Which of the following is always irrational?

Square root of a non-perfect square integer

Which of the following is equal to 5\/6?

0.8333 (finite to four decimals)

Which statement about pi is correct?

pi is irrational and transcendental

pi is the square root of a rational number

Which statement about algebraic and transcendental numbers is correct?

Every rational number is algebraic

All irrational numbers are transcendental

Every rational number is transcendental

All algebraic numbers are rational

Rational and Irrational Numbers Practice Quiz

Use this 20‑question quiz to practice identifying rational and irrational numbers in real examples. You will sort decimals, fractions, square roots, and constants like pi, with instant built‑in checks. Review at your own pace and catch gaps before a test.

Rational Numbers - Rational numbers can be expressed as a fraction p/q where p and q are integers and q ≠ 0. Their decimals always terminate or repeat in a pattern, like 0.75 for 3/4 or 0.333… for 1/3. This makes them predictable and neat to work with.
Irrational Numbers - Irrational numbers refuse to fit into simple fractions, and their decimals go on forever without repeating. Think of π (3.14159…) or √2 (1.41421…); they keep surprising you without a pattern. Exploring them feels like chasing an endless treasure hunt.
Operations with Irrationals - Adding or multiplying two irrationals might result in either a rational or irrational outcome. For instance, √2 + (2 - √2) equals the tidy rational 2, while √2 + π stays delightfully irrational. It's like a rollercoaster that sometimes returns cleanly and sometimes loops forever.
Identify by Decimals - To spot whether a number is rational or irrational, peek at its decimal form. If it terminates (like 0.125) or repeats (like 0.666…), you're dealing with a rational number. If it never settles into a pattern, you've found an irrational adventurer.
Rational + Irrational - Mixing a rational number with an irrational one always gives you an irrational number. For example, 2 + √3 remains irrational, flavoring math with endless surprises. This rule helps you quickly classify sums and products.
Simplifying Irrational Expressions - Learn to tame wild expressions like √(7 + 4√3) by using clever algebraic tricks. Simplifying these radicals is essential before diving into more epic equations. It's like giving your math problems a neat haircut!
Proofs of Irrationality - Prove famous numbers irrational by contradiction: assume they're rational and uncover a logical twist. The classic √2 proof is a perfect starter that feels like a detective mystery. Tackling these proofs boosts your mathematical detective skills.
Rationalize the Denominator - Rationalizing the denominator removes pesky roots from the bottom of fractions. Multiply numerator and denominator by a conjugate to get a cleaner expression. This trick keeps your solutions looking polished.
Practice Core Operations - Sharpen your skills by adding, subtracting, multiplying, and dividing irrational numbers. For example, (√5 + √2)(√5 - √2) simplifies neatly to 3, showing a surprising rational outcome. Regular practice turns these operations into second nature.
Number Set Hierarchy - Explore how natural numbers, whole numbers, integers, rationals, and irrationals fit into the grand number universe. Visualizing these relationships is like creating a map to guide your number adventures. It helps you quickly classify any number you meet.