1.1-Introduction to Real Numbers
1.1-Introduction to Real Numbers Important Formulae
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Real numbers are a fundamental concept in mathematics, encompassing a wide range of numbers used in everyday calculations and advanced mathematics. They include all the rational and irrational numbers, providing a complete system for measurement, calculation, and representation of quantities.
1. Definition of Real Numbers
Real numbers can be defined as all the numbers that can be found on the number line. They include:
- Rational Numbers: Numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Examples include $\frac{1}{2}, -3, 0.75$.
- Irrational Numbers: Numbers that cannot be expressed as a simple fraction. Their decimal expansions are non-terminating and non-repeating. Examples include $\pi$ and $\sqrt{2}$.
2. Representation of Real Numbers
Real numbers can be represented on a number line, where:
- Rational numbers can be located at specific points.
- Irrational numbers are also located on the number line, but cannot be precisely expressed as fractions.
This visualization helps in understanding the density of real numbers, as between any two distinct real numbers, there exists another real number.
3. Classification of Real Numbers
Real numbers can be classified into several subsets:
- Naturals (N): The set of positive integers starting from 1. $\{1, 2, 3, \ldots\}$
- Whole Numbers (W): The set of natural numbers including 0. $\{0, 1, 2, 3, \ldots\}$
- Integers (Z): The set of whole numbers and their negative counterparts. $\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$
- Rational Numbers (Q): Numbers that can be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers, and $q \neq 0$. For example, $\frac{3}{4}$.
- Irrational Numbers: As mentioned earlier, these cannot be expressed as fractions, such as $\sqrt{3}$ or $e$.
4. Properties of Real Numbers
Real numbers possess several important properties:
- Closure Property: The sum or product of any two real numbers is also a real number.
- Commutative Property: For addition and multiplication, the order of the numbers does not affect the result. That is, $a + b = b + a$ and $ab = ba$.
- Associative Property: For addition and multiplication, the grouping of numbers does not affect the result. That is, $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$.
- Distributive Property: This property connects addition and multiplication. That is, $a(b + c) = ab + ac$.
5. Operations with Real Numbers
Real numbers can be added, subtracted, multiplied, and divided (except by zero) following the rules of arithmetic. These operations are essential for solving equations, evaluating expressions, and performing calculations in various fields.
6. Decimal Representation
Real numbers can be represented in decimal form. Rational numbers have either terminating or repeating decimal expansions, while irrational numbers have non-terminating, non-repeating decimal expansions. For example:
- Terminating: $0.75$
- Repeating: $0.333\ldots = \frac{1}{3}$
- Irrational: $\pi \approx 3.14159\ldots$
Understanding real numbers is crucial for further studies in mathematics, as they serve as the foundation for algebra, calculus, and other mathematical concepts.
The real numbers and subsets thereof. The image may be misleading in that it implies there are real numbers that are neither rational or irrational. Could also mislead one about the relative "sizes" of these sets.
Damien Karras, Public domain, via Wikimedia Commons
