Numbers: Mastering Real, Rational, and Irrational Numbers
What Are Numbers?
Numbers form the foundation of Mathematics, classifying quantities and measurements into different types for specific uses.
Types of Numbers
Real Numbers (
)
)Real numbers include all numbers on the number line, covering both rational and irrational numbers.
Rational Numbers (
)
)Rational numbers are expressed as fractions (
) where 
, and 
. Examples: 
.
Irrational Numbers
Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions. Examples: 
.
Properties of Numbers
Closure Property
- Rational numbers are closed under addition, subtraction, multiplication, but not division by zero.
- Irrational numbers are not closed under arithmetic operations.
Density Property
Between any two real numbers, there exists another real number. For example, between 
and 
, 
exists.
Operations on Real Numbers
Addition and Subtraction
Real numbers follow associative, commutative, and distributive properties.
Example:
![\[ 2.5 + 3.5 = 6, \quad (-5) + 8 = 3 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-dd6c16a34c3afe1470c4c6fc2c626265_l3.png "Rendered by QuickLaTeX.com")
Multiplication and Division
Multiplication distributes over addition. Division by zero is undefined.
Example:
![\[ (2 \cdot 3) + (5 \cdot 3) = 3(2 + 5) = 21 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-1b83e8ec77f5783de1f4d9a667faf2cb_l3.png "Rendered by QuickLaTeX.com")
Applications of Numbers
Real Numbers in Geometry
- Used for measurements like lengths, areas, and volumes.
Rational Numbers in Finance
- Represent fractions of currency or proportions.
Irrational Numbers in Engineering
- Used in calculating circular dimensions (
) or root-related calculations.
) or root-related calculations.Example Problem
Classify the following numbers as rational or irrational:
![\[ 0.333\ldots, \sqrt{16}, \pi, -5 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-2855b62329b7a44d3969a9b047cdc7d1_l3.png "Rendered by QuickLaTeX.com")
Solution:
is rational (
).
is rational.- is irrational.
is rational.
is rational (
).
is rational.
is rational.Common Mistakes in Number Calculations
- Assuming all roots are irrational (e.g.,
is rational). - Misclassifying repeating decimals as irrational.
- Dividing by zero, which is undefined.
is rational).Practice Questions
- Classify the following as rational or irrational:
. - Explain why is irrational.
- Provide an example of a number that is real but not rational.
.
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