1. A real number is either a rational number (can be expressed as a fraction of integers) or an irrational number (cannot be expressed as a fraction and has a non-repeating, non-terminating decimal expansion).

2. The set of real numbers, denoted by $\mathbb{R}$, includes all these numbers and is expressed as $( \mathbb{R} = { x : x \text{ is a rational number or an irrational number} } )$.

3. Properties of Real Numbers:

• Trichotomy Property: For any two real numbers ( a ) and ( b ), exactly one of the following is true: ( a < b ), ( a = b ), or ( a > b ).
• Transitive Property: For any three real numbers ( a ), ( b ), and ( c ), if ( a < b ) and ( b < c ), then ( a < c ).

4. A real number between a and b can be found using their average $\frac{(a+b)}{2}$.

5. Intervals denote sets of real numbers between two endpoints:

• Closed Interval $[a, b]$: Includes both endpoints.
• Half-Open Interval $[a, b)$: Includes ( a ) but not ( b ).
• Right-Open Interval $(a, b]$: Excludes ( a ) but includes ( b ).
• Unbounded Interval $(a, ∞)$: Extends infinitely to the right.

6. The absolute value of a number $ x $, denoted $ |x| $, is the non-negative value of $ x $. Thus,

7. Exponents and Radicals:

• Exponent $( a^n )$ denotes $ a $ multiplied by itself $ n $ times.
• Zero Exponent: $( a^0 = 1 )$ if $( a \neq 0 )$.
• Negative Exponent: $( a^{-n} = \frac{1}{a^n} )$.
• Radicals: $( \sqrt[n]{b} )$ represents the $( n )-th$ root of $( b )$.

8. Simplifying Expressions with Exponents:

• Use the laws of exponents for simplification:
  • $( a^n \times a^m = a^{n+m} )$
  • $( \frac{a^n}{a^m} = a^{n-m} )$
  • $( (a \times b)^n = a^n \times b^n )$

9. Addition and Subtraction of Radicals:

• Combine like radicals (same radicand and index) by adding or subtracting their coefficients:
  • like: $( 7\sqrt{27} – 12\sqrt{48} + 6\sqrt{75} )$ simplifies to $( 3\sqrt{3} )$.

10. Approximation:

• Rounding a number is the process of approximating a number to its nearest specified place value for simplicity or convenience.
• Significant Figures: Indicate the precision of measurements.
• Upper and Lower Bounds: Define ranges of values considering measurement precision.

11. Scientific Notation: Expresses numbers as $( a \times 10^n )$ where $( 1 \leq a < 10 )$ and $( n )$ is an integer.

• like: $( 0.000000234 = 2.34 \times 10^{-7} ); ( 567,200,000,000 = 5.672 \times 10^{11} )$.

12. Rationalization means converts irrational denominators to rational ones by multiplying :

• like: $( \frac{2}{\sqrt{5}} )$ rationalized to $( \frac{2\sqrt{5}}{5} )$.

This summary encapsulates key concepts about real numbers, their properties, operations involving exponents and radicals, and methods for approximation and notation.