Let's go through each question and answer it.

Question 1:

What is the base of the binary system?

Answer:
The base of the binary system is 2. In binary, only two digits are used: 0 and 1.

Question 2:

How can we determine the base of any number system?

Answer:
To determine the base of a number system, identify the unique digits used in that system:

• The base of a number system is the number of unique digits it uses.
• For example, the decimal system has a base of 10 because it uses ten digits (0 through 9).
• The hexadecimal system has a base of 16, using digits 0–9 and letters A–F.

Question 3:

Identify whether the following number representations are possible in the specified number system:

To determine if each representation is possible, the digits used must be valid for the given base (all digits in a base-N number must be between 0 and N-1).

1. 10110₂

   • Base 2: Valid digits are 0 and 1.
   • Answer: Possible.

2. 57120₂

   • Base 2: Valid digits are 0 and 1.
   • Answer: Not possible (5, 7, and 2 are invalid in base 2).

3. 57120₁₀

   • Base 10: Valid digits are 0 to 9.
   • Answer: Possible.

4. 10110₁₀

   • Base 10: Valid digits are 0 to 9.
   • Answer: Possible.

5. 32110₄

   • Base 4: Valid digits are 0 to 3.
   • Answer: Not possible (digit 4 is invalid in base 4).

6. 77720₈

   • Base 8: Valid digits are 0 to 7.
   • Answer: Not possible (8 and 9 are invalid in base 8).

7. 11186₈

   • Base 8: Valid digits are 0 to 7.
   • Answer: Not possible (digit 8 is invalid in base 8).

8. 50505₆

   • Base 6: Valid digits are 0 to 5.
   • Answer: Possible.

Question 4:

What is the relationship between digit, base, and position number in the decimal system?

Answer:
In any positional number system (including decimal):

• Each digit represents a value based on its position.
• The base determines the value of each position (power of the base).
• For a number $d_nd_{n-1} \ldots d_1d_0.d_{-1}d_{-2} \ldots$ in base $b$: $\text{Value} = d_n \times b^n + d_{n-1} \times b^{n-1} + \ldots + d_0 \times b^0 + d_{-1} \times b^{-1} + \ldots$
• In the decimal system (base 10), each position represents a power of 10, with the rightmost digit in the integer part representing $10^0$, the next one to the left representing $10^1$, and so on.

Question 5:

For the following decimal numbers, determine what value the highlighted digit represents and write this value as the product of the digit and a power of 10.

The highlighted digit appears to be 9 in each number listed.

1. 44931.95₁₀ (highlighted digit: 9)

   • Position: Hundreds place.
   • Value: $9 \times 10^2 = 900$.

2. 44931.95₁₀ (highlighted digit: 9)

   • Position: Hundreds place.
   • Value: $9 \times 10^2 = 900$.

3. 44931.95₁₀ (highlighted digit: 9)

   • Position: Hundreds place.
   • Value: $9 \times 10^2 = 900$.

4. 44931.95₁₀ (highlighted digit: 9)

   • Position: Hundreds place.
   • Value: $9 \times 10^2 = 900$.

Would you like more detailed explanations for any question? Here are some related questions to explore further:

1. What is the hexadecimal system, and what digits does it use?
2. How does the positional system differ in bases other than 10?
3. How are fractions represented in binary and other non-decimal bases?
4. Can irrational numbers be represented in non-decimal bases?
5. How do we convert numbers from one base to another?

Tip: When checking if a number is valid in a particular base, ensure all digits are less than the base itself (e.g., digits in base 4 must be 0, 1, 2, or 3).