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How many fractions are between 1 and 2?
Since the number scale is continuous, there is an infinite amount.
How many points are needed to determine a straight line?
You need two points to determine a line. A single point can have an infinite number of lines passing through it.
How many perpendicular lines does a heptagon have?
An infinite number. Each point on a line, however small the line is, can have a perpendicular through it. And since there are an infinite number of points on any line, the answer is an infinite number.
How do you recognize when an equation has no real solution or an infinite number of solutions?
It depends on the equation. Also, the domain must be such that is supports an infinite number of solutions. A quadratic equation, for example, has no real solution if its discriminant is negative. It cannot have an infinite number of solutions. Many trigonometric equations are periodic and consequently have an infinite number of solutions - provided the domain is also infinite. A function defined as follows: f(x) = 1 if x is real f(x) = 0 if x is not real has no real solutions but an infinite number of solutions in complex numbers.
Are there fewer rational numbers than irrational numbers?
For any given subset, yes, because there are an infinite number of irrational numbers for each rational number. But for the set of ALL real numbers, both are infinite in number, even though the vast majority of real numbers would be irrational.
