Data Representation and Binary Arithmetic

Data Representation and Binary Arithmetic Notes

Conversion Between Number Bases

• Binary to Decimal Conversion:

  • Example: Convert binary value 10101 into decimal

  • Options:

    • a. 20

    • b. 22

    • c. 21

• Decimal to Binary Conversion:

  • Example: Convert decimal 58 into binary

  • Options:

    • d. 111010

    • e. 101110

    • f. 111001

• Hexadecimal Number System:

  • Definition: Hexadecimal is referred to as Base 16.

  • Options:

  • g. Base 2

  • h. Base 10

  • i. Base 16

• Hexadecimal to Decimal Conversion:

  • Example: Convert hexadecimal 2A into decimal:

  • Calculation:

    • $2 \times 16^1 + 10 \times 16^0 = 32 + 10 = 42$

  • Result: 42

• Decimal to Hexadecimal Conversion:

  • Example: Convert decimal 35 into hexadecimal:

  • Calculation:

    • $35 \div 16 = 2 \text{ remainder } 3$

  • Result: 23

Learning Outcomes

• By the end of this lesson, students will be able to:

  • Add 2 binary numbers together.

  • Perform binary shifts on 8-bit binary numbers.

  • Complete various binary conversions.

Binary Addition

• Addition of binary numbers follows specific rules:

  1. Work right to left when adding.

  2. Add the unit's place first and carry over if needed:

  • Example of simple decimal addition:

    • $1 + 1 = 10$ in binary, carries 1 to the next column.

Rules of Binary Addition

• The following rules apply for binary addition:

  1. $0 + 0 = 0$

  2. $0 + 1 = 1$

  3. $1 + 0 = 1$

  4. $1 + 1 = 0$ (carry 1)

  5. $1 + 1 + 1 = 1$ (carry 1)

• Example of binary addition demonstrating various cases:

  • Calculation:
    1011

  • 1010
    Result:

  • Carrying at bit positions leads to a final result of 10101.

Binary Shifts

• Binary Shift Left:

  • Shifting bits to the left (L)

  • This operation doubles the value of the binary number.

  • Example:

    • 1000 = 8 (in decimal)

    • Shift left once results in 10000 = 16 (in decimal)

• Binary Shift Right:

  • Shifting bits to the right reduces the binary value.

  • This operation halves the value of the binary number.

  • Example 1:

    • Start with 1000 = 8

    • Right shift once gives you 100 = 4

    • Right shift again gives you 10 = 2

  • Example 2:

    • Start with 1001 = 9

    • Right shift once gives you 100 = 4

    • Right shift again gives you 10 = 2

• Logical Shift Operations:

  • Left shifts multiply a number by 2 for each shift;

  • Right shifts divide a number by 2.

  • Example: shifting 1110 yields values (in binary):

  • Left shift 1110 = 11100 (multiplies the number).

  • Right shift 1110 = 111 (divides the number).

• Loss of Accuracy:

  • Important to note that shifts may result in loss of accuracy, especially where fractional values are represented.

  • Example: The value 22 / 4 is not exactly 5.

Worksheet Task and Exercises

• Practical Exercises: Students will perform calculations and compare answers with a partner:

  1. Calculate: 1001 0101 + 0010 0110

  2. Add 0100 0011 to the result of the previous question.

  3. Calculate the left shift of 0110 1011.

  4. Explain the effect of a left shift on the binary value.