Beginning Activity Beginning Activity 2: Variables
Not all mathematical sentences are statements. For example, an equation such as
is not a statement. In this sentence, the symbol \(x\) is a variable. It represents a number that may be chosen from some specified set of numbers. The sentence (equation) becomes true or false when a specific number is substituted for \(x\text{.}\)
1.
Does the equation \(x^2 - 25 = 0\) become a true statement...
(a)
if \(- 5\) is substituted for \(x\text{?}\)
(b)
if \(\sqrt{5}\) is substituted for \(x\text{?}\)
Definition.
A variable is a symbol representing an unspecified object that can be chosen from a given set \(U\text{.}\) The set \(U\) is called the universal set for the variable. It is the set of specified objects from which objects may be chosen to substitute for the variable. A constant is a specific member of the universal set.
Some sets that we will use frequently are the usual number systems. Recall that we use the symbol \(\mathbb{R}\) to stand for the set of all real numbers, the symbol \(\mathbb{Q}\) to stand for the set of all rational numbers, the symbol \(\mathbb{Z}\) to stand for the set of all integers, and the symbol \(\mathbb{N}\) to stand for the set of all natural numbers.
2.
What real numbers will make the sentence “\(y^2 - 2y - 15 = 0\)” a true statement when substituted for \(y\text{?}\)
3.
What natural numbers will make the sentence “\(y^2 - 2y - 15 = 0\)” a true statement when substituted for \(y\text{?}\)
4.
What real numbers will make the sentence “\(\sqrt{x}\) is a real number” a true statement when substituted for \(x\text{?}\)
5.
What real numbers will make the sentence “\(\sin ^2 x + \cos ^2 x = 1\)” a true statement when substituted for \(x\text{?}\)
6.
What natural numbers will make the sentence “\(\sqrt{n}\) is a natural number” a true statement when substituted for \(n\text{?}\)
7.
What real numbers will make the sentence
a true statement when substituted for \(y\text{?}\)