represent the same amount of data in binary and decimal systems.

1 kibibyte (KiB) is 1024 bytes and is used in binary systems, while 1 kilobyte (KB) is 1000 bytes and is used in decimal systems.

Submit

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1. Some exam boards at GCSE require knowledge of these terms. ↩