Unit 1: Exam Review Notes

Collaboration and Teamwork

• Teamwork: Primarily focuses on individuals getting along and working together harmoniously.

• Collaboration: A deeper process that involves:

  • Creating new understandings for a problem or task.

  • Embracing disagreement and finding ways to combine opposing points of view.

  • Going beyond passively agreeing; avoiding conflict may hinder genuine collaboration.

  • Requiring participants from diverse backgrounds and with varied perspectives to foster richer solutions.

Bits and Bytes: Fundamentals of Digital Information

• Bit: The most basic unit of information, representing either a $0$ or a $1$.

• Byte: A unit of digital information comprising $8$ bits.

  • An $8$-bit system (1 Byte) is capable of storing values up to $255$.

  • It can represent a total of $256$ unique number options, including $0$.

Binary Number Operations

Sequential ID Assignment

• Problem: If student IDs increase sequentially by $1$, and the last ID assigned was binary $1001 hinspace 0011$, what is the next ID?

• Solution Steps:

  1. Convert the given binary to decimal: $(1001 hinspace 0011)<em>2 = (1 \cdot 2^7) + (0 \cdot 2^6) + (0 \cdot 2^5) + (1 \cdot 2^4) + (0 \cdot 2^3) + (0 \cdot 2^2) + (1 \cdot 2^1) + (1 \cdot 2^0) = 128 + 0 + 0 + 16 + 0 + 0 + 2 + 1 = 147</em>{10}$.

  2. Add $1$ to the decimal value: $147 + 1 = 148_{10}$.

  3. Convert the new decimal value back to binary:

     • Find the largest power of $2$ less than or equal to $148$ ($128 = 2^7$).

     • Subtract $128$ from $148$ (remainder $20$).

     • Next power of $2$ less than $20$ is $16 = 2^4$.

     • Subtract $16$ from $20$ (remainder $4$).

     • Next power of $2$ less than $4$ is $4 = 2^2$.

     • Subtract $4$ from $4$ (remainder $0$).

     • The binary representation is formed by placing a $1$ at the positions corresponding to the powers of $2$ used, and $0$ otherwise: $1001 hinspace 0100_2$.

• Answer: The next ID will be binary $1001 hinspace 0100$.

Ordering Numeric Values

• Problem: Order the following values from least to greatest: Binary $1011$, Binary $1101$, Decimal $5$, Decimal $12$.

• Solution Steps:

  1. Convert all binary numbers to decimal for consistent comparison:

     • $(1011)<em>2 = (1 \cdot 2^3) + (0 \cdot 2^2) + (1 \cdot 2^1) + (1 \cdot 2^0) = 8 + 0 + 2 + 1 = 11</em>{10}$.

     • $(1101)<em>2 = (1 \cdot 2^3) + (1 \cdot 2^2) + (0 \cdot 2^1) + (1 \cdot 2^0) = 8 + 4 + 0 + 1 = 13</em>{10}$.

  2. List all values in decimal: $5, 11, 12, 13$.

  3. Order them from least to greatest:

     • $5$ (Decimal)

     • $11$ (Binary $1011$)

     • $12$ (Decimal)

     • $13$ (Binary $1101$)

• Answer: Decimal $5$, binary $1011$, decimal $12$, binary $1101$.

Data Representation Errors

• Fixed Number of Bits: Computers represent numbers using a fixed number of bits, which limits the range and precision of values that can be stored.

• Overflow Error: Occurs when a computer attempts to represent a number that is too large for the allocated number of bits.

  • Example 1: With $8$ bits:

    • Maximum representable value is $2^8 - 1 = 255$.

    • The smallest number that would cause an overflow error is $256$.

  • Example 2: With $6$ bits:

    • Maximum representable value is $2^6 - 1 = 63$.

    • The smallest number that would cause an overflow error is $64$.

  • Hint: An overflow error is caused by the numerical value that would require the

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