Class 8 Math: Real Numbers
By Pratap Sanjay Sir
Introduction to Real Numbers
Real numbers include all numbers on the number line, both rational and irrational. They are fundamental in mathematics and used in everyday calculations and scientific measurements.
Definitions
Number: A mathematical object used to count, measure, and label. Examples: 1, 2, 3, 4.
Real Numbers: Numbers that can be found on the number line, including rational and irrational numbers. Examples: 2, 3.14.
Imaginary Numbers: Numbers that can be written as a real number multiplied by the imaginary unit \(i\), where \(i^2 = -1\). Examples: 3i, -2i.
Rational Numbers: Numbers that can be expressed as a fraction of two integers. Examples: 1/2, -3/4.
Irrational Numbers: Numbers that cannot be expressed as a simple fraction. Examples: √2, π.
Complex Numbers: Numbers in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. Examples: 3 + 4i, 2 - 5i.
Integers: Whole numbers and their negatives. Examples: -1, 0, 1.
Fractions: Parts of a whole, expressed as a numerator over a denominator. Examples: 1/2, 3/4.
Decimals: Another way to express fractions. Examples: 0.5, 0.75.
Positive Integers: Numbers greater than zero. Examples: 1, 2, 3.
Negative Integers: Numbers less than zero. Examples: -1, -2, -3.
Zero: A number that is neither positive nor negative. Example: 0.
Non-Negative Numbers: All positive numbers and zero. Examples: 0, 1, 2.
Non-Positive Numbers: All negative numbers and zero. Examples: 0, -1, -2.
Whole Numbers: All natural numbers and zero. Examples: 0, 1, 2.
Natural Numbers: Counting numbers starting from 1. Examples: 1, 2, 3.
Prime Numbers: Numbers with only two divisors: 1 and themselves. Examples: 2, 3, 5.
Composite Numbers: Numbers with more than two divisors. Examples: 4, 6, 9.
Co-prime Numbers: Two numbers are co-prime if their greatest common divisor (GCD) is 1. Examples: 8 and 15, 9 and 28.
Twin-Prime Numbers: Pairs of prime numbers that differ by 2. Examples: (3, 5), (11, 13).
Properties of Real Numbers
Real numbers have several important properties:
- Closure: The sum or product of any two real numbers is a real number.
- Commutative Property: a + b = b + a and ab = ba for any real numbers a and b.
- Associative Property: (a + b) + c = a + (b + c) and (ab)c = a(bc) for any real numbers a, b, and c.
- Distributive Property: a(b + c) = ab + ac for any real numbers a, b, and c.
- Identity Property: There exists an additive identity (0) and a multiplicative identity (1).
- Inverse Property: Every real number a has an additive inverse (-a) and a multiplicative inverse (1/a, a ≠ 0).
Examples of Real Numbers
Example 1: Natural Numbers
1, 2, 3, 4, 5, ...
Example 2: Whole Numbers
0, 1, 2, 3, 4, ...
Example 3: Integers
..., -3, -2, -1, 0, 1, 2, 3, ...
Example 4: Rational Numbers
1/2, -3/4, 5/1, 0.75
Example 5: Irrational Numbers
√2, π, e
Example 6: Complex Numbers
3 + 4i, 5 - 2i
Example 7: Prime Numbers
2, 3, 5, 7, 11
Example 8: Composite Numbers
4, 6, 8, 9, 10
Example 9: Co-prime Numbers
8 and 15, 9 and 28
Example 10: Twin-Prime Numbers
(3, 5), (11, 13), (17, 19)
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