Number System Identification

To identify the number system for the given sequence (0, 1, 2, 3, 10, 11, 12, 13, 20, …), we need to understand how different number systems count. Each number system is defined by its base, which determines the number of unique digits it uses before rolling over to the next place value.

Let's carefully examine the provided number sequence: 0, 1, 2, 3, 10, 11, 12, 13, 20, …

Analyzing the Number Sequence

The most crucial observation in the given sequence is how the numbers progress.

• The sequence starts with 0, 1, 2, 3.
• Immediately after the digit 3, the sequence continues with 10. This is a key indicator. In the standard decimal (base-10) system, after 3, we would expect 4, 5, 6, 7, 8, 9, and then 10. The fact that the sequence jumps directly from 3 to 10 suggests that the number system does not use digits beyond 3.
• If a number system uses digits up to N-1, its base is N. Since the highest single digit observed is 3, it implies that the base of this number system is \(3 + 1 = 4\).

Exploring Different Number Systems

Let's compare the given sequence with how numbers are represented in the common number systems listed in the options:

• Decimal System (Base-10): The decimal system uses ten unique digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). The sequence in decimal would be: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … This does not match our given sequence because it includes digits 4 through 9, which are absent before '10' in the given sequence.
• Octal System (Base-8): The octal system uses eight unique digits (0, 1, 2, 3, 4, 5, 6, 7). The sequence in octal would be: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, … This also does not match our given sequence because after '3', it would show '4', '5', '6', '7' before '10'.
• Hexadecimal System (Base-16): The hexadecimal system uses sixteen unique digits (0-9, A-F). This system clearly does not match the given sequence as it would use many more single digits before '10'.
• Quaternary System (Base-4): The quaternary system uses four unique digits (0, 1, 2, 3). Let's see how numbers progress in the quaternary system:
  • \(0_4 = 0_{10}\)
  • \(1_4 = 1_{10}\)
  • \(2_4 = 2_{10}\)
  • \(3_4 = 3_{10}\)
  Since 3 is the largest digit in base-4, to represent the next number, we "roll over" to the next place value, similar to how 9 is followed by 10 in decimal.
  • \(10_4 = (1 \times 4^1) + (0 \times 4^0) = 4 + 0 = 4_{10}\)
  • \(11_4 = (1 \times 4^1) + (1 \times 4^0) = 4 + 1 = 5_{10}\)
  • \(12_4 = (1 \times 4^1) + (2 \times 4^0) = 4 + 2 = 6_{10}\)
  • \(13_4 = (1 \times 4^1) + (3 \times 4^0) = 4 + 3 = 7_{10}\)
  Again, after \(13_4\), we have used all possible digit combinations for the units place with '1' in the 'fours' place. To get the next number, we increment the 'fours' place (the '1' becomes '2') and reset the units place to '0'.
  • \(20_4 = (2 \times 4^1) + (0 \times 4^0) = 8 + 0 = 8_{10}\)
  This progression 0, 1, 2, 3, 10, 11, 12, 13, 20 perfectly matches the given sequence.

Quaternary System Explained

The Quaternary (Base-4) system is a positional numeral system with base 4. It uses four distinct digits: 0, 1, 2, and 3. Each position in a quaternary number represents a power of 4.

Here is a table showing the conversion of the numbers in the sequence from Quaternary to Decimal:

Based on this detailed analysis, the number system for the given sequence is undeniably Quaternary.