Properties of Real Numbers

Properties of Real Numbers

• Commutative Property of Addition and Multiplication: The order of numbers being added or multiplied does not change the result.
• Associative Property of Addition and Multiplication: The order in which numbers are added or multiplied does not change the result when there are more than two numbers.
• Distributive Property: The multiplication of a single number with the sum of two or more numbers is equal to the sum of the multiplication of that number with each of the addends.
• Existence of Additive and Multiplicative Identities: The existence of 0 (additive identity) and 1 (multiplicative identity) which do not change the value of a number when added to or multiplied by it.
• Existence of Additive Inverses: For each real number, there exists an additive inverse (opposite) which when added to the number results in 0.
• Existence of Multiplicative Inverses: For each non-zero real number, there exists a multiplicative inverse (reciprocal) which when multiplied by the number results in 1.

Inequalities

• Properties of Inequalities:
  • Transitive Property: If a > b and b > c, then a > c.
  • Additive Property: If a > b, then a + c > b + c.
  • Multiplicative Property: If a > b and c > 0, then ac > bc.
• Solving Inequalities: Inequalities can be solved by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value, as long as the direction of the inequality is maintained.

Applications of Real Numbers

• Geometry: Real numbers are used to represent lengths, widths, and heights of geometric shapes.
• Trigonometry: Real numbers are used to represent angles and trigonometric functions.
• Calculus: Real numbers are used to represent limits, derivatives, and integrals.
• Statistics: Real numbers are used to represent data and probabilities.

Intervals

• Interval Notation: A way of representing intervals using brackets and parentheses to indicate inclusion or exclusion of endpoints.
• Types of Intervals:
  • Open Interval: (a, b), where a and b are not included.
  • Closed Interval: [a, b], where a and b are included.
  • Half-Open Interval: [a, b) or (a, b], where one endpoint is included and the other is not.
• Operations on Intervals:
  • Union: The combination of two or more intervals.
  • Intersection: The overlap of two or more intervals.

Properties of Real Numbers

• Real numbers obey the commutative property of addition and multiplication, meaning the order of numbers being added or multiplied does not change the result.
• The associative property of addition and multiplication also applies, indicating that the order in which numbers are added or multiplied does not change the result when there are more than two numbers.
• The distributive property allows for the multiplication of a single number with the sum of two or more numbers to be equal to the sum of the multiplication of that number with each of the addends.
• There exist additive and multiplicative identities, 0 and 1, which do not change the value of a number when added to or multiplied by it.
• For each real number, there exists an additive inverse (opposite) that when added to the number results in 0.
• For each non-zero real number, there exists a multiplicative inverse (reciprocal) that when multiplied by the number results in 1.

Inequalities

• The transitive property of inequalities states that if a > b and b > c, then a > c.
• The additive property of inequalities allows for the addition of the same value to both sides of an inequality, maintaining the inequality's direction.
• The multiplicative property of inequalities allows for the multiplication of both sides of an inequality by a positive value, maintaining the inequality's direction.
• Inequalities can be solved by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value, as long as the direction of the inequality is maintained.

Applications of Real Numbers

• Real numbers are used to represent geometric measurements, such as lengths, widths, and heights of shapes.
• Real numbers are used to represent angles and trigonometric functions in trigonometry.
• Real numbers are used to represent limits, derivatives, and integrals in calculus.
• Real numbers are used to represent data and probabilities in statistics.

Intervals

• Interval notation is used to represent intervals using brackets and parentheses to indicate inclusion or exclusion of endpoints.
• There are three types of intervals: open, closed, and half-open intervals.
• Open intervals, denoted by (a, b), do not include the endpoints a and b.
• Closed intervals, denoted by [a, b], include the endpoints a and b.
• Half-open intervals, denoted by [a, b) or (a, b], include one endpoint and exclude the other.
• Intervals can be combined using union and intersection operations.
• The union of two or more intervals is the combination of all values in the intervals.
• The intersection of two or more intervals is the overlap of all values in the intervals.