Real Numbers
Classify and apply properties of real numbers: naturals, whole numbers, integers, rationals, and irrationals form nested subsets of the real number system used throughout algebra.
Key Concepts
The set of real numbers consists of rational numbers (which can be written as fractions like $\frac{1}{2}$) and irrational numbers (which cannot, like $\pi$). Rational numbers include integers, whole numbers, and natural numbers.
a. The number $33$ is a real, rational, integer, whole, and natural number. b. The number $ 15$ is a real, rational, and integer. c. The number $\sqrt{13}$ is a real and irrational number.
Think of numbers as a big family tree. The 'Real Numbers' are the great grandparents. Their children are the 'Rationals' (the predictable ones) and 'Irrationals' (the wild ones). Each generation is a subset of the one before, creating a perfectly organized number family with its own unique members.
Common Questions
What are the subsets of the real number system?
Natural numbers (1, 2, 3...) are a subset of whole numbers (0, 1, 2...), which are a subset of integers (...-2,-1,0,1,2...), which are a subset of rational numbers (expressible as p/q). Rational and irrational numbers together form the real numbers.
What is the difference between 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. Their decimal expansions terminate or repeat. Irrational numbers like pi or sqrt(2) cannot be expressed as fractions; their decimal expansions are non-terminating and non-repeating.
Why is it important to know which subset a number belongs to in algebra?
Knowing a number's subset affects which properties apply. For example, the closure property for integers guarantees that adding two integers always gives an integer. But dividing two integers may give a rational that is not an integer. Misclassifying a number can lead to invalid conclusions about its properties.