Number Base - SS3 Mathematics PDF

Number Base - SS3 Mathematics Comprehensive Note 1. Introduction to Number Bases A number base is a way to represent numbers using a specific set of digits and place values. It defines the number of unique digits, including zero, used in a numbering system. For example: Base 10 (Decimal) uses digi...

Number Base - SS3 Mathematics Comprehensive Note 1. Introduction to Number Bases A number base is a way to represent numbers using a specific set of digits and place values. It defines the number of unique digits, including zero, used in a numbering system. For example: Base 10 (Decimal) uses digits 0–9. Base 2 (Binary) uses digits 0 and 1. Base 8 (Octal) uses digits 0–7. Base 16 (Hexadecimal) uses digits 0–9 and A–F. 2. Place Value System In any base system, the value of a digit depends on: Its face value (the actual digit). Its place value, which is a power of the base. For example, in base 10, the number 345 means: 345 = ( 3 × 1 0 2 ) + ( 4 × 1 0 1 ) + ( 5 × 1 0 0 ) 345=(3×10 2 )+(4×10 1 )+(5×10 0 ) 3. Conversion Between Number Bases (i) Conversion from Base 10 to Other Bases Example 1: Convert 45 (Base 10) to Base 2. Divide by 2 and record the remainders. Continue until the quotient is 0. Write remainders from bottom to top. 45 ÷ 2 = 22 remainder 122 ÷ 2 = 11 remainder 011 ÷ 2 = 5 remainder 15 ÷ 2 = 2 remainder 12 ÷ 2 = 1 remainder 01 ÷ 2 = 0 remainder 1 45 ÷ 2=22 remainder 122 ÷ 2=11 remainder 011 ÷ 2=5 remainder 15 ÷ 2=2 remainder 12 ÷ 2=1 remainder 01÷2=0 remainder 1 Answer: 4 5 10 = 10110 1 2 45 10 =101101 2 (ii) Conversion from Other Bases to Base 10 Example 2: Convert 1011 (Base 2) to Base 10. 101 1 2 = ( 1 × 2 3 ) + ( 0 × 2 2 ) + ( 1 × 2 1 ) + ( 1 × 2 0 ) 1011 2 =(1×2 3 )+(0×2 2 )+(1×2 1 )+(1×2 0 ) = 8 + 0 + 2 + 1 = 1 1 10 =8+0+2+1=11 10 4. Arithmetic Operations in Number Bases (i) Addition Example: Add 110 1 2 + 101 1 2 1101 2 +1011 2 in Base 2. 1101 + 1011 ------ 11000 Answer: 1100 0 2 11000 2 (ii) Subtraction Example: Subtract 110 1 2 − 101 1 2 1101 2  −1011 2. 1101 - 1011 ------ 010 Answer: 1 0 2 10 2 5. Applications of Number Bases Computing Systems: Computers use binary (Base 2) for processing data. Octal and Hexadecimal: Used in programming and digital circuits to simplify binary numbers. Measurements: Time and angles are measured in base 60. 6. Base Conversion Tricks Binary → Octal: Group digits in 3s starting from the right. Binary → Hexadecimal: Group digits in 4s starting from the right. Octal/Hexadecimal → Binary: Expand each digit into binary form. Example: Convert 10110 1 2 101101 2 to Octal: Group: 101 ∣ 101 = 5 ∣ 5 101∣ 101=5∣ 5 → 55₈ 7. Exercises Convert 12 3 8 123 8 to Base 10. Convert 4 5 10 45 10 to Base 5. Add 110 1 2 + 101 1 2 1101 2 +1011 2. Subtract 1011 0 2 − 110 1 2 10110 2  −1101 2. Convert 2 F 16 2F 16 to Base 10. 8. Answers to Exercises 12 3 8 = ( 1 × 8 2 ) + ( 2 × 8 1 ) + ( 3 × 8 0 ) = 8 3 10 123 8  =(1×8 2 )+(2×8 1 )+(3×8 0 )=83 10 4 5 10 45 10 → Divide repeatedly by 5 → 140_5 110 1 2 + 101 1 2 = 1100 0 2 1101 2 +1011 2 =11000 2 1011 0 2 − 110 1 2 = 1000 1 2 10110 2 −1101 2 =10001 2 2 F 16 = ( 2 × 1 6 1 ) + ( 15 × 1 6 0 ) = 4 7 10 2F 16 =(2×16 1 )+(15×16 0 )=47 10 This note should cover all the essential concepts and operations needed to understand Number Bases at the SS3 level.