Rational or irrational?
Write twelve numbers, from √9 and 22/7 to π and √2, as decimals and as p/q, and sort them into integers, rational numbers and irrational numbers.
Core concept
A rational number can be written as p/q (q ≠ 0) and its decimal ends or repeats; an irrational number cannot, and its decimal never ends and never repeats; integers are rational too
Topic: Real numbers
The real numbers are all the numbers on the number line. A rational number can be written as a fraction: A part of a whole, written like ¾: the bottom number says how many equal parts, the top how many you take. All key words p/q of two integer: A whole number that can be positive, negative or zero, like −3, 0 or 7. All key words, with q not 0: 3/4, −5/2, and every integer too, since 3 = 3/1. As a decimal: A number with a point, like 2.5, where the digits after the point are tenths, hundredths … All key words, a rational number either ends (3/4 = 0.75) or repeats a pattern for ever (1/3 = 0.333…). An irrational number cannot be written as p/q. Its decimal never ends and never repeats: √2 = 1.41421356…, π = 3.14159265…. A square root is rational only when the number is a perfect square: √9 = 3, but √2 and √5 are irrational. We often use 22/7 for π and 1.414 for √2, but these are only rational approximations.
What you will learn
- Write rational numbers as p/q and as decimals.
- Tell an ending or repeating decimal from one that never ends and never repeats.
- Classify numbers as integers, rational numbers or irrational numbers.
- Know that 22/7 and 1.414 are approximations of π and √2.
Before you start
- Know integers, fractions, decimals and square roots.
Materials
- A notebook
Equipment
- A calculator
Procedure
- Pick a number's card and write it as a decimal.
- Try to write it as p/q.
- Do this for at least eight numbers, then sort them all.