PHYSICS140 LECTURE 3

  • Numbers in Binary

    • Previously covered concepts: adding binary numbers, multiplying them, and representing negative numbers using two's complement.

    • Two's Complement Representation:

      • Example: +13 is represented as 00001101.

      • To convert it to -13, invert the bits and add one:

      • Step 1: Invert 00001101 → 11110010.

      • Step 2: Add 1 → 11110010 + 1 = 11110011, which is -13 in binary (5-bits).

    • Example Exercise (2's Complement)

      - Find -13 in 5-bit two's complement:

      +13 in binary → 01101 (with leading zeros)

      • Negative Conversion: Inverting gives 10010 (from 01101) and adding one gives 10011, confirming representation.

      • Most significant bit denotes positives/negatives.

    • Multiplication of Signed Numbers

      • Process for multiplying signed numbers:

      • Turn negatives to positives with two's complement before multiplication.

        • Example for Multiplication:

        • Multiply -3 by 4.

        • Step 1: Represent -3 in binary:

          • +3 is 00011.

          • Convert to -3: Invert → 11100, Add 1 → 11101.

        • Step 2: Represent 4 in binary → 00100.

        • Step 3: Perform multiplication:

          • 11101 (which represents -3)

          • × 00100 (which represents 4)

          • Result in binary is 1111100, which translates to -12 in decimal.

        • Determine sign of result based on multiplication rules (negative × positive = negative).

    • Introduction to Hexadecimal

      • Hexadecimal as a compact way to represent binary values (base 16).

      • Each hexadecimal digit corresponds to four binary digits.

      • Example: C3 in hex equals 11000111 in binary.

      • Counting in hexadecimal:

      • Counts range from 0 to F, where A=10, B=11, C=12, D=13, E=14, F=15.

      • Example Conversion from Hex to Decimal:

        • Hex C3:

        • Calculate using place value based on powers of 16:

          • C × 16^1 + 3 × 16^0

          • (12×16 + 3×1 = 192 + 3 = 195).

      • Hex to Binary: Each hex digit translates to four binary digits.

      • Example: C = 1100, 3 = 0011 → Overall: 11000011.

    • Storing Information in Binary

      • Distinction between codes and arithmetic.

      • Binary Coded Decimal (BCD):

      • Represents decimal digits using 4-bit binary codes; each digit is separate (e.g., 4563 → 0100 0101 0110 0011).

      • Highlighted the non-arithmetic nature of stored information versus representational numbers.

      • Practical application in systems like telephones to register key presses, not compute values.

    • Next Lecture Preview

      • Focus on further codes, particularly how to store letters and plain text in computing systems.

      • Encouraged to practice binary and hexadecimal exercises on Canvas.