Original file size: The original file contains 6 numbers, each requiring a certain amount of storage (e.g., 4 bytes if each number is a 32-bit integer). Let's assume each number takes 4 bytes, so the original file size is 6 * 4 = 24 bytes.

Compressed file size: The compressed file contains the string "2 4 8 16 32 64; 2; 2; 2; 2; 2". This consists of 12 characters. Assuming each character takes 1 byte, the compressed file size is 12 bytes.

Compression Ratio: Compression Ratio = (Original File Size / Compressed File Size) = 24 bytes / 12 bytes = 2. This means the compressed file is 1/2 the size of the original file.

Explanation: The compression method achieves a higher compression ratio because it exploits the redundancy in the data. Instead of storing each number individually, it stores the number of times the number repeats (the '2' in this case) and then the number itself. This is more efficient when there are many repetitions of the same value. The original method required storing each number separately, regardless of whether it was a repetition or not. RLE reduces the amount of data needed to represent the same information.

The fundamental trade-off between lossless and lossy data compression is the relationship between file size and data fidelity. Lossless compression achieves smaller file sizes but cannot reduce the file size below its original size. Lossy compression achieves significantly smaller file sizes but at the cost of some data loss.

Here's a breakdown of when to choose each method:

• Choose Lossless Compression When:

• Data Integrity is Critical: When it's essential that the original data is perfectly reconstructed. This is crucial for text documents, software code, and financial records.
• Small Size Reduction is Acceptable: When a modest reduction in file size is sufficient and preserving all data is paramount.

• Choose Lossy Compression When:

• File Size is a Priority: When minimizing file size is more important than preserving every detail of the data. This is common for multimedia files like images, audio, and video.
• Minor Data Loss is Unnoticeable: When the data loss is imperceptible to the human eye or ear. For example, a slight reduction in image quality is often acceptable for photographs viewed on a screen.

In summary, the choice between lossless and lossy compression depends on the specific application and the relative importance of file size versus data fidelity.

Two common data compression methods are lossless compression and lossy compression.

Lossless Compression: This method reduces file size without losing any of the original data. It works by identifying and eliminating redundancy in the data. The compressed file can be perfectly reconstructed to its original state. It is suitable for data where accuracy is paramount.

• Principle: Identifying and replacing repeating patterns with shorter codes. For example, if a character appears frequently, it can be represented by a shorter code than if it appears infrequently.
• Example File Format: ZIP is a widely used lossless compression format for archiving files. PNG is a lossless image format often used for graphics with sharp lines and text.

Lossy Compression: This method achieves higher compression ratios by discarding some of the original data. The discarded data is deemed less important to the overall quality of the file. This is acceptable for multimedia where a slight loss in quality is not noticeable.

• Principle: Removing data that is considered perceptually less important. For images, this might involve reducing the number of colors or removing high-frequency details. For audio, it might involve removing frequencies that are masked by louder sounds.
• Example File Format: JPEG is a lossy image format commonly used for photographs. MP3 is a lossy audio format widely used for music. MPEG is a lossy video format used for streaming and video recording.