Irrational Numbers Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Classify each number as rational or irrational: 16\sqrt{16}, 5\sqrt{5}, 0.750.75, ฯ€\pi.

Solution

  1. 1
    16=4\sqrt{16} = 4, which is an integer, so it is rational. 0.75=340.75 = \frac{3}{4}, also rational.
  2. 2
    5\sqrt{5} is not a perfect square, so 5\sqrt{5} is irrational. ฯ€\pi is a well-known irrational number.
  3. 3
    Rational: 16\sqrt{16}, 0.750.75. Irrational: 5\sqrt{5}, ฯ€\pi.

Answer

Rational:ย 16,โ€…โ€Š0.75Irrational:ย 5,โ€…โ€Šฯ€\text{Rational: } \sqrt{16},\; 0.75 \qquad \text{Irrational: } \sqrt{5},\; \pi
A number is rational if it can be written as ab\frac{a}{b} with integers a,ba, b (bโ‰ 0b \neq 0). Square roots of non-perfect-squares are irrational, and ฯ€\pi cannot be expressed as any fraction.

About Irrational Numbers

An irrational number is a real number that cannot be expressed as a ratio of two integers pq\frac{p}{q}; its decimal expansion goes on forever without repeating any fixed block of digits.

Learn more about Irrational Numbers โ†’

More Irrational Numbers Examples

Example 2 medium

Show that [formula] lies between [formula] and [formula], then estimate it to one decimal place.

Example 3 medium

Prove that [formula] is irrational by contradiction.

Example 4 easy

Is [formula] rational or irrational? Explain.

Example 5 easy

Is [formula] rational or irrational?

โ† All Irrational Numbers Examples Irrational Numbers Concept