Real Numbers Quiz

Real Numbers

• Real numbers encompass all numbers that can be represented on a number line, including rational and irrational numbers.
• Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Examples include integers, fractions, and terminating or repeating decimals.
• Irrational numbers cannot be expressed as a fraction of two integers. Examples include √2, π, and e. They have non-repeating, non-terminating decimal representations.
• Real numbers are ordered and have a well-defined addition and multiplication operations.
• The set of real numbers is denoted by the symbol 'ℝ'.

Properties of Real Numbers

• Closure: If a and b are real numbers, then a + b and a * b are also real numbers.
• Commutative Property: a + b = b + a and a * b = b * a
• Associative Property: (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c)
• Distributive Property: a * (b + c) = a * b + a * c
• Identity Property: There exists a unique real number 0 such that a + 0 = a for all real numbers a. There exists a unique real number 1 such that a * 1 = a for all real numbers a.
• Inverse Property: For every real number a, there exists a unique real number -a such that a + (-a) = 0. For every non-zero real number a, there exists a unique real number 1/a such that a * (1/a) = 1.

Subsets of Real Numbers

• Natural Numbers (N): The set of positive integers. {1, 2, 3,...}
• Whole Numbers (W): The set of natural numbers and zero. {0, 1, 2, 3,...}
• Integers (Z): The set of whole numbers and their negative counterparts. {..., -3, -2, -1, 0, 1, 2, 3,...}
• Rational Numbers (Q): Numbers that can be expressed as a fraction p/q where p and q are integers and q ≠ 0.
• Irrational Numbers (Q'): Numbers that cannot be expressed as a fraction of two integers.

Real Number Line

• The real number line is a graphical representation of real numbers.
• Each point on the line corresponds to a unique real number.
• The real number line extends infinitely in both positive and negative directions.
• The order of real numbers is directly reflected on the number line. A number to the right on the number line is greater than a number to its left.

Absolute Value

• The absolute value of a real number a, denoted as |a|, is the distance of a from zero on the number line.
• |a| is always non-negative.
• |a| = a if a ≥ 0 and |a| = -a if a < 0

Intervals

• Open intervals (a, b) contain all real numbers between a and b, but not including a and b.
• Closed intervals [a, b] contain all real numbers between a and b, including a and b.
• Half-open intervals [a, b) and (a, b] contain some of the endpoints.

Order Properties

• If a > b, then a is to the right of b on the number line.
• If a < b, then a is to the left of b on the number line.
• Transitive Property of Order: If a > b and b > c, then a > c. If a > b, then a + c > b + c. If a > b and c > 0, then ac > bc

Real Number Operations

• Addition, subtraction, multiplication, and division are fundamental operations on real numbers.
• The order of operations (PEMDAS/BODMAS) applies to expressions involving real numbers.
• Rules for adding, subtracting, multiplying, and dividing real numbers follow specific procedures.