Introduction to the Number System
The Number System is a fundamental concept in mathematics that categorizes numbers into distinct types, such as natural numbers, integers, and rational numbers.
- Natural Numbers (\( \mathbb{N} \)): Counting numbers like 1, 2, 3...
- Whole Numbers (\( \mathbb{W} \)): Natural numbers including 0.
- Integers (\( \mathbb{Z} \)): Positive and negative whole numbers, including 0.
- Rational Numbers (\( \mathbb{Q} \)): Numbers that can be expressed as a fraction \( \frac{p}{q} \) where \( q \neq 0 \).
- Irrational Numbers: Numbers that cannot be expressed as fractions, like \( \pi \) and \( \sqrt{2} \).
- Real Numbers (\( \mathbb{R} \)): All rational and irrational numbers.
HCF and LCM
The HCF (Highest Common Factor) of two numbers is the greatest number that divides both without leaving a remainder.
The LCM (Lowest Common Multiple) of two numbers is the smallest number that is a multiple of both.
Example:
- HCF of 12 and 18 is 6.
- LCM of 12 and 18 is 36.
Prime and Composite Numbers
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 more than two factors (e.g., 4, 6, 8, 9).
Divisibility Rules
Divisibility rules help us determine whether one number is divisible by another without performing the division.
- A number is divisible by 2 if its last digit is even.
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- A number is divisible by 5 if its last digit is 0 or 5.
Decimal and Fraction Representation
Numbers can be represented as fractions or decimals.
Example:
- \( \frac{1}{2} \) as a decimal is 0.5.
- \( 0.75 \) as a fraction is \( \frac{3}{4} \).
Quiz: Test Your Knowledge
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Which of the following is a prime number?
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What is the HCF of 12 and 18?
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Which of the following is divisible by 5?
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Convert \( \frac{3}{4} \) to a decimal.
Quiz: Test Your Knowledge
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What is the HCF of 24 and 36?
Your current score: 0
Quiz: Test Your Knowledge
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Which of these is a prime number?
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What is the LCM of 8 and 12?
Score: 0/0