AI Tutor
Imagine slicing a pizza: you can easily share 3 slices among 4 friends as the exact fraction 34\frac{3}{4}, or 0.750.75. But what if you run into a number that is impossible to write as any fraction, no matter how huge the numbers are?
Every real number on our number line falls into one of two rival teams: rational numbers or irrational numbers.

The Rational Team

A rational number is any number you can write as a simple fraction ab\frac{a}{b}, where aa and bb are integers and bb is not zero. An integer is any positive or negative whole number, including zero, like βˆ’3,0,-3, 0, or 55.
When you turn a rational number into a decimal, it always does one of two things: it terminates (ends cleanly, like 12=0.5\frac{1}{2} = 0.5) or it repeats in a pattern forever (like 13=0.333β‹―=0.3β€Ύ\frac{1}{3} = 0.333\dots = 0.\overline{3}).
What happens if a decimal never ends and never repeats a pattern?

The Irrational Team

An irrational number cannot be written as a fraction of two integers. Its decimal expansion goes on forever without settling into a repeating loop.
In the 5th century BCE, the Greek thinker Hippasus discovered that the diagonal of a 1Γ—11 \times 1 square, 2β‰ˆ1.41421356…\sqrt{2} \approx 1.41421356\dots, could never be written as a fraction, stunning the mathematical world.
Famous irrational numbers include Ο€β‰ˆ3.14159…\pi \approx 3.14159\dots and square roots of numbers that are not perfect squares, like 3,5,\sqrt{3}, \sqrt{5}, and 10\sqrt{10}.
How can you tell instantly when looking at roots and fractions on a test?