Real Numbers and the Number Line

• Real numbers encompass all rational and irrational numbers.
• The number system is a structure for representing numbers, each with unique properties and uses.
• Real numbers can be visualized on a number line.

Real Numbers

• Real numbers include both rational and irrational numbers.
• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q is not zero.
• Irrational numbers cannot be expressed as a fraction and have non-repeating, non-terminating decimal representations (e.g., √2, π).
• Real numbers can be positive, negative, or zero.
• Examples of real numbers are: -5, 0, 1/2, √2, π, 7.
• Real numbers obey properties like commutativity, associativity, and distributivity.

Number Line

• The number line is a visual representation of real numbers as points on a line.
• Zero is the origin, with positive numbers extending to the right and negative numbers to the left.
• Each point on the number line corresponds to exactly one real number.
• The number line is continuous, meaning there are no gaps between the numbers.
• It illustrates the order of real numbers; numbers to the right are greater than those to the left.
• The distance between a number and zero on the number line is its absolute value.
• The number line is used to visualize operations on real numbers and solve inequalities.

Representation of Real Numbers on the Number Line

• Every real number has a unique position on the number line.
• Rational numbers can be located by dividing the line segment between integers into equal parts.
• Irrational numbers can be located using geometric constructions or approximations.
• For example, √2 can be located using the Pythagorean theorem by constructing a right triangle with sides of length 1.
• Decimal representations of real numbers determine their position with increasing accuracy.
• Successive magnification can be used to visualize the decimal expansion of a real number on the number line.

Rational Numbers

• Rational numbers can be written in the form p/q, where p and q are integers and q ≠ 0.
• Rational numbers have either terminating or repeating decimal expansions.
• Examples include 1/2 = 0.5 (terminating) and 1/3 = 0.333... (repeating).
• The set of rational numbers is denoted by Q.
• Between any two rational numbers, there exists another rational number.
• To find a rational number between two given rational numbers, you can take their average.

Irrational Numbers

• Irrational numbers cannot be expressed in the form p/q.
• They have non-terminating and non-repeating decimal expansions.
• Examples include √2, √3, π, and e.
• The set of irrational numbers is notated differently depending on context.
• Irrational numbers fill the gaps between rational numbers on the number line.
• The sum or product of two irrational numbers can be either rational or irrational.
• For example, √2 * √2 = 2 (rational), but √2 + √3 is irrational.

Decimal Expansions of Real Numbers

• Every real number has a decimal expansion.
• Rational numbers have either terminating or repeating decimal expansions.
• Irrational numbers have non-terminating and non-repeating decimal expansions.
• A terminating decimal expansion can be written as a fraction with a power of 10 in the denominator.
• A repeating decimal expansion can be converted to a fraction using algebraic manipulation.
• The decimal expansion provides a way to approximate real numbers to a desired level of accuracy.

Operations on Real Numbers

• Real numbers can be added, subtracted, multiplied, and divided (except division by zero).
• These operations follow the commutative, associative, and distributive properties.
• The sum of two rational numbers is always rational.
• The product of two rational numbers is always rational.
• The sum or product of a rational and an irrational number is always irrational (if the rational number is non-zero).
• The set of real numbers is closed under addition, subtraction, multiplication, and division (excluding division by zero).

Absolute Value

• The absolute value of a real number x is its distance from zero on the number line.
• It is denoted as |x|.
• If x is positive or zero, then |x| = x.
• If x is negative, then |x| = -x.
• The absolute value is always non-negative.
• Properties of absolute value include: |x| ≥ 0, |-x| = |x|, |xy| = |x||y|.
• The absolute value can be used to define distance between two points on the number line: d(x, y) = |x - y|.

Inequalities

• Inequalities compare two real numbers using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
• Inequalities can be represented on the number line using intervals.
• Solving inequalities involves finding the set of real numbers that satisfy the inequality.
• When multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed.
• The number line provides a visual aid for understanding and solving inequalities.
• Absolute value inequalities can be solved by considering two cases.

Intervals

• An interval is a set of real numbers between two given endpoints.
• Intervals can be open, closed, half-open, or infinite.
• An open interval (a, b) includes all numbers between a and b, but not a and b themselves.
• A closed interval [a, b] includes all numbers between a and b, as well as a and b.
• A half-open interval (a, b] or [a, b) includes one endpoint but not the other.
• Infinite intervals extend to infinity in one or both directions.
• Intervals can be represented on the number line using parentheses or brackets.
• Intervals are used to describe solution sets of inequalities.