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NUMBER SYSTEM
1. Types of Numbers
- Natural Numbers (N): Counting numbers starting from 1 (1, 2, 3, …).
- Whole Numbers (W): Natural numbers including zero (0, 1, 2, 3, …).
- Integers (Z): Whole numbers and their negatives (… -3, -2, -1, 0, 1, 2, 3 …).
- Rational Numbers (Q): Numbers that can be expressed as the quotient of two integers (e.g., 1/2, -3/4).
- Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., √2, π).
- Real Numbers (R): All rational and irrational numbers combined.
2. Number Systems
- Decimal System: Base-10 number system using digits 0 through 9.
- Binary System: Base-2 number system using digits 0 and 1.
- Octal System: Base-8 number system using digits 0 through 7.
- Hexadecimal System: Base-16 number system using digits 0-9 and letters A-F.
3. Conversions
- Decimal to Binary: Use division by 2, and record remainders.
- Binary to Decimal: Use powers of 2 to convert each binary digit to decimal.
- Decimal to Octal: Use division by 8, and record remainders.
- Octal to Decimal: Use powers of 8.
- Decimal to Hexadecimal: Use division by 16, and record remainders.
- Hexadecimal to Decimal: Use powers of 16.
4. Number Properties
- Prime Numbers: Numbers greater than 1 that have no divisors other than 1 and themselves (e.g., 2, 3, 5, 7).
- Composite Numbers: Numbers greater than 1 that have divisors other than 1 and themselves.
- Even and Odd Numbers: Even numbers are divisible by 2; odd numbers are not.
- Divisibility Rules: Shortcuts for determining if a number is divisible by another (e.g., divisible by 2 if it ends in 0, 2, 4, 6, or 8).
5. Factors and Multiples
- Factors: Numbers that divide a given number exactly.
- Multiples: Products of a number and integers.
- HCF (Highest Common Factor): Greatest number that divides two or more numbers exactly.
- LCM (Least Common Multiple): Smallest number that is a multiple of two or more numbers.
6. Fractions and Decimals
- Simplifying Fractions: Reducing fractions to their simplest form.
- Operations on Fractions: Addition, subtraction, multiplication, and division.
- Decimal Conversion: Converting fractions to decimals and vice versa.
7. Percentage Calculations
- Finding Percentages: Calculating percentages of numbers.
- Percentage Increase/Decrease: Finding how much a number has increased or decreased in percentage terms.
8. Ratios and Proportions
- Ratios: Comparing two quantities.
- Proportions: Equating two ratios.
9. Square Roots and Cube Roots
- Square Roots: Finding the number which, when multiplied by itself, gives the original number.
- Cube Roots: Finding the number which, when cubed, gives the original number.
10. Exponents and Powers
- Laws of Exponents: Rules for multiplying and dividing powers (e.g., am×an=am+na^m \times a^n = a^{m+n}am×an=am+n).
11. Logarithms
- Basic Logarithms: Understanding the logarithmic functions and their properties.
12. Number Theory Problems
- Common Problems: Typically include problems related to divisibility, prime factors, and modular arithmetic.
1. Types of Numbers
- Natural Numbers: 1, 2, 3, 4, 5, …
- Whole Numbers: 0, 1, 2, 3, 4, …
- Integers: … -3, -2, -1, 0, 1, 2, 3, …
- Rational Numbers: 1/2, -3/4, 7 (which is 7/1)
- Irrational Numbers: √2 (approximately 1.414), π (approximately 3.14159)
- Real Numbers: Any number that can be found on the number line, including both rational and irrational numbers.
2. Number Systems
- Decimal to Binary:
- Convert 13 to binary:
- Divide 13 by 2: quotient is 6, remainder is 1
- Divide 6 by 2: quotient is 3, remainder is 0
- Divide 3 by 2: quotient is 1, remainder is 1
- Divide 1 by 2: quotient is 0, remainder is 1
- Reading the remainders from bottom to top, 13 in decimal is 1101 in binary.
- Binary to Decimal:
- Convert 1011 to decimal:
- (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (1 × 2^0)
- = 8 + 0 + 2 + 1
- = 11 in decimal.
3. Conversions
- Decimal to Hexadecimal:
- Convert 255 to hexadecimal:
- Divide 255 by 16: quotient is 15, remainder is 15 (F in hexadecimal)
- Thus, 255 in decimal is FF in hexadecimal.
- Hexadecimal to Decimal:
- Convert A3 to decimal:
- A in hexadecimal is 10.
- (10 × 16^1) + (3 × 16^0)
- = 160 + 3
- = 163 in decimal.
4. Number Properties
- Prime Numbers:
- First few prime numbers: 2, 3, 5, 7, 11.
- Composite Numbers:
- 4, 6, 8, 9, 10 (each has divisors other than 1 and itself).
- Divisibility Rule for 3:
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- For 123: 1 + 2 + 3 = 6, which is divisible by 3, so 123 is divisible by 3.
5. Factors and Multiples
- Finding Factors:
- Factors of 12: 1, 2, 3, 4, 6, 12.
- Finding Multiples:
- First few multiples of 4: 4, 8, 12, 16, 20.
- HCF (Highest Common Factor) of 24 and 36:
- Prime factorization of 24: 2^3 × 3
- Prime factorization of 36: 2^2 × 3^2
- Common factors: 2^2 × 3
- HCF = 12.
- LCM (Least Common Multiple) of 4 and 6:
- Prime factorization of 4: 2^2
- Prime factorization of 6: 2 × 3
- LCM = 2^2 × 3 = 12.
6. Fractions and Decimals
- Simplifying Fractions:
- Simplify 24/36:
- Divide numerator and denominator by their HCF (12):
- 24 ÷ 12 = 2, 36 ÷ 12 = 3
- Simplified fraction is 2/3.
- Decimal Conversion:
7. Percentage Calculations
- Finding 25% of 80:
- 25% of 80 = (25/100) × 80 = 20.
- Percentage Increase:
- Increase from 50 to 60:
- Increase = 60 – 50 = 10
- Percentage increase = (10/50) × 100% = 20%.
8. Ratios and Proportions
- Ratios:
- Ratio of 30 to 45:
- Simplify by dividing both by their HCF (15):
- 30 ÷ 15 = 2, 45 ÷ 15 = 3
- Ratio = 2:3.
- Proportions:
- If 4/5 = x/20, find x:
- Cross-multiply: 4 × 20 = 5 × x
- 80 = 5x
- x = 16.
9. Square Roots and Cube Roots
- Square Root:
- √49 = 7 (since 7 × 7 = 49).
- Cube Root:
- ∛27 = 3 (since 3 × 3 × 3 = 27).
10. Exponents and Powers
11. Logarithms
- Basic Logarithms:
- If log base 10 of 1000 = x, then 10^x = 1000
- Since 10^3 = 1000, x = 3.
12. Number Theory Problems
- Modular Arithmetic:
- Find 29 mod 5:
- 29 divided by 5 is 5 with a remainder of 4.
- Thus, 29 mod 5 = 4.
